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
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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.
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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
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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
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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
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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
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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<4pm. Steamo (.01 s Pat =s 7 MPa) d < 0.3 pm.
Correction factor K (Cunningham's correction) in Eq. (1.1.5).
m
RepR> 1. See Fig. 2 for corresponding velocities V and particle diameters d in water and air.
Empirical correlations for drag coefficient
Modification References
Notes
a -c
CD-
Infinite extent
Homogeneous
Particle motion: Rem < 1
CD.
Not important in most flow tracing situations because the particle concentration is less than 1 :30,000 parts by volume. A possible exception is where hydrogen bubbles are used. Does not apply when the experimental fluid is a liquid.
Mockros and Lain find for unsteady particle motion that if the history integral is included in the equation of motion, Stokes' drag law may be valid for RepR= lW. For calculations of RepRin turbulent fluid flows, see Section 1.1.2.2.6.
Steady
Unsteady with u << 1.
Unsteady at all values of u.
Nature of Particle: No internal fluid motions in bubbles and drops.
Always. Impurities, which are never absent, change the surface tension, leading to internal motions.'
Undeformed bubbles and drops.
d > 1 mm (based on examinaation of experimental data)
Spherical
Allowable maximum deformation:d Length/diameter = 0.9 (discoid), = 1.05 (needlelike).
None, but include history integral in the equation of motion. None, but inertia of added mass ( x ) included in the equation of motion. x = 0.5 used in producing Fig. 3.
r
Correction factor K (HadamardRybczynski) in Eq. (1.1.5) when the experimental fluid is a liquid. See next line in this table.
m
Use drag diameter dd in Eq. (1.1.5).
d,n,v,w
1
The exact procedure for taking account of unsteady flow is uncertain,' but it is assumed here that including both the history integral and the added mass in the equation of motion will permit an accurate assessment of particle motion with unsteady flow.
Most of the experimental data applies to particles moving under gravity in a stationary experimental fluid, but experiments by Baker and Chao' suggest that the rule may also be applied when the experimental fluid is in motion. Errors involved in ignoring this correction can be very serious. (Continued)
TABLE11. (Continued) (I
D. J. Carlson and R. F. Hoglund, A I M J. 2, 1980 (1964).
* C. T. Crowe, AIAA J. 5, 1201 (1%7).
W . J. Yanta, AIAA Fluid Plasma Dyn. Con$, 6th. 1973 AIAA Paper No. 73-705 (1973). J. Happel and H. Brenner, “Low Reynolds Number Hydrodynamics.” Prentice-Hall, Englewood Cliffs, New Jersey, 1965. T. Greenstein and J. Happel, J. Fluid Mech. 34, 705 (1968). V. Fidleris and R. C. Whitmore, Br. J. Appl. Phys. 12,490 (l%l). p J. S. McNown, H.M. Lee, M. B. McPherson, and S . M.Engez, Proc. Znt. Cong. Appl. Mech., 7rh, Vol. 2, p. 17 (1948). * S. L.*So0 and C. L. Tien, J . Appl. Mech. 82, 5 (1%0). H. C . Brinkman, Appl. Sci. Res., Sect. A 1, 27 (lW7). C. K. W . Tam, J. Fluid Mech. 38, 537 (1969). N. Zuber, Chem. Eng. Sci. 19, 897 (1964). I P. N. Rowe, Trans. Ins?. Chem. Eng. 39, 175 (l%l). H.Lamb, “Hydrodynamics,” 5th ed. Cambridge Univ. Press, London and New York, 1930. ” C. N. Davies, “The Sedimentation of Small Suspended Particles,” p. 25. Inst. Chem. Eng., London, 1947. D. J. Ryley, Engineer 198, 74 (1954). L. F. Mockros and R. Y. S. Lai,Proc. A m . Soc. Civ. Eng. 95, 629 (1%9). L. B. Torobin and W. H. Gauvin, Can. J. Chem. Eng. 37, 224 (1959). H. L. Dryden, F. D. Murnaghan, and H. Bateman, “Hydrodynamics.” Dover, New York, 1953. * R. E. Davis and A. Acrivos, Chem. Eng. Sci. 21,681 (1%6). J. F. Harper, Adv. Appl. Mech. 12, 59 (1972). J. L. L. Baker and B. T. Chao, AIChE J. 11, 268 (1%5). N. A. Fuchs, “The Mechanics of Aerosols” (rev. ed.). Pergamon, Oxford, 1964. C. N. Davies, Nature (London) 2Q1,905 (1964). L. B. Torobin and W. H. Gauvin, Can. J. Chem. Eng. 37, 129 (1959). ” L. B. Torobin and W. H. Gauvin, Can. J. Chem. Eng. 37, 167 (1959).
‘
1.1.
15
TRACER METHODS
This is a very simple result, which leads to a linear form of the BBO equation. Its validity for low values of the relative particle Reynolds number, such as those usually associated with flow tracing particles, has been satisfactorily verified by experiment. However, it depends on a number of assumptions. These are the following: (i) the fluid medium is incompressible; (ii) the fluid medium is of infinite extent; (iii) there is no slip at the particle surface between the experimental fluid medium and the particle, i.e., the experimental fluid is homogeneous; (iv) the relative particle Reynolds number is less than unity; (v) the particle motion is steady; and (vi) the particle is rigid, i.e., there are no internal motions within the particle, and it retains its spherical shape under all hydrodynamic conditions. If any of the above assumptions do not apply, Stokes’ drag law must be modified. The usual practice is to introduce a multiplying factor (say, K ) into Eq. (1.1.5) that allows for the departure from the conditions required by Stokes’ law. The necessary modifications and the conditions under which they should be used are summarized in Table 11. 1.1.2.2.3. PARTICLE SIZEAND DENSITY. The motion of a sample of flow tracing particles will undoubtedly have a statistical character even though the fluid velocity is not a random variable of time. This is because the dynamic characteristics of the individual members of such a sample rarely, if ever, have the same magnitude, but instead have random magnitudes which should therefore be expressed in statistical terms. This is an aspect of the application of flow tracing particles to fluid velocity measurement that has received very little attention,* and it is only possible to give a simplified discussion. Nevertheless, the topic is of sufficient importance to warrant more extensive consideration at some future time. The principles involved in determining the statistical dynamic characteristics of flow tracing particles are best demonstrated using the equation of motion associated with the type I11 approximation (see Section 1.1.2.2.1); the extension to the complete equation of motion is straightforward. Suppose we have a sample of flow tracing particles with n particles having mass m,, and diameter d. If we sum over each of the individual equations of motion for each of the particle sizes, we obtain
[c nmp(dUp/dt)]/Cnd + 37Tp~(CUpnd)/C
nd
=
~T/.LFUF. (1.1.6)
The left-hand side of this equation involves two special mean quantities la P. K. Khosla and S. Lederman, “Motion of a Spherical Particle in a Turbulent Flow” PIBAL Rep. No. 73-22. Polytechnic Institute of New York, 1973.
* The work of Khosla and Ledermanlaisthe only example known to the author where the problem has been considered.
16
1.
MEASUREMENT OF VELOCITY
which must be determined from experimental measurement of the statistical distribution of particle masses, diameters, and velocities. * Measurements of the type indicated in Eq. (1.1.6) have never, so far as the author is aware, been made and would, in fact, be very difficult. However, it may be possible to simplify the problem. If it can be assumed that particle density and particle velocity up do not depend on particle size, then Eq. (1.1.6) can be written (1.1.7) ~ * ~volume~ where dm = [(Z nd3)/(C nd)I1l2 and has been ~ a l l e d *the diameter mean diameter. This indicates, provided the assumptions are accepted, that with a type I11 approximation to the equation of motion, only this mean diameter should be used to calculate the velocity of flow tracing particles having a distribution of sizes. Correspondingly, the method used to measure the particle sizes should, preferably, provide this mean diameter directly. According to the simplified approach outlined above, an accurate knowledge of the sizes and densities of flow tracing particles is a prerequisite to the estimation of their dynamic characteristics. However, this cannot be determined from a knowledge of the method by which they are produced. The processes of forming small particles are much too complex (see Section 1.1.2.6). It is therefore necessary to measure these quantities after the flow tracing particles are produced. The purpose of the discussion presented here is to review those methods of particle size and density measurement that are particularly suitable for application to flow tracing partic1es.t Since the particle properties are generally repreD. P. Kessler and J. L. York, AIChE J. 16, 369 (1970). L. York and H. E. Stubbs, Trans. Am. Soc. Mech. Eng. 74, 1157 (1952). *l G. M. Benson, M. M. El-Wakil, P. S. Myers, 0. A. Uyehara, J. A m . Rocker Soc. 30, sn J.
447 (1960).
C. Orr, Jr., “Particulate Technology.” Macmillan, New York, 1959.
* Such measurements, when the particles are not accelerating, have been made by Kessler and York,ls York and Stubbs,Poand Benson et Data of this type are most appropriately obtained by techniques (see Table 111) where the measurements are made while the particles are flowing in the test section. t The literature on particle size measurement comprises hundreds of references and it is hoped that the material presented here will be helpful in guiding the experimenter interested in such measurements to the procedures, that are most relevant to his problem. A number of important bookssc*z4-27 have been published on the techniques of particle size measurement and, of these, Herdan’s book” should be consulted by anybody seriously concerned with the statistical aspects of particle size measurement. References 25, 26, and 27 are es-
1.1. TRACER METHODS
17
sented by a statistical distribution, this topic is considered first, followed by methods of measuring particle density and size. The section concludes with a brief discussion of particle sampling, which is an essential preliminary to the measurement of their properties. Any nonnegative function having a finite integral over the range of particle sizes may be used as a size distribution function f to represent n in Eq. (1.1.6) as a function of diameter d or mass m D . However, the choice is limited because it is advisable to avoid using functions in which it is difficult to determine the numerical values of the parameters in the functional relation.* For analytical purposes it is convenient to use a continuous distribution function f such that the number of particles having property between z and z + dz is given by n = f dz. The functions which have been found useful in representing particle size distributions can be divided into those of an exponential type and those which are related to the normal distribution function. The latter R. A. Mugele and H. D. Evans, Ind. Eng. Chem. 43, 1317 (1951). G . Herdan, “Small Particle Statistics.” Butterworth, London, 1960. 25 R. D. Cadle, ”Particle Size Determination.” Wiley (Interscience), New York, 1955. 26 C. Orr,Jr., “Fine Particle Measurement.” Macmillan, New York, 1959. ” T. Allen, “Particle Size Measurement.” Chapman & Hall, London, 1968. L. T. Work and K. T. Whitby, in “Encyclopedia of Chemical Technology” (R. E. Kirk and D. F. Othmer, eds.), Vol. 12, p. 472. Wiley (Interscience), New York, 1954. 28 P. G. W. Hawksley, in “Progress in Coal Science” (D. H. Bangham, ed.), Vol. 1, 127. Wiley (Interscience), New York, 1950. G . L. Beyer, Tech. Org. Chem. 1, Part I, 191 (1959). 31 N . Dombrowski and G. Munday, Biuchern. B i d . Eng. Sci. 2, 209 (1968). 3z J. M. Dallavalle, “Micromeritics,” 2nd ed. Pitman, New York, 1948. H. L. Green and W. R. Lane, “Particulate Clouds: Dusts, Smokes and Mists,” 2nd ed. Van Nostrand-Reinhold, Princeton, New Jersey, 1964. 34 P. G. W.Hawksley, Br. Cual UIJ. Res. Assoc., M o n . Bull. 15, 105 (1951). 35 J. R. Hodkinson, in “Aerosol Science” (C. N. Davies, ed.), p. 287. Academic Press, New York, 1966. 23
24
pecially strong on the practical aspects of size measurement. Short reviews and summaries have been prepared by Work and Whitby,28 Hawksley,2u Beyer,30 and Dombrowski and M ~ n d a y . ~Other ] short discussions, forming part of longer treatises on particles, have been given by D a l l a ~ a l l eand ~ ~ Green and Lane.33An important review article by Hawksley3’ discusses the physical processes involved in particle size analysis with a particular emphasis on light scattering techniques. A much more up-to-date review of the latter material has been given in an excellent review prepared by H ~ d k i n s o n . ~ ’ * So far as the author is aware, no work has been done on distributions for particle density data; consequently, the remainder of this discussion is restricted to analytical forms that are known to be suitable for particle size distributions only. Of course, subsequent investigation may show that some o r all of these distributions are also applicable to density data.
18
1. MEASUREMENT OF VELOCITY
have the general form, on a number basis,* of
f d z = [ u s ~ ( ~ T ) ” ~exp[]-’
(Z
-
Z ) 2 / 2 ~ 3dz,
( 1.1.8)
where the general variable z, the normalizing constant a, the mean Z, and the dispersion parameter sn characterize the distribution. The exponential distributions have the general form, when using number of particles as the dependent variable,
f dz
= uzp exp( - bzq) dz.
(1.1.9)
The quantities a , 6, p, and q are empirical constants which have the values and interrelations shown in the reference^^^'^^-^' cited earlier. The physics of particle formation processes tends to lead to a preferential size. As a result, particle size frequency distributions usually have a maximum value at the favored size. However, the distribution function is almost always skewed, so the normal probability function very rarely fits particle size data. If the variable z is log d, where d is the particle diameter, one obtains the commonly used log normal distribution, which is skewed toward the larger sizes. Any experimental data on particle sizes should first be tested to determine how well they fit a log normal distribution. The exponential distributions, mentioned above, are probably best used for data that are so highly skewed that they do not follow even the log normal distribution. Unfortunately, it is difficult to obtain averages so it from exponential type functions without complicated is desirable to avoid this analytical form. Information on particle density can be obtained in three ways: (a) by using tabulated data, (b) by measuring the particle material in bulk before it is dispersed as individual flow tracing particles, and (c) by measuring the density of an appropriate sample of flow tracing particles. The first method is probably quite satisfactory for gaseous and liquid materials, unless it is suspected, particularly with liquids, that the material is different from that for which the tabulated data applies. Should the latter circumstances arise, the density of the material should be measured. With solid particles, the use of tabulated data is unsatisfactory, and it is essential to measure the density of samples of individual flow tracing particles. This is because the density of a small solid particle differs from
* Details of forms using the volumes of particles or the surface areas of particles as the dependent variable will be found in the references cited earlier;ac**4-*7 likewise for the integrated or cumulative forms. These references also contain further information on the origin and use of the various functions.
1.1.
TRACER METHODS
19
that of the bulk material and between individual particles due to the inhomogeneities (cracks, trapped air bubbles, occluded liquid) which they all have. In addition, solid particles sometimes have a tendency to form irregularly shaped clumps made up of a number of different particles which have been drawn together under the action of surface tension or electrical forces. This means that particles which are spherical and which have a predictable density can act as irregularly shaped particles with a very uncertain density. Whytlaw-Gray and Patterson3s cite examples of particles produced by combustion with densities which are one tenth or one twentieth of the density of the parent material in bulk. Similar, but not such extreme values, have also been reported by Lane and Stone3‘ for polystyrene particles. The empirical nature of the size distribution functions is illustrated by the lack of dimensional homogeneity of the functions employed. The measurement of density is among the most precise of physical measurements. Even with quite simple equipment it is possible, according to a review by Bauer and L e ~ i n ,to ~ *achieve a precision of f 0.1%, and precisions of 1 part in l@ are attained with ease. The article by Bauer and Lewin is the best source of information on the various techniques. The papers of BaueFO and Blank and Willardmare also excellent; the latter reference deals with microanalytic methods which are appropriate for the measurement of the density of flow tracing particles. The definition of flow tracing particle size is complicated by the fact that some particles have a regular shape, usually spherical, and other particles have an irregular shape. Gas bubbles, liquid droplets, and certain solid particles, e.g., those made of polystyrene latex, are spherical, but a large group of flow tracing particle materials, e.g., metal flakes, powders, and naturally occurring particles, are very irregular. The size of a spherical particle is completely defined by its diameter, but the size of an irregularly shaped particle cannot be defined geometrically and must be expressed in terms of some size dependent property. According to H a ~ k s l e y most , ~ ~ particle size measurement processes depend on the evaluation of one or more basic properties of the particle. These are its volume, its projected area, and its resistance to motion through a fluid. 96
R. Whytlaw-Gray and H. S. Patterson, “Smoke.” Arnold, London, 1932.
W. R. Lane and B. R. D. Stone, in “Mechanism of Corrosion by Fuel Impurities” (H. R. Johnson and D. J . Littler, eds.), p. 417. Buttenvorth, London, 1%3. 58 N . Bauer and S . Z. Lewin, in “Technique of Organic Chemistry,” Vol. I, Part I, 3rd ed., p. 131. Wiley (Interscience), New York, 1959. 38 N . Bauer, in ”Encyclopedia of Chemical Technology” (R. E. Kirk and D. F. Othrner, eds.), Vol. 4, p. 875. Wiley (Interscience), New York, 1949. 40 E. W. Blank and M. L. Willard, J . Chem. Educ. 10, 109 (1933). 37
20
1.
MEASUREMENT O F VELOCITY
For an irregularly shaped particle, we may conveniently express the magnitudes of these properties by the diameters of spherical particles that have the same value of one or more of the properties. Therefore, corresponding to the three basic properties are three diameters, which are called by HawksleyS4the volume diameter d, ,the projected area diameter d,, and the drag diameter dd (see Table I11 for definitions of these quantities) . The three basic diameters may be related by what are called shape factor^.'^,^^ For spherical particles these can be calculated, but for irregularly shaped particles empirical methods must be used. A useful table of shape factors has been compiled by H e y ~ o o d . ~ ~ Davies17 has criticized the use of shape factors on physical grounds and, in view of the lack of a theoretical foundation for the shape factor, it is probably better used as a general guide rather than in exact calculations of the particle dynamic characteristics. The problem of nonspherical particles and their hydrodynamic characteristics has already been considered in Table I1 of Section 1.1.2.2.2. There are many available methods of particle size measurement which are discussed in the general references cited earlier. Not all of these methods are suitable for use with flow tracing particles, and the selection of an appropriate method should involve a number of considerations, which will be briefly reviewed. (i) The chosen technique should allow measurements to be made under conditions as close as possible to those in which the particle is to be used, because the results produced by different methods are found to be sensitive to the conditions under which the measurement is made. In particular, the measurements should preferably be made: (a) at the low concentrations (say, 1 part by volume of particles to 30,000 parts by volume of the experimental fluid) found in flow tracing applications; (b) in a flowing fluid using the fluid which is to be used in the experimental system; (c) at the approximate temperature, pressure, and velocities to be encountered in the intended application. (ii) The technique should be able to provide the volume-diameter mean diameter d,, directly or to allow its direct calculation. (iii) The precision and accuracy of the technique should be appropriate for the proposed application of the particle size data (the important effect of sampling errors should be noted). (iv) Different measurement methods have different ranges of particle “ H. Heywood, in “Symposium on Particle Size Analysis,” p. 14. Inst. Chem. Eng., London, 1947.
1.1. TRACER METHODS
21
size for which they are most suitable, and the choice of a technique for flow tracing particle size measurement must take this into account. (v) Flow tracing particles may be solids, liquids, or gases, but some of the size measurement methods are restricted in the particle/fluid material combinations for which they may be used. Preliminary microscopic examination of a sample of the flow tracing particles is helpful in applying the preceding criteria. This will indicate the following important features: the nature of the particles, presence of mixed constitutents, size range, general shape of the particles, extent of agglomeration, effects of dispersion, and occurrence of permanent agglomerates. Particle size measurement techniques which appear to fit the preceding criteria have been listed in Table 111. These are arranged in order of decreasing preference, depending on how well they fit the criteria described above, their reliability, and their level of development. Some of the methods are listed rather cryptically in Table I11 and will be briefly described. Photomicrography involves photographing the particles flowing with the experimental fluid in the test section of the flow system, using a suitable camera-microscope combination. The resulting photograph is enlarged and can be subjected to hand or automatic analysis (seventh entry in Table 111). In the photosedimentation method, the steady velocity of particles moving in a fluid under gravity is measured, and this is then related to their size. The microscope (third entry in Table 111) can be used directly to measure particle size, using appropriately engraved graticules in the field of view.42 However, this approach is extremely fatiguing for the user, particularly when small particles are being measured. A much better method is to project the microscope image onto a large screen and make the size measurements from this e n l a ~ g e m e n t . ~The ~ . ~principle ~ of the photosedimentation method is extended in the dynamic methods of particle size measurement, where the velocity of the particle under a variety of accelerating forces is measured and the particle size is then inferred from its velocity by an appropriate relation. Measurements of this sort have been made by accelerating the particles through a in fluids undergoing periodic motion,4sand, if G. L. Fairs, Chem. Ind. (London) 62, 374 (1943). a E. T. Dunn, Jr., Ind. Eng. Chem., Anal. Ed. 2, 59 (1930). I . Hvidberg, Kolloid-Z. 72, 274 (1935).
W.J. Yanta, AIAA Fluid Plasma Dyn. Conf.,6th, Palm Springs, California, 16-18July 1973 AIAA Paper No. 73-705 (1973). 4a H. M. Cassel and H. Schultz, in Proc. U . S . National Conference on Air Pollution ( C . L. McCabe, ed.), p. 634. McGraw-Hill, New York, 1952.
TABLE111. Methods for Measuring the Size of Flow Tracing Particles Method
Conditions of measurement
Particle sizes measured in use Photosedimentation Close to use conditions but fluid stationary Requires removal of Microscope sample from flow system Dynamic Close to use Characteristics conditions
Measured diameter"
Requires removal of sample from flow system
Electrical fluid conductance counter
Flow conditions quite different from use conditions
Automatic scanning Same as phote microscopy micrography Light scattering Flow conditions quite different from use conditions Holographic Particle sizes measured in use microscopy Laser Particle sizes measured in use interferometer
Uncertainty
Number
Photomicrography
Sieving
Distribution
0.5-100
Number
da
Number
Size rangeb (pm)
3- 100
Maximum
0.5- 100
2 15%"
da
Types of particlesb
Comments
All
c -i
All
k -0
Not gas bubbles
Number
+5%
1-20
All
Weight
2 1%'
3- 10,Ooo
Number
2 1%W
Only solids and liquids which can solidify Only conCommercial instru. ducting mentm liquid experimental fluids All
0.5-200
da
Number
0.5- 100
d.
Number
0.5-20 5-400
da
Number
da
Number
23""
References
Uncertainty in the model introduces uncertainty into the size measurement
2-70
?
10- 120
?
nn.00
Commercial instrument." Restricted to spherical particles ss Restricted to spherical particles tt -vv Allows use of laser velocimeter sys-
Imaging
Particle size measured in use
d,
Number
Transient heat transfer
Particle size measured in use
d,
Number
t10%""
5-lW
3-1000 400-1200
All
L(P)/G(F) G(P)/L(F)
tem. Restricted to spherical particles Commercially avail- uu able linescan cameras may be suitable Uncertainty in the oo.xx--u model introduces uncertainty into the size measurement
The particle diameters are defined as follows: d., the area diameter, is the diameter of a circle having the same projected area as the projected area of the particle. d , , the volume diameter, is the diameter of a sphere having the same volume as the particle. d d , the drag diameter, is the diameter of a sphere having the same resistance to motion as the particle in a fluid of the same viscosity and at the same velocity. d., the Stokes diameter = (d:/dd)1'2,is the diameter of a sphere having the same density and the same acceleration as the particle in a fluid of the same density and viscosity. L(P) is a liquid particle droplet; G(P), a gaseous particle or bubble; G(F), a gaseous experimental fluid; and L(F), a liquid experimental fluid. N. Dombrowski and P. C. Hooper, Chem. Eng. Sci. 17, 291 (1%2). S. M. DeCorso J . Eng. Power 82, 10 (1%0). J. L. York and H. E. Stubbs, Trans. ASME 74, 1157 (1952). D. P. Kessler and J. L. York, AIChE J. 16, 369 (1970). G. M. Benson, M. M. El-Wakil, P. S. Myers, and 0. A. Uyehara, J . Am. Rocket Soc. 30, 447 (1960). G. Herdan, "Small Particle Statistics." Butterworth, London, 1%0. B. B. Morgan, Research (London) 10,271 (1957). 'A. S. G. Curtis, in "Photography for the Scientist" (L. E. Engel, ed.), p. 483. Academic Press, New York, 1%7. W. B. Kunkel and J. W . Hansen, Rev. Sci. Instrum. 21, 308 (1950). V. D. Hopper and T. H. Laby, Proc. Soc. London, Ser. A 178, 243 (1941). E. E. Dodd, J. Appl. Phys. 24, 73 (1953). C. J. Stairmand, Engineering 154, 141 (1942). C. J. Stairmand, Engineering 154, 181 (1942). G. L. Fairs, J. R. Microsc. Soc. [3] 71, 209 (1951). q G. L. Fairs, Chem. Ind. (London) 62, 374 (1943). M. Cole, Microscope 19, 87 (1971). W. N . Charman, J. R. Microsc. Soc. [3] 82, 81 (1%3). H. Heywood, Bull., Inst. Min. Metall. 477, 19 (1946). R. P. Loveland, Am. Soc. Test. Muter., Spec. Tech. Publ. 234, 57 (1958). C. P. Saylor, Appl. Opt. 4, 477 (1%5). 1o L. T. Work and K. T. Whitby, in "Encyclopedia of Chemical Technology" (R. E. Kirk and D. F. Othmer, eds.), Vol. 2, p. 472. Wiley (Interscience), New York, 1954. (Continued)
TABLE111. (Continued) C. G. Sumner, “Clayton’s Theory of Emulsions and Their Technical Treatment,” 5th ed. McGraw-Hill (Blakiston), New York, 1954. T. Allen, “Particle Size Measurement.” Chapman & Hall, i London, 1968. * C. Orr and J. M. Dallavalle, “Fine Particle Measurement, Size, Surface, and Pore Volume.” Macmillan, New York, 1959. R. D. Cadle, “Particle Size.” CRC Press, Cleveland, Ohio, 1%5. bb W. J. Yanta, AIAA Fluid Plasma Dyn. Conf.,6th. 1973 AIAA Paper No. 73-705 (1973). EE H. Binark and W. E. Ranz, Ind. Eng. Chem. 51,701 (1959). dd H. M. Cassel and H. Schultz, in “Air Pollution” (C. L. McCabe, ed.), p. 634. McGraw-Hill, New York, 1952. P. V. Wells and R. H. Gerke, J. Am. Chem. SOC. 41, 312 (1919). N. Fuchs and J. Petjanov, Kolloid-2. 65, 171 (1933). * K. T. Whitby, Am. SOC. Test. Muter., Spec. Tech. Publ. 234, 3 (1959). ’* S. Kiesskalt, Z . Erzbergbau Mettallhuettenwes. 8, B63 (1955). H. Heywood, Bull., Inst. Min. Metall. 477, 19 (1946). R. R. Irani and C. C. Callis, “Particle Size: Measurement, Interpretation and Application.” Wiley, New York, 1%3. kk J. R. Joyce, J . Inst. Fuel 22, 150 (1949). ‘I A. P. R. Choudhury, C. G. Lamb, and W. F. Stevens, Trans., Indian Inst. Chem. Eng. 10,21 (1957-1958). m m D. Hasson and J. Mizrahi, Trans. Inst. Chem. Eng. 39,415 (l%l). In R. H. Berg, Am. SOC. Test. Muter., Spec. Tech. Publ. 234, 245 (1959). .o R. Davies,Am. Lab. 5(12), 17 (1973); 6(1), 73 (1974); 6 (2), 47 (1974). pp H. L. Green and W. R. Lane, “Particulate Clouds: Dusts, Smokes, and Mists,” 2nd ed. Van Nostrand-Reinhold, Princeton, New Jersey, 1964. (A C. R. Adler, A. M. Mark,W. R. Marshall, Jr., and R. J. Parent, Chem. Eng. Prog. 50, 14 (1954). J. R. Hodkinson, in “Aerosol Science” (C. N. Davies, ed.), p. 287. Academic Press, New York, 1966. J. B. DeVelis and G. 0. Reynolds, “Theory and Applications of Holography.” Addison-Wesley, Reading, Massachusetts, 1967. W. M.Farmer and D. B. Brayton, Appl. Opt. 10,1319 (1971). W. M. Farmer, Appl. Opt. 11, 2603 (1972). W. M. Farmer, Appl. Opt. 13, 610 (1974). Iw R. G. Knollenberg and W. E. Neisch, in “Proceedings of the Technical Program-Electro-Optical Systems Design Conference (K. A. Kopetzky, ed.), p. 594. Ind. Sci. Cod. Manage., Chicago, Illinois, 1970. r z V. W. Goldschmidt, J . Colloid Sci. 20, 617 (1%5). uu C. A. A. van Paasen, Inr. J. Heat Mass Transfer 17, 1527 (1974). s. C. Chuang and V. W. Goldschmidt, in “Turbulence Measurements in Liquids” (G. K. Patterson and J. L. Zakin, eds.), p. 88. University of Missouri, RoIla, 1971. aaa F. T. Gucker, Jr. and D. G. Rose, Br. J. Appl. Phys.. Suppl. 3, 138 (1954). JJ
1.1.
TRACER METHODS
25
the particles can be charged, by studying the motion of the particles in an electric field.47-52 Individual particles can be arranged to flow in succession through a sensing zone where one of the following characteristics can be measured: light ~ c a t t e r i n g , ~ ~electrical * ~ l * ~ ~conductance of the suspending fluid and p a r t i ~ l e , ~or l - heat ~ ~ t r a n ~ f e r . ~The ~ . ~automatic ~ scanning of microscope images and photomicrographs by a collimated light beam can also be considered under the sensing zone type of measurements. Two general observations can be made about the methods selected for flow tracing particle size measurement. First, it is extremely difficult to measure with an uncertainty less than * 5 % the sizes of particles which are smaller than about 2 pm. Second, particle size measurement is an extremely expensive process. It involves the examination of a very large number of particles if large sampling errors are to be avoided. This means that with those methods which are essentially nonautomatic in character (photomicrography, photosedimentation, microscopy, some dynamic methods, and sieving), the length of time it takes a human operator to make a series of measurements can be quite lengthy. Fully automatic methods, the sensing zone measurements, require elaborate and very expensive electronic equipment for pulse height analysis. Particle density and particle size determination start with some type of sampling process. These fall into three general categories, viz., sampling by probe, sampling by observation, and the sampling of undispersed material, but these are not universally applicable with all particle size measurement techniques, and Table IV indicates the relationship between sampling methods and measurement methods. The most important consideration in selecting a sampling method concerns sampling accuracy and precision, since no size or density analysis can be more accurate than the sampling procedure that precedes it. Sampling precision is improved by increasing the number of samples. Sources of fixed error, which affect sampling accuracy, are listed in Table V, together with methods for minimizing the indicated error and useful B . Kunkel and J. W. Hansen, Rev. Sci. Insfrum. 21, 308 (1950). V. D. Hopper and T. H. Laby, Proc. R . Soc. London, Ser. A 178,243 (1941). P. V. Wells and R. H. Gerke, J . A m . Chem. SOC. 41, 312 (1919). N. Fuchs and J. Petianov, Kolloid-Z. 65, 171 (1933). 5* R. Davies, A m . Lab. 5 (12), 17 (1973). R. Davies, Am. Lab. 6 ( I ) , 73 (1974); 6 (2), 47 (1974). R. H. Berg, in Symp. Particle Size Measurement, A m . Soc. Test. Muter.. Spec. Tech. Pub. No. 234, p. 245 (1959). sI V. W. Goldschmidt, J . Colloid Sci. 20, 617 (1965). ss C. A. A . van Paasen, l n t . J . Heat Mass Transfer 17, 1548 (1974). " W.
26
1. MEASUREMENT OF VELOCITY TABLEIV. Sample and Size Measurement Techniques Sampling technique
Size measurement technique“ Photomicrography Photosedimentation Microscopy Dynamic Characteristics Sieving Electrical fluid conductance Automatic scan microscopy Light scattering Holographic microscopy Laser interferometer Imaging Transient heat transfer References
Observationb
UndispersedC material
Xd X X X Xd X
-
X X X X X X X
X
X X X X
Probeb
e-k
See Table I11
j-l
See Table 111 for details of these measurement techniques. Used where the particles are dispersed in the experimental fluid. Used where the particles are available in bulk, i.e., before being introduced into the experimental fluid. Particles are collected for examination by impingement on one or more collecting surfaces or by entrapment in a filter. A. B. Akers and W. D. Won, in “An Introduction to Experimental Biology” (R. L. Dimmick et al.. eds.), p. 59. Wiley (Interscience), New York, 1969. ’R. D. Cadle, P. L. Magill, A. A. Nicholl, H.C. Ehrmantraut, and G. W. Newell, in “Air Pollution Handbook” (P. L. Magill et a / . . eds.), p. 10-1. McGraw-Hill, New York, 1956. N. Dombrowski and G. Munday, Biochem. Biol. Eng. Sci. 2,209 (1968). * J. A. Browning, Adv. Chem. Ser. 20, 136 (1958). J. M. Pilcher and R. E. Thomas, Adv. Chem. Ser. 20, 155 (1958). R. D. Cadle, “Particle Size Determination.” Wiley (Interscience), New York, 1955. T. Allen, “Particle Size Measurement.” Chapman & Hall, London, 1968. ’ G. Herdan, “Small Particle Statistics,” 2nd ed. Buttenvorth, London, 1960.
‘ ’
references. Of the listed references, the monograph by HerdanZ4is the most complete and authoritative account of sampling and sampling errors. 1.1.2.2.4. THEORETICAL DETERMINATION OF PARTICLE DYNAMIC CHARACTERISTICS. Having discussed the special assumptions involved in Eq. (1.1.2), it is appropriate to turn to the problem of its solution. In view of the fact that there is a different solution for each experimental situation, the practical approach is to present the results in the most general possible form. Then, when necessary in the subsequent discussions of this chapter, special solutions can be obtained from this general solution.
1.1.
27
TRACER METHODS
TABLEV. Fixed Errors of Sampling Minimization Sampling technique Probe
Observation
Undispersed material
Sources of error Nonrandom sampling in space and/or time.
Sampling probe disturbs particle size distribution. Particle distribution altered between sampling point and size measurement device due to particle deposition and agglomeration in connecting ducts. Nonrandom sample. Sample not representative of of particles in test section. Nonrandom sample. Sample not representative of particles in test section.
Technique
References
Samples should be drawn from various points in the test section at different times. Points and times should take account of system characteristics, e.g., sampling from a vertical duct minimizes sedimentation effects. Use isokinetic sampling.
-
a -d
Make size measurements as close to sampling point as possible.
e
Samples taken from random locations in test section.
f-i
Use sampling techniques described in the references.
bj-1
H. H. Watson, Am. Ind. Hyg. Assoc., Q . 15, 21 (1954). R. D. Cadle, ”Particle Size Determination.” Wiley (Interscience), New York, 1955. R. D. Cadle, P. L. Magill, A. A. Nicholl, H. C. Ehrmantraut, and G. W. Newell, in “Air Pollution Handbook” (P. L. Magill et a / . . eds.), p. 10-1. McGraw-Hill, New York, 1956. * R. Dennis, W. R. Samples, D. M. Anderson, and L. Silverman, Ind. Eng. Chem. 49, 294 (1957). B. Y. H. Liu and J. K . Agrawal, Aerosol Sci. 5 , 145 (1974). I S . M. De Corso, J . Eng. Power 82, 10 (l%O). V . W. Goldschmidt, J. Colloid Sci. 20, 617 (1965). C. R. Adler, A. M. Mark, W. R. Marshall, Jr., and R. J. Parent, Chem. E n g . Prog. 50, 14 (1954). * G. Herdan, “Small Particle Statistics,” 2nd ed. Butterworth, London, 1960. G. L. Fairs, J . R. Microsc. Soc. [3] 71, 209 (1951). N. Dombrowski and G. Munday, Biochem. Biol. Eng. Sci. 2,209 (1968). C. Orr, Jr. and J . M. Dallavalle, “Fine Particle Measurement.” Macmillan, New York, 1959.
’ ’
Likewise, for special solutions not considered here, the reader will also be able, in principle, to employ the same general approach. The general solution of Eq. (1.1.2), i.e., the general form for the particle dynamic characteristics in rectilinear flow, is given by the transfer func-
28
1.
MEASUREMENT OF VELOCITY
tion H, where iip(iw) = H(iw)iiF(iw).
(1.1.10)
The Fourier transform of the quantity u (or the Laplace transform if we replace io by s, where s is the Laplace transform variable) is given by ii. Applying the Fourier transform to Eq. (1.1.2), we obtain H(iw) = [Biw
+ A + C(~iw)”~]/[iw+ A + C ( ~ i o ) ” ~ ](1.1.11) .
The transfer function in Eq. (1.1.11) can be rearranged and written in terms of real and imaginary parts; thus H ( N s , a)= [1 +fi(N,,
d l + if2(Nsr d,
(1.1.12)
where N , , f i , and& are defined in the List of Symbols. In terms of the amplitude function (qz = liip12/IiiF12= IH(iw)I2)and the phase angle a we can’write Eq. (1.1.12) as*
H ( N , , u) = r] exp(ia) IH(iw)I2 = qz = ( 1
a = tan-’lfZ/(l
+
+jl
+ fl)].
( 1.1.13a)
(1.1.13b) ( 1.1.13~)
The amplitude ratio r] and the phase angle a are plotted in Fig. 3 as a function of the Stokes number = v / ( o d 2 ) . Note that when interpreting Fig. 3, the Stokes number is inversely proportional to the exciting frequency w of the fluid, so that frequency w increases in the negative direction on the Stokes number axis. The amplitude ratio r] can also be interpreted as the sensitivity of the flow tracing particles when viewed as one element of an instrument for measuring fluid velocities. The static sensitivity when the fluid frequency w is zero, is unity from Eqs. (1.1.1 l), (1.1.12), or (1.1.13a). This can also be seen from Fig. 3, where it corresponds to the condition at large values of the Stokes number. The dynamic sensitivity corresponds to the condition for nonzero values of the fluid frequency, and from Fig. 3 it can be seen that the sensitivity is greater than one if the particle is less dense than the fluid. This is, perhaps, an unexpected result, and could be a significant factor in some experimental situations since it allows an effective magnification or amplification of the fluid velocity. The temporal dynamic characteristics of the flow tracing particles are * Solutions to the problem of the motion of a particle when the fluid has a rectilinear periodic motion have been given by a number of authors (see Table VII).
Ns
Ns
(a1
(b)
FIG. 3. (a) and (b) Amplitude ratio for flow tracing particles. u.00
FIG.3c. Phase angle for flow tracing particles, for various particle/fluid density ratios.
30
1.
MEASUREMENT OF VELOCITY
represented by the phase angle a. For values of the density ratio CT less than unity, the phase angle is positive, so that the particle motion leads the motion of the fluid. Correspondingly, when the particle is denser than the fluid, i.e., when the density ratio CT is greater than one, the particle lags behind the fluid. 1.1.2.2.5. EXPERIMENTAL DETERMINATION OF THE DYNAMIC CHARACTERISTICS. The preceding section presents results which could be of practical value in the assessment of the dynamic characteristics of flow tracing particles, but experimental verification is essential because of the uncertainties involved in the theoretical model. These are principally in the choice of the drag coefficient and in assigning a value to the particle diameter. Practical methods for the experimental study of particle dynamic characteristics are described in this section. It is clear from the discussion of the previous section that the description of the particle dynamic characteristics in terms of the frequency response is desirable. This will provide information in the most general form, since it leads directly to the transfer function for the particles. Table VI provides a compact list of various measurements that have been made of the response of small particles to a periodically varying fluid velocity. Of these, the only investigation that appears to have specifically involved flow tracing particles is due to Mazumder, Hoyle, and Their results for relative particle Reynolds numbers much less than unity are compared in Fig. 4 with predictions based on Eq. (1.1.10). The agreement is seen to be good, indicating that the theory of Section 1.1.2.2.4can be applied with confidence to the assessment of flow tracing particles, provided the conditions of the model can be met. Frequency response testing is difficult when the experimental fluid is a liquid. The necessary apparatus is much more cumbersome and expensive than the electrically driven systems that can be used when air or some other gas is the experimental fluid (see Table VI for some examples). In addition, the comparatively heavy moving parts limit the frequency bandwidth of the fluid velocity. For these reasons, consideration must also be given to the possibility of testing the response of a flow tracing particle to a step change in fluid velocity or to a step change in an imposed body force. However, it should be recognized that step function testing may not provide such good information on the particle dynamic characteristics as frequency testing, because the accuracy of the former is influenced by difficulties in determining the zero time point. Information
M. K. Mazumder, B. D. Hoyle, and K . J. Kirsch, in Proc. Second Inr. Workship on Laser Velocimeiry, 2nd (H. D. Thompson and W. H. Stevenson, eds.), Purdue Univ. Eng. Exp. Stn. Bull. No. 144, p. 234 (1975)
1.1. TRACER METHODS
31
TABLEVI. Techniques for Experimentally Studying the Dynamic Characteristics of Flow Tracing Particles Liquid experimental fluid Fluid velocity Periodic variations Step change (shock)” Steady, turbulent
Gaseous expenmental fluid
Liquid droplets
Gas bubbles
Solid particles
Liquid droplets
-
-
I
I
b,c,e -
-
-
-
d
Solid particles
f-i j,k m
In the experiments described in Ref. k , it was found that the result given by Eq. (1.1.14) was inapplicable, and it was concluded that this was a consequence of the distortion of the particle due to the very high acceleration rates that were encountered. This distortion affected the particle drag coefficient, which could no longer be expressed by Eq. (1.1.4). E. B. Denison, W. H. Stevenson, and R. W. Fox, AIChE J. 17,781 (1971). N . S . Berman and E. E. Cooper, in “Turbulence Measurements in Liquids” (G. K. Patterson and J. L. Zakin, eds.), p. 10. University of Missouri, Rolla, 1971. L. F. Jernqvist and T. G. Johansson, J. Phys. E 7, 246 (1974). E. B. Tunstall and G. Houghton, Chem. Eng. Sci. 23, 1067 (1%8). E. N. da C . Andrade, Proc. R. Soc. Lond, Ser. A 134, 445 (1932). 0. Brandt, H. Freund, and E. Heidmann, 2. Phys. 104, 511 (1937). F. T. Gucker and G. J. Doyle, J. Phys. Chem. 60, 989 (1956). 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. Eng. Exp. Stn., Bull. 144, 234 (1975). H. D. von Stein and H. J. Heifer, Appl. Opt. 11, 305 (1972). W. J. Yanta AIAA Fluid Plasma Dyn.Conf.,6th, 1973 AIAA Paper No. 73-705 (1973). G. Birkhoff and T. E. Caywood, J. Appl. Phys. 20, 646 (1949). L. B. Torobin and W. H. Gauvin, AIChEJ. 7, 406 (l%l).
’
on experimental methods for measuring step response will be found in the references listed in Table VI. The theoretical response of a flow tracing particle to a step change in fluid velocity can be obtained by applying the Laplace transform to the equation of motion with the initial condition ~ ~ (=7u ) F ~ ( T ) at 7 = 0, where h is the Heaviside unit step function. The result is*
-p
where a and
exp(P27) erfc[P(T)1’2}, (1.1.14)
P are defined in the List of Symbols.
* Actually this is the solution for u < 8. For u = 8, a = 4 = p, and for u > 8, a and B are complex. The explicit forms of the solutions for up/uFin these two cases can be obtained from the calculations of Hjelmfelt” (see Table VII).
32
1.
08
-
06
-
MEASUREMENT OF VELOCITY
9
0.4 -
02
10-1
10-e
10-3
d-&/2=6r/2NS
I
(CWs')
FIG.4. Comparison of theoretical prediction (line) of particle amplitude ratio with experimental measurements (circles). (Reproduced, with the authors' permission, from Mazumder et
1.1.2.2.6. APPLICATION OF DYNAMIC CHARACTERISTICS. It is proposed in this section to discuss the quantitative use of the flow tracing particle dynamic characteristics. Information of this type can be used in two ways. First, it can form one aspect of the assessment of candidate flow tracing particles in which the particle velocity is calculated for a given fluid velocity. Second, experimental data can be numerically corrected for the effects of the particle dynamic characteristics by computing the time history of the fluid velocity from the time history of the particle velocity. The calculations are carried out as follows: the Fourier (if periodic functions are involved) or Laplace (if transient functions are involved) transform of the given time history is substituted into Eq. (1.1.10) together with the particle frequency response (transfer function H) and then solved for the Fourier or Laplace transform of the unknown element. Thus iip = HiiF we have: (particle assessment) iiF
=
i p / H
(data analysis).
( 1 . 1 . 1 5a)
( 1 . 1 .1 5b)
The desired time history ( u p or u p ) is then determined by taking the inverse transform. Some of the few analytical solutions of Eq. (1.1.10) that are available are listed in Table VII. Not all time histories can be represented by explicit mathematical for-
33
1.1. TRACER METHODS TABLEVII. Solutions of the BBO Equation with Stokes’ Drag Law
UF
Approximation (see Table I) -
Constant“
Linear variation along axis of duct Periodic Turbulent
Reference
Notes
b -e
Ref. d , e consider all values of the density ratio CT
-
I I1 I11 I I1
-
IV -
-
11 111
-
-
The motion of a particle released from rest in a fluid moving at a constant velocity is analogous to the motion of a particle moving under gravity in a fluid at rest if the quantity (V - V,)/(V, - V,) in the latter solution is substituted for up/uF in the former (see Table VIII). A. T. Hjelmfelt, Jr., Behavior of a sphere accelerating in a viscous fluid. Ph.D. Thesis, Northwestern University, Evanston, Illinois (1965). L. M. Brush, H. W. Ho,and B. C. Yen, Proc. A m . SOC.Civ. Eng. W(HYI), 149(1964). * H. Villat and J. Kravtchenko, “Lecons sur les fluides visqueax.” Gauthier-Villars, Pans, 1943. A Fortier, “Mecanique des suspensions.” Masson, Pans, 1967. W. Davis and R. W. Fox, J . Basic Eng. 89, 77 1 ( 1967). S . J. Lukasik and C. E. Grosch, “Velocity Measurements in Thin Boundary Layers,” Memo No. 122. T . M. Davidson Lab., Stevens Institute of Technology, Hoboken, New Jersey, 1959. S . K. Friedlander, A f C h E J. 3, 381 (1957). ‘ F. H. Wright, “The Particle-Track Method of Tracing Fluid Streamlines,” h o g . Rep. No. 3-23. JPL, Caltech, Pasadena, California, 1951. J. Faure, Houille Blanche 18 (3), 298 (1%3). .+ C. J. Chen and R. J. Emrich, Phys. Fluids 6, 1 (1963). ’ M . Gilbert, L. Davis, and D. Altman, Jet Propul. 25, 26 (1955). * S. R. Gutti, Proc. A m . SOC. Civ. Eng. W (HY4), 1073 (1968). ” S. R. Gutti, Proc. A m . SOC. Civ. Eng. 97 (HY7), 1117 (1971). W. Koenig, Ann. Phys. (Leipzig) [3] 42, 353 (1891). C. M. Tchen, Mean values and correlation problems connected with the motion of small particles suspended in a turbulent fluid. Ph.D. Thesis, Delft, Netherlands (1947). a F. A. Schraub, S. J. Kline, J. Henry, P. W. Runstadler, Jr., and A. Littell,J. Basic Eng. 87,429 (1965). V. C. Liu, J. Meteorol. 13, 399 (1956). a B. T. Chao, Oesterr. fng.-Arch. 17-18, 7 (1964). ‘ A. T. Hjelmfelt, Jr. and L. F. Mockros, Appl. Sci. Res. Sec. A 16, 149 (1%6). ” J . 0. Hinze, “Turbulence.” McGraw-Hill, New York, 1959. ” P. K. Khosla and S. Lederman, “Motion of a Spherical Particle in a Turbulent Flow,” Rep. PIBAL No. 73-22. Polytechnic Institute of New York, 1973. D. M. Levins and J. R. Glastonbury, Trans. fnst. Chem. Eng. 50, 32 (1972). S. L. SQO, Chem. Eng. Sci. 5 , 57 (1956).
’
34
1. MEASUREMENT OF VELOCITY
mulas, and approximate numerical methods may be ~ e q u i r e d . ~ ’ -Me~ chanical, electrical, and optical devices are also available.B1 There are difficulties associated with recovering the fluid velocity from measurements of the particle velocity. First, the particle velocity must be such that it may be differentiated unambiguously from the system noise (Brownian motion; see Section 1.1.2.4). Second, the process of recovery is tedious and liable to introduce errors. However, in principle, it is possible to avoid such computations by choosing the particle material so that its density is equal to that of the experimental fluid, in which case the density ratio u is unity, the amplitude ratio r) is unity, and the phase angle a is zero (see Fig. 3). With a density ratio other than unity, the amplitude ratio and the phase angle are strongly dependent on frequency, at least for low values of the Stokes number (see Fig. 3). Nevertheless, by an appropriate choice of materials for the particle and the experimental fluid, it may be possible for a limited frequency range, corresponding to conditions of maximum interest to the experimenter, to obtain an amplitude ratio and a phase angle that are close to zero. Thus, for example, with a density ratio of zero (m = 0), which would correspond to a gas bubble used as a flow tracer in water, the amplitude ratio is unity for a Stokes number of 6 and the phase angle is close to zero for a Stokes number of 2. A particularly important application of particle tracking is in the measurement of turbulent fluid velocities. Although there has been some work in this area using stroboscopics2-M and chronophotographic methodsBSfor observing particle motions, it is only with the introduction of the laser velocimeter technique in 1964 by Yeh and CumminsBgthat the use of particle tracking in this application has become really significant. Because of the inherent complexity of turbulent flows, only simple systems have been analyzed mathematically. One such system concerns s7 E. T. Whittaker and G . Robinson, “The Calculus of Observations,” 4th ed. Blackie, Glasgow & London, 1944 (reprinted: Dover, New York, 1967). C. R. Huss and J . J. Donegan, Nut. Adv. Comm. Aeronaut. Tech. Notes, 3598 (Jan.
1956). 58
C. R . Huss and J. J. Donegan, Nut. Adv. Comm. Aeronaut. Tech Notes 4073 (Oct.
1957).
J. J . Donegan and C. R. Huss, Nut. Adv. Comm. Aeronaut. Tech. Notes 3701 (1956). N . F. Barber, “Experimental Correlograms and Fourier Transforms.” Pergamon, Oxford, 1961. Bp A. Fage and H. C. H. Townend, Proc. R . Soc. London, Ser. A 135,656 (1932). Bs A. Fage, Philos. Mag. [7] 21, 80 (1936). A. Fage, in “50 Jahre Grenzschichtforschung” (H. Gortler and W. Tollmein, eds.), p. 132, Vieweg, Braunschweig, 1955. E. R. Corino and R. S. Brodkey, J . Fluid Mech. 37, 1 (1969). (Is Y. Yeh and H. Z. Cummins, Appl. Phys. Lert. 4, 176 (1964).
‘‘
1.1.
35
TRACER METHODS
the motion of small particles in a homogeneous, isotropic, steady flow field. The fluid velocity then depends only on time and not on the coordinates of the point under consideration. This situation has been considered previously by a number of authors (see Table VII). It corresponds to the conditions at the center of a pipe or in the atmosphere. Because of these assumptions, only qualitative conclusions can be drawn regarding the use of flow tracing particles in other types of turbulent flows, e.g., agitated tanks or boundary layers. However, even under these limitations, the technique which is described may allow experimental data from a wide variety of turbulent flow situations to be interpreted with greater confidence than would otherwise be the case. To represent the fluid ( V , ) and particle (Up)velocities in the turbulent flow, we will follow the usual procedure and assume we can write up(t) =
Op
+ Up(t),
UF(f) =
OF
+ UF(f),
( 1.1.16)
where Up, and OF represent the time averaged values of Up and VF, respectively, and up and uF the fluctuating portions of the velocity, are so defined that their time averages are zero. The quantities that are usually required in the study of turbulent flows are* the time mean velocity of the fluid OF, the time mean square of the fluid velocity fluctuations 3, the time mean product ~ (= i i in isotropic turbulent flow), and the energy spectrum density function FF . The relation between the measured quantities $ and FF, and the desired quantity can be showns7-ssto be
z,
Thus, the desired relation between u', and can be obtained from a knowledge of the transfer function of the flow tracing particles and the spectral density function FFof the turbulent velocity field. The equation of motion for a small particle in a turbulent fluid is closely related to Eq. (1.1.2) governing its motion in laminar flows. There are, however, some differences between the equations of laminar and turbulent flow, and this has been discussed in detail by Corrsin and L ~ m l e y , ' ~ 87
V . C. Liu, J . Meteurul. 13, 399 (1956). S. K . Friedlander, AIChE J . 3, 381 (1957). B. T. Chao, Oesterr. Ing.-Arch. 18, 7 (1964). S. Corrsin and J . Lumley, Appl. Sci. R e s . , Sect. A 6, 114 (1956).
* The correlation function is sometimes measured, but in practice it is easier to measure the spectral density function FF. Since the correlation function and the spectral density function are a cosine transform pair, the conversion of the experimental data on the spectral density function to a correlation function is accomplished quite easily.
36
1.
MEASUREMENT OF VELOCITY
who show that an equation identical in form to Eq. (1.1.2b) is obtained. However, this equation is actually different, in that the variables urn and urn are random functions of time which can only be described statistically, i.e., they are stochastic variables. Because of the identity of form, it is possible to use the transfer function for the laminar case, Eq. (1.1.1 l), in Eq. (1.1.17), which applies when the fluid motion is turbulent. The formulation of the equation of motion for the particle in a turbulent fluid involves a number of assumptions which are discussed by Hinze.” It seems probable that all of these except one can be satisfied in most experimental situations. The assumption that seems most doubtful is that the flow tracing particle is surrounded by the same particles of fluid at all times. In view of the very disordered nature of turbulent flow, this does not seem very likely. However, Hinze’l has suggested that if this assumption is to be satisfied at all, it is most likely to be when the densities of the particle and the fluid are about equal. This is the condition under which, from the point of view of its dynamic characteristics, it would be most appropriate to use a flow tracing particle. We would therefore anticipate that the uncertainties associated with this assumption should be minimized in any experiment in which careful attention has been given to the dynamic characteristics of the flow tracing particles. One further assumption which must be verified in a given experimental situation is the validity of Stokes’ law. The equation of motion has been set up under the assumption that this applies for the drag experienced by the particle in the turbulent fluid. This is an attractive assumption because it simplifies the analysis of the dynamics of the flow tracing particles. The usual procedure which is employeds0 to justify this assumption is to suppose that if the relative particle Reynolds number RepR based on the rms relative velocity (p)112is unity or less, i.e., RepR = (F)”’d/V S 1,
( 1.1.18)
then Stokes’ law may be used. The method for determining the mean square relative velocity v2 can be deduced by following an argument analogous to that used in connection Doing this, we obtain with formulating Eq. (1.1.17) for
z.
-
=
IHR(i@)l*FF(W)
( 1.1.19)
is the absolute value of the relative amplitude ratio, and the where other quantities have been defined earlier. The relative amplitude ratio can be obtained from Eq. (1.1.2~)(with n J . 0. Hinze, “Turbulence.” McCraw-Hill, New York, 1959.
1.1.
37
TRACER METHODS
1
(a)
(b)
FIG. 5. (a) and (b) Relative amplitude ratio for flow tracing particles for various particle/fluid density ratios.
D
= 0) if the quantities V and uFare treated as stochastic variables. The result is lHR(Ns
9
m)lz =
f: -k 6 ,
(1.1.20)
where fi and fi have been introduced before. The quantity vR(= IHRJ) is plotted as a function of the Stokes number N s in Fig. 5 . The assessment of the suitability of flow tracing particles for use in a given turbulent flow requires a knowledge of the fluid spectral density F F , which must be obtained either from subsidiary experiments, or from published data which refer to a similar experimental situation. Because of the considerable experimental difficulties associated with making Lagrangian measurements, practically all turbulence data are Eulerian. They are obtained using fixed detectors, such as a hot wire anemometer, or, more recently, a laser velocimeter. The use of Eulerian
38
1.
MEASUREMENT OF VELOCITY
data in conjunction with an amplitude ratio H R obtained from a Lagrangian equation of motion raises important questions regarding the interpretation of the ratio uT/ut and the quantity p. This is discussed in greater detail by Somer~cales.'~However, Chaoaehas suggested that for very small particles with a density close to that of the experimental fluid, the Lagrangian and Eulerian mean square velocities of the particles should be approximately equal. This may serve as a justification for ignoring the whole problem and using Eqs. (1.1.17) and (1.1.19) without regard to the inherent differences in the various quantities which appear in these equations. 1.1.2.3. Loading Error. It is known that the addition of particles to a fluid can affect its viscosity, and it might also be suspected that the presence of particles may modify a turbulent flow field. The resulting errors in fluid velocity measurement which could arise from this source will be called loading errors, in analogy with the effect of a transducer on the quantity to be measured. Thus a thermocouple may distort the temperature that it is measuring, and it is then said to load the system. The modifications, due to added particles, of the fluid viscosity have been extensively investigated, and a good review has been prepared by Happel and Brenner." It is clear from this work that at the particle concentrations of interest in flow tracing (say, 1 part of particles to 30,000 parts of fluid by volume), the change in the apparent fluid viscosity due to the particles is negligible. A fluid which is in turbulent motion and contains flow tracing particles must exchange energy with the particles. The energy supplied to the particles will influence the turbulent motion. Only a limited amount of work has been done on this problem, and a paper by Goldschmidt et ~ 1 . ' pro~ vides a recent review of the experimental investigations. They conclude that although the question has not been finally settled, the effect of low concentrations (say, less than 1 : 1000 by volume) on the turbulence is negligible. 1.1.2.4. Limit of Sensitivity. It was noted in Section 1.1.2.2.4 that the sensitivity of flow tracing particles increases as the density ratio u and diameter d decrease, but this improvement in sensitivity cannot go on without limit. A condition is eventually reached at which the sensitivity E. F. C. Somerscales, in Proc. Inr. Workshop Laser Velocimetry, 2nd (H. D. Thompson and W. H. Stevenson, eds.), p. 216. Purdue Univ. Eng. Exp. Sin. Bull. No. 144 (1975). 7s J. Happel and H. Brenner, "Low Reynolds Number Hydrodynamics." Prentice-Hall, Englewood Cliffs, New Jersey, 1965. " V. W. Goldschmidt, M. K. Householder, G . Ahmadi, and S. C. Chang, Prog. Heat Mass Transfer 6, 487 (1972).
1.1.
TRACER METHODS
39
of the particle is so great that in addition to responding to the velocity of the fluid, it also responds to the forces imposed by collisions between it and the molecules of the experimental fluid. This is the well known phefor a good review), which in nomenon of Brownian motion (see certain cases may be sufficiently large so as to mask the displacement of the particle due to the mean motion of the experimental fluid, i.e., the desired signal, the gross particle motion, is buried in noise, the Brownian motion of the particle. Brownian motion in flow tracing particles will also introduce uncertainty into the measurement of the particle displacements. In turbulent fluid flow it can also manifest itself in a spuriously large rms velocity, which should actually be corrected by removing the Brownian motion effects. The theory* of Brownian motion is due to Einstein,77 S m o l u ~ h o w s k i , ~ ~ and L a n g e ~ i nand , ~ ~it shows that the mean square displacement of the particle in the x direction in a time t is given by -
x2 = ( 2 k B T / 3 r d p F ) t .
(1.1.21)
The magnitude of Brownian motion as given by this equation has been verified by e~periment.'~'+'~ In Fig. 6, the root mean square Brownian displacement during Is, as obtained from Eq. (1.1.21), is plotted against particle diameter. If we take 0.5 p m as the minimum practical observable particle displacement in Is, it can be seen from Fig. 6 that particles of diameter smaller than - 2 p m will exhibit Brownian motion in water. In air, particles smaller than 100 p m diameter will have an observable Brownian motion. However, in this case, the effects of the gravitational field will be much more important Longmans, Green, ' 5 E. F. Burton, "The Physical Properties of Colloidal Solutions." New York, 1938. " P. Langevin, C. R . Hebd. Seances Acud. Sci. 146,530 (1908). '' A. Einstein, "lnvestigations on the Theory of Brownian Movement" (R.Fiirth, ed.). Dover, New York, 1956. " M. von Smoluchowski, Ann. Phys. (Leipzig) [4] 21, 756 (1906). 78 J. B. Perrin, "Brownian Movement and Molecular Reality" (F. Soddy, transl.). Taylor & Francis, London, 1910. B' H. Fletcher, f h y s . Rev. 33, 81 (191 1). H. Fletcher, f h y s . Rev. [2] 4, 440 (1914). R. M. Elrick, I1 and R. J. Emrich, Phys. Nuids 9, 28 (1966).
* Langevin's approach fits in very closely with ideas which have been employed here (Section 1.1.2.2) to assess the dynamic characteristics of flow tracing particles. Langevin solved the equation of motion for a small particle [Eq. (1.1.2a)l under the assumption that the type 111 approximation could be used and that the particle was subject to a random force arising from the molecular collisions. Details of the calculation can be found
1. MEASUREMENT
40
OF VELOCITY
8a t
5
s W
>
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 )
<#
I +
=# >#
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.
MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
(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.
MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
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
MEASUREMENT OF VELOCITY
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
1. MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
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).
1. MEASUREMENT OF VELOCITY
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
1. MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
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
MEASUREMENT OF VELOCITY
\ 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 *
and OD
= o d e s - eo) (U/c)/[ 1 - es
(U/c)].
The particle velocity U is such that it may reasonably be assumed* that (U/c) << 1, and so
es
OD
2 :
(oO/C)U
*
(1.1.45a)
(es - Q)
or WD
2 :
(27rn/ho)U * (es - eo) = ko(es - eo) U.
According to Eqs. (1.1.45), if we know
ooand
(1.1.45b)
the directions repre-
'On J . W. Foreman, Jr., R . D. Lewis, J . R. Thornton, and H. J . Watson, Proc. fEEE 54, 424 (1966).
* This is equivalent to assuming ko ks or (wo - w s ) / c << 1. Foreman et a/.zonestimate that if IUI = 3 x lo3 ms, then ko and k" will differ by about 1 part in 106. Compared to the other uncertainties connected with this technique of fluid velocity measurement (see Section 1.1.4.4.5). this error is negligible. 3
102
1.
MEASUREMENT OF VELOCITY
sented by the vectors es and eo, and if we can measure the Doppler frequency shift wD,then we can determine the velocity U or, more precisely, the component of U in the direction of the vector es - eo. Equations (1.1.45) show that a system that measures the Doppler frequency shift oD provides a measurement of the fluid velocity U that depends only on the wavenumber ko of the incident light and the relative directions of the velocity vector U and the vector es - e,,. These are quantities that can be determined by the choice of light source and the selected system geometry, so the measurement of the velocity U by a laser Doppler velocimeter is an absolute measurement, i.e., the velocimeter does not require calibration. In addition, Eqs. (1.1.45) show that the laser Doppler velocimeter provides a linear relation between the particle velocity U and the Doppler frequency shift wD. Often we are interested in measuring the components Uj (j = 1, 2, 3) of the particle velocity in three appropriately chosen coordinate directions. If we use a Cartesian coordinate system, we can write compactly 3
U
njUj,
(es - eo) =
( 1.1.46)
J= 1
where n j = el - eoj, and e l , eoj are the direction cosines of the unit vectors es and eo, respectively. Equation (1.1.46) indicates that measurements must be made in three directions, given by n (j = 1,2, 3), in order to determine the magnitude (but not the direction; see Section 1.1.4.4.8) of each of the components Uj of velocity. This topic is discussed in further detail in Section l . l .4.4.8. If there is only one component of velocity, e.g., in laminar flow in a duct, where there is only an axial component, the determination of that component from Eq. (1.1.46) can be greatly simplified by an appropriate choice of e,, and es. Thus, suppose we have the situation shown in Fig. 13, where the vectors U,e,,, and eslie in the plane of the paper. Choosing the x axis (j = 1) to be coincident with the vector U, we have from Eq. ( 1.1.46)
u
(es - eo) = (eS
- eol)U1,
where eol = cos(90 - 4 2 ) " and es = cos(90
+ 42)".
Then
U
*
(es - eo) = 2U1 sin(a/2),
(1.1.47)
wD = 2koU1sin(a/2).
(1.1.48)
for, from Eq. (1.1.45b),
1.1.
TRACER METHODS
I03
FIG.13. Velocity vector I/ and light propagation vectors eo and es: single velocity component.
This very simple result is useful in demonstrating some of the characteristics of the laser Doppler velocimeter. The determination of the parameters of a turbulent flow from the laser Doppler velocimeter has been considered by Durst and white la^.^^^ If the fluid is in turbulent flow, we can write
uj = uj + uj
(j= 1, 2, 3).
( 1.1.49)
Then Eq. (1.1.45a) becomes, using Eq. (1.1.46), ( 1.1.50)
Put
Hence the mean particle velocity components can be determined from time average measurements of the Doppler frequency shift in three mutually orthogonal directions. Now
u)8
F. Durst and J . H. Whitelaw, Prog. Hear Mass Transfer 4, 311 (1971).
1. MEASUREMENT
104
OF VELOCITY
The time mean square of the frequency fluctuations is
(1.1.51) Taking measurements in the six directions indicated by the quantities nf, nl and their products, it is possible to obtain the time mean square of the velocity fluctuations and the Reynolds stresses.210*211 For isotropic turbulence in which ufuj = 0 and u1 = ul = u , Eq. (1.1.49) becomes (1.1.52) In the case of a flow with only one velocity component (Fig. 13), we put j = 1 in Eq. (1.1.52)and obtain (WD
-6DY =
(OO/c~~n~.
Now nr = (eol - e1)2 = 4 sin2(a/2); hence [(OD
- i&)2]1'2= (~~/c)2[sin(a/2)](U2)~/~
(1.1.53)
In principle, according to Eq. (1.1.53) the evaluation of the laser Doppler velocimeter signal in the presence of isotropic turbulence is straightforward compared to the nonisotropic case shown in Eq. (1.1.51). In practice, the measurement of turbulence (isotropic or nonisotropic) with the laser Doppler velocimeter can involve difficulties that are not apparent in Eqs. (1.1.51), (1.1.52),and (1.1.53). In those measurement situations involving a high density of flow tracing particles, the photodetector output signal is inherently noisy regardless of the character of the flow (see Section 1.1.4.4.7.1). Consequently it is necessary to differentiate between the intrinsic signal noise (the so-called ambiguity noise) and the random nature of the turbulent contribution to the velocimeter output; this is discussed in Section 1.1.4.4.7.1. 1.1.4.3. Receiving Techniques. The laser light scattered by the moving particle has had its frequency shifted relative to the frequency of the incident light by an amountfD that depends on the velocity of the scatzll
C. Greated, J . Phys. E 3,753 (1970). P. J. Bourke, C. G. Brown, and L. E. Drain, Disa lnf. No. 12, p. 21 (1971).
1.1.
TRACER METHODS
105
tering flow tracing particle. It is the function of the element of the laser Doppler velocimeter that receives the scattered light, to determine the frequency shift and hence provide information at its output on the particle velocity. The scattered light incident at the receiver may be viewed as a frequency modulated carrier, with a carrier frequency equal to the incident light frequency fo and the modulating frequency being the Doppler frequency shift f,. The receiver is required to demodulate the input signal. The demodulation process involves the solution of a problem in light spectroscopy. However, the spectroscopic measurements require an unusually high resolution fD/&. This has a strong influence on the method that is adopted for demodulation. Thus from Eq. (1.1.48), under typical conditions (say, Q = 45", A,, = 6328 nm, helium-neon laser) a flow tracing particle velocity of 1 cm/s corresponds to a Doppler frequency shiftf, of 1 kHz. The frequency& of the light incident at the flow tracing particle (the carrier frequency) is about 4 x l O I 4 Hz, so the required resolution fD/& of the spectroscopic measurement is The available techniques of light spectroscopy can be broadly classified as212.213. . (a) systems using a slit, (b) Fabry-Perot interferometers, and (c) light beating or optical mixing methods. We will briefly consider each of these in turn from the point of view of their attainable resolution relative to the required resolution. Spectroscopic methods that use a slit are exemplified by the wellknown prism spectrograph. This device has the disadvantage of having to meet the requirements indicated insufficient resolution cfD/&= above. In addition, the slit, which is a necessary element in the device, limits the amount of light available for analysis. Since this is already at a very low level, having been scattered by the flow tracing particle, it may be difficult to attain an adequate output signal from the velocimeter. The Fabry-Perot interferometer, which has been used widely in the high resolution spectroscopy of light, has a resolution of about lo-'. Such a resolution would be appropriate where the Doppler frequency shift was greater than about 1 MHz, corresponding to a velocity of about 300 m/s. Clearly, the application of the Fabry-Perot interferometer to the laser Doppler velocimeter is limited to very high velocities. However, this is no disadvantage because the light mixing methods (described below) are not suitable for determining a Doppler frequency shift of more than about 1 MHz. Therefore, in practice, the Fabry-Perot interferometer is used for very high speed flows, and the light mixing techniques P. Jacquinot, Rep. Prog. Phys. 23, 267 (1960). A. Girard and P. Jacquinot, in "Advanced Optical Techniques" (A. C. S. van Heel, ed.), p. 71. North-Holland Publ., Amsterdam, 1967. z12
'13
106
1. MEASUREMENT O F VELOCITY
(see below) are applied to the more commonly encountered low speed flows. The extensive history of heterodyne detection in the radio frequency portion of the electromagnetic spectrum suggests a way in which Doppler frequency shifts of less than 1 MHz might be detected. In radio frequency heterodyne detection, the signal of interest is shifted to a lower frequency by mixing it in a square law device with a reference signal of fixed known frequency. At the resulting lower frequencies, narrow band filters are available, which can be swept across the signal spectrum to pick out the modulating frequency. Optical mixing detection applies these techniques to light waves so that the frequency modulated signal has an apparent center frequency equal to the modulating frequency fD. In the course of the detection process, the light waves are converted to an electrical signal, so the spectral analysis is carried out in the electrical frequency region of the electromagnetic spectrum. For a center frequency fD of 1 MHz, filters with a minimum bandwidth Afof about 1 Hz are available, so the apparent resolution Aflfo of optical mixing spectroscopy is This easily meets the resolution requirements described at the beginning of this section. At very low light levels, where the signal may be buried in the shot noise of the photodetector, temporal autocorrelation methods are superior to the use of filters in determining the frequency of the light. This is because the autocorrelation analysis of the signal is an enhancement process that suppresses noise effects and emphasizes the signal. Because low light levels can be treated by a photon, or discrete pulse model it is both convenient, and more accurate than analog techniques, to use digital methods for carrying out the correlation analysis. This method of spectrally analyzing light is known as photon counting correlation. The low light levels cause the data to arrive at very low rates, and, in consequence, the measuring time is probably greater than with the other, previously mentioned, spectroscopic methods. The remainder of this discussion of laser Doppler velocimeters will be concerned with the optical mixing and Fabry-Perot techniques for obtaining the desired Doppler frequency shift. In particular, Section 1.1.4.4 will discuss various aspects of the optical mixing method, and Section 1.1.4.5 will deal with the Fabry-Perot interferometer. 1.1.4.4. Optical Mixing 1.1.4.4.1. PHOTODETECTOR CURRENT.At radio frequencies, heterodyne detection is carried out by mixing the information bearing wave (corresponding to the scattered light in the laser velocimeter) with another wave, the reference wave, of slightly different frequency, and the mixed
1.1. TRACER METHODS
I07
waves are then passed through an element, such as a diode or a crystal, having an output proportional to the square of its input. When the input to such a square law device is two mixed periodic waves, the output consists of a component having double the original frequencies together with components having frequencies which are, respectively, the sum of the two original frequencies and the difference of the two original frequencies. By appropriate filtering, the desired component can be separated from the other output components. For a laser Doppler velocimeter, the appropriate square law element is either a photomultiplier tube or a photodiode (the respective characteristics of these are discussed in Section 1.1.4.4.6). The nature of these devices is such that only the component equal to the difference of the two input frequencies appears at the output of the detector, i.e., the detector also acts as a filter.* Good general reviews of optical mixing are provided by Ross,21sde Lange,217and Pratt.218 A more advanced treatment has been given by Cummins and S ~ i n n e y . ~ l ~ Although the preceding discussion has emphasized the idea of processes that occur in time, it is worthwhile noting that optical heterodyne detection can also be viewed as interference between electromagnetic fields, a process that occurs in space. The interference arises as a result of the phase difference between the light scattered by the flow tracing particles and the reference beam. This phase difference is time dependent because the flow tracing particle varies its position relative to the photodetector, and, as a result, the interference is also a time dependent process. Consequently, if the interference pattern at the photodetector were observed, it would be seen to move as the flow tracing particle A . T. Forrester, W. E. Parkins, and E. Gerjuoy, Phys. Rev. 72,728 (1947). A. T. Forrester, R. A. Gudmundsen, and P. 0. Johnson, Phys. Rev. 99, 1691 (1955). ‘16 M. Ross, “Laser Receivers.” Wiley, New York, 1966. 21‘ 0. E. de Lange, IEEE Spectrum 5, 77 (1968). 218 W. K. Pratt, “Laser Communication Systems.” Wiley, New York, 1979. 21B H. Z. Cummins and H. L. Swinney, Prog. Opt.8, 135 (1970). 2“
‘I5
and * Optical mixing detection appears to have been first proposed by Forrester ef demonstrated by F o r r e s t e P in 1955. However, because these experiments were carried out prior to the invention of the laser, it was necessary to use a mercury arc light filtered to have an output in the vicinity of 5461 A. In addition, the optical characteristics of the light emitted by this source are such that it had to be condensed by a lens onto a pinhole (see Section 1.1.4.4.3). As a result of the spectral filtering and the need to introduce a pinhole (spatial filtering) between the source and the detector, only a part of the light emitted by the source was available at the detector; this resulted in a small signal-to-noise ratio and hence led to a large measurement of uncertainty. The laser avoids this difficulty and provides the opportunity to use optical mixing detection without elaborate techniques to ensure a measurable signal. This has led, among other things, to the laser Doppler velocirneter.
108
1.
(e)
SOURCE
MEASUREMENT OF VELOCITY
-
FIG. 14. Schematic arrangement of various laser Doppler optical configurations. -light direct from source; ---, light scattered from flow tracing particle. BD is the beam divider; BC, the beam combiner, FTP, the flow tracing particle. This diagram shows all configurations operating in the forward scattering mode. However, all configurations can be “folded” about the flow tracing particle so as to operate in the backward scattering mode. The backscattering mode is useful where it is physically impossible to locate the photodetector so as to receive forward scattered radiation or where the photodetector would be located inconveniently far from the transmitter (a common occurrence in the measurement of atmospheric flows). Beam splitters and beam combiners are only shown schematically. Durrani and Greated**Oshould be consulted for practical arrangements. (a) Single incident beam heterodyne configuration.”’ Other name: reference beam configuration. (b) Single incident beam homodyne configuration.’m*m Other names: differential Doppler, differential heterodyne. (c) Dual incident beam heterodyne configuration.’” Other names: reference
1.1.
TRACER METHODS
I09
passes through the illuminating beam of light. Although the interference explanation of the operation of the laser Doppler velocimeter will only be used occasionally in this paper, it does point out that the device is part of the larger field of interferometry. The laser Doppler velocimeter based on optical heterodyne detection was first demonstrated in 1964 by Yeh and Cummins.ss They employed an optical system (see Fig. 14) in which light from the source was divided by a beam splitter; one beam was incident on the moving flow tracing particle, and the other-the reference beam-was arranged to fall directly on the photodetector. The light scattered from the flow tracing particle was mixed with the reference beam at the photodetector. Subsequently, other optical configurations (see Fig. 14), in which all the relevant light beams passed through the measuring volume, were developed by Goldstein and Kreid,2z2 Lehmann,223Bond (see MaForeman ef af.,221 zumder and W a n k ~ l m ~ ~Durst ~ ) , and white la^,^^^ Schwar,206 and R ~ d d . ~The ~ various ~ - ~ laser ~ ~ Doppler velocimeter configurations can be classed* as either single incident beam or dual incident beam configurations, where single and dual refer to the number of light beams incident on the measuring volume from the light source (see Fig. 14). In the single incident beam configuration, light scattered in two directions, S1 and S2, is 220 T . S. Durrani and C. A. Greated, "Laser Systems in Flow Measurement." Plenum, New York, 1977. 221 J . W. Foreman, E. W. George, and R. D. Lewis, Appl. Phys. Lett. 7 , 77 (1965). 222 R. J . Goldstein and D. K. Kreid, J . Appl. Mech. 34, 813 (1967). 223 B. Lehmann, in "Electricity from MHD 1968," Vol. 111, p. 1341. IAEA, Vienna, 1968. IP4 M. K. Mazumder and D. L. Wankum, Appl. Opt. 9, 633 (1970). 225 M . J . Rudd, J. Phys. E 2, 55 (1979). 226 M . J . Rudd, Nature (London) 224, 587 (1969). **' M. J. Rudd, in "Optical Instruments and Techniques" (J. H. Dickson, ed.), p. 158. Oriel Press, London, 1970.
* This terminology has been developed in the preparation of this paper. Some configurations are known in the literature by a number of names (see Fig. 14), and it seemed worthwhile, to avoid ambiguity, to derive a uniform system for use in this paper. beam velocimeter, local oscillator heterodyne configuration. (d) Dual incident beam homodyne c ~ n f i g u r a t i o n . ~ ~Other ~ * ~ ~names: ' dual scatter, dual beam, fringe anemometer, symmetrical heterodyne. Although a symmetrical arrangement is shown, the scattered light (the dashed line) is often collected at an oblique angle to the line joining the source and the flow tracing particle. In this way (i) the spatial resolution is improved (see Fig. 19). and (ii) background light caused by flare from optical components and the walls of the test section (if present is minimized). However, because this reduces the amount of scattered light, it may be necessary to use special signal processing procedures, e.g., photon counting correlation (see Section 1.1.4.4.7.7). (e) Yeh and Cummins configuration.w
110
1.
MEASUREMENT O F VELOCITY
directed onto the photodetector. With two incident beams, the light scattered in a single direction out of the two incident beams 01 and 02, is arranged to fall on the photodetector. The two classes of laser Doppler velocimeter have an inverse relation to one another (Wang228),as can be seen from Fig. 14, i.e., any single incident beam configuration can be obtained from a dual incident beam arrangement, and vice versa, by interchanging the light source S and the photodetector D. However, for reasons to be explained below (see Section 1.1.4.8), the single incident beam arrangements are no longer used, and they have only been included here for the sake of completeness. The main objective of this section is to calculate the photodetector output current for the various optical configurations and, on the basis of this, to deduce some of the characteristics of the laser Doppler velocimeter. For a point on the photodetector, the output current i is proportional to the time averaged square of the incident electric field at that point, and is given by*
where a is the radiant sensitivityt of the photodetector, T is related to the effective bandwidth (or time constant) of the photodetector and any devices used to process and analyze the output from the photodetector, and R is the distance between the flow tracing particle and the photodetector. The evaluation of the integral in Eq. (1.1.54a) is simplified if we use the complex exponential notation for the electric fields. Then we have 2323283
=* C. P. Wang, Appl. Phys. L e f f .18, 522 (1971). B. J. Thompson, in “Optical Transforms” (H. Lipson, ed.), p. 27. Academic Press, New York, 1972. 230 M . V. Klein, “Optics.” Wiley, New York, 1970. 231 H. H. Hopkins, in “Advanced Optical Techniques” (A.C. van Heel, ed.), p. 190. North-Holland Publ., Amsterdam, 1967. M. FranGon, “Optical Interferometry.” Academic Press, New York, 1966. M.Born and E. Wolf, “Principles of Optics,’’ 4th ed. Pergamon, Oxford, 1970. 228
* A more elegant approach to the calculation of the photodetector current uses the theory of random processes.zzs~230This combines the effects of the temporal characteristics of the photodetector, the spectral characteristics of the light, and the geometrical characteristics of the velocimeter. However, in the method adopted here, which follows hop kin^,^^^ these various factors are considered separately; this makes it easier to relate the calculated photodetector current to the physical conditions in the velocimeter. t The radiant sensitivity a is defined as follows: a = q q / h f , where 7 is the quantum efEciency of the photodetector (depends on frequency but a typical value could be 10%); the electronic charge q = 1.6018 x A . s/electron, Planck’s constant h = 6.6237 x W . s*/photon, and f is the frequency of the light in hertz.
1.1.
111
TRACER METHODS
i(R) = a ( E ( R , t)E*(R, t ) ) ,
(1.1.54b)
where the star indicates the complex conjugate, and the sharp brackets indicate the time average shown in Eq. (1.1.54a). To carry out the calculation indicated in Eq. (1,1.54b), the nature of the electric field E at the photodetector must be determined. For either class of velocimeter optical configuration, this can be considered to consist of the reference beam electric field Eo and the two scattered fields El and E2. It will be understood that El = E: for the single incident beam case, and El = Eil (superscript s indicates scattered light) for the dual incident beam configuration, with a similar identification for E z . The indicated substitutions for El and E2 will be made at appropriate points in the calculation of the photodetector current (for an example, see Table XVII). This slightly clumsy procedure has been adopted in preference to the more general approach, which would deal with an arrangement involving two incident beams and two scattered beams, which leads to an expression for the photodetector current that is very long, involving fifteen terms, and is, in any case, only of interest when written in the special forms appropriate to particular optical configurations. Accordingly, the total electric field E at the photodetector will be, by superposition, ( 1.1.5 5a) E = Eo + El + E2, where, from Eq. (1.1.39),
EO = lAol exp[-j(wot +
EO
+ ko 191,
( 1.1 .55 b)
and from Eqs. (1.1.43) Ei
=
()ASI/R)exp[-j(wft - PRO +
E ~ ) ]
(i
=
1, 2), (1.1.55~)
where the variation with R of IASI/R will be assumed small in subsequent calculations. Then we have from Eq. (1.1.54b) i(R) = a[Zo + Zl -t Z2
+ 2% ( E a t ) + 2% (E,,E$) + 2% (EIE,*)], (1.1.56)
where % indicates the real part of the quantity enclosed in sharp brackets, the monochromatic intensity Z is given by Z, = (EfEF)( i = 0, 1, 2), and for the last three terms we obtain periodically varying functions 2%(EfEj*)= 2(iil~)~,~sinc[(w~ - wj)(T/2)] cos[(w, - oj)(t+ T / 2 ) + 61 ( i = 0, 1 ; j = 1, 2),
(1.1.57)
where wt - wj is the frequency of the photodetector output signal, which, by virtue of Eqs. (1.1.44), is proportional to the velocity U , and 6 is an op-
TABLEXVII. The Quantities 2 R < E,E,* > for the Various Optical Configurations” ~
Homodyne
FT<
0
~~
2 R <&El* > =
2 R < &&* > =
0
0
2 R < E,&* > =
Heterodyne
- O ( E , = 0)
20
f./ F TP
a rs is the optical path length between the light source and the photodetector. Ro is the optical path length between the flow tracing particle and the photodetector at t = 0, which can conveniently be taken as the time at which the particle enters the measuring volume. As mentioned in the body of the text, the terminology used to classify the various configurations is different from that commonly found in the literature.
I14
1.
MEASUREMENT OF VELOCITY
tical path difference that depends on the location, relative to the photodetector, of the light source rs and/or the illuminated measuring volume Ro (see Table XVIJ). The first three terms in Eq. (1.1.56) do not contain information on the Doppler frequency shift, but they do contain information on the spatial distribution of light in the illuminated measuring volume. These terms have the effect of raising the mean value of the last three terms in Eq. (1.1.56) above zero (see Fig. 17); for this reason these three terms, when consolidated together, are called the pedestal. The sinc function in Eq. (1.1.57) is plotted in Fig. 15, and from this it can be seen that the energy incident on the photodetector is zero when (wl - w,)(T/2) = T . It increases again as Tincreases; however, it disap-
n
I-, 3-
3ZJ 0 C .-
u)
FIG. IS. Sinc function. The absolute value of the sinc function should be used in Eq. (1.1.57) because the energy incident on the photodetector cannot be negative. Portions of
the curve lying below the absicissa are interpreted as indicating a phase reversal in the interference fringe contrast. The unbroken line represents sinc(o, - w,)T; the broken line, Isinch, - wJ)71exp &. where fiu = 0 when sinc(o, - ~ J )>T 0, and flu = n when sinc(o, - o,)T < 0.
1.1.
TRACER METHODS
1 I5
pears again for (wi- w1)(T/2) = 27r. For a satisfactory signal from the photodetector, (01 - wl)( T/2) must therefore be restricted to particular values. If (cot - wj)(T/2) does not exceed n/4, we see that sinc[(wf - o,)(T/2)] is about 0.9. Arbitrarily taking this as defining the maximum permissible value of T that gives a satisfactory signal, we have (of - q ) T S 7r/2. Since T is fixed by the characteristics of the photodetector and the signal processing equipment, the above condition specifies the maximum detectable frequency difference wf - w1 and hence the maximum measurable particle velocity. Typically, T = 1O-O s; then, according to the above criterion, mi - u1must be less than lo0 s-' or 2 X lo8 Hz. Using Eq. (1.1.48), we have 2 x lo8 = (2Ul/A) sin(a/2). For a helium-neon laser (A, = 623.8 nm) and assuming, for example, we have 2 x lo8 = (2U1/A) sin(a42). For a helium-neon laser (A? = 623.8 nm) and assuming, for example, a = 45", we have U , = 200 m/s. This corresponds to approximately the minimum useful frequency at which the Fabry-Perot interferometer can be used for measuring the velocity of the flow tracing particles (see Section 1.1.4.5). We will assume in subsequent calculations that the conditions in the laser Doppler velocimeter are such that the sinc function can be taken as unity, i.e., the particle speeds are substantially smaller than 200 m/s. In addition, since T is much less than the running variable t, and using Eq. (1.1.44a) to introduce the velocity, we can write Eq. (1.1.57) as
The terms corresponding to i , j = 1 , 2, 3 for the different optical configurations used in laser Doppler velocimeters are shown in Table XVII. Note that the path difference terms have been replaced by a characteristic path difference ko6. On combining Eqs. (1.136) and (1.1.58), we have a general equation that represents the photodetector output current for all the optical configurations used for laser Doppler velocimeters with detection by optical mixing; or, to be more exact, this represents the output current from a small element of the photodetector for a single, infinitesimally small flow tracing particle illuminated by a point source of perfectly monochromatic light. A more complete expression must allow for the sizes of the flow tracing particle, the light source, and the photodetector; the spectrum of the light emitted by the source; and the presence of more than one flow tracing particle in the measuring volume. However, a number of important characteristics of the laser Doppler velocimeter can be demonstrated even if these points are neglected, so the derivation of the more exact expression will be postponed until Sections 1.1.4.4.3and 1.1.4.4.6.
116
1.
MEASUREMENT OF VELOCITY
1.1.4.4.2. HETERODYNE A N D HOMODYNE CONFIGURATIONS. There are three terms of the form shown in Eq. (1.138) which appear in Eq. (1.1.57),and these are listed in Table XVII for the various optical configurations. The four conclusions listed below can be drawn from an examination of this table: (a) Two of the terms, 2% ( E a r ) and 2% (E&$), involve optical mixing of the reference beam (subscript 0) and the scattered light (subscripts 1 and 2); the third arises from the mixing of light scattered in two directions by the flow tracing particle. Because the mixed light originates from different sources in the first two terms, they will be classed as hererodyne contributions to the photodetector current, and the third, because it involves mixing of light from the same source, will be called a homodyne contribution, following the terminology* used by Cummins and S ~ i n n e y .Actually, ~~~ in the sense that Cummins and Swinney use the term homodyne, it applies to optical mixing between light scattered from two or more particles located in the measuring volume (the sources in this case are the various flow tracing particles; see Section 1.1.4.4.6). (b) The fist of the heterodyne terms 2%(EoEf) carries no information on the particle velocity U , and can therefore be included with the first three terms of Eq. (1.136) as part of the pedestal. (c) Both the heterodyne terms 2% (E&$) and the homodyne term 2%(E,E;) are not simultaneously required in the laser Doppler velocimeter, so one may be suppressed.t This can be done in two ways: (i) the
* This definition of homodyne differs from that of some authors,217who restrict it to heterodyne systems in which the illuminating beam and the reference beam originate from the same source (as in the laser Doppler velocimeter). A heterodyne system is then defined (in accord with radio communication practice) as one in which the two beams (waves) originate from different sources (as far as can be ascertained, a heterodyne configuration of this latter type has not been used in a laser Doppler velocimeter). It should be noted that this definition of homodyne and heterodyne really only applies to cases where the measuring volume contains a single flow tracing particle. If there is more than one flow tracing particle in the measuring volume, there can be heterodyning between light waves scattered from different particles (see Section 1.1.4.4.6). Hence the heterodyne and homodyne terminology introduced here is not entirely accurate in situations involving multiple flow tracing particles; however, it is believed that the context should clarify the usage. t However, Rudd2*66-227 has proposed an optical configuration that apparently represents a combination of the single and dual incident beam, homodyne configurations. Presumably, this just results in a signal that is stronger than that obtained from the simpler configurations because it increases the amount of scattered light incident on the photodetector. Nevertheless, the signal-to-noise characteristics of this configuration may be unsatisfactory, because unscattered laser light is directly incident on the photodetector so that laser noise is introduced into the signal.
1.1.
117
TRACER METHODS
reference beam can be eliminated, i.e., lois put equal to zero; or (ii) the relative strengths of the reference beam Zo and the scattered light can be adjusted so that the former is stronger than the latter, resulting in the suppression of the homodyne contribution to the photodetector current. The first approach forms the basis of the optical configurations demonstrated by L e h m a n ~ Mazumder ~ , ~ ~ ~ and W a n k ~ m and , ~ ~Durst ~ and white la^.^^^ Since this optical configuration involves the mixing of light scattered in two directions relative to the incident light direction, it has been called a homodyne detection method in Table XVII. The second method for eliminating one of the information carrying terms results, in practice, from the requirement that the photodetector aperture (for the dual incident beam configuration) or the light source aperture (for the single incident beam configuration) must be small if a satisfactory signal is to be obtained from the velocimeter (see Section 1.1.4.4.3 for a discussion of the reasons for this requirement). (d) The homodyne contribution is often conveniently interpreted as arising from light scattered by the flow tracing particles as they pass through the interference fringes formed in the measuring volume at the intersection of the two incident beams.227 This can be shown if we consider the Doppler frequency shift in this case, which is given by [see the term 2% (E,E,*)for the dual beam homodyne configuration and Eq. (1.1.48)] wD = ko(eol - eoz)* U
=
2koU1 sin(a/2),
(1.1.59)
where it is assumed that the particle trajectory coincides with the x axis of the coordinate system. The distribution of energy = i/a produced by the superposition of two plane light waves, such as occurs in the laser Doppler velocimeter measuring volume, is given by
6
= 21A01'{1
5
=
+ cos[ko(eol - em)
r])
( 1.1.60a)
or 21.4,12{1
+ c0s[2k0x sin(a/2)].}
(1.1.60b)
This represents a'periodic variation with x in the energy distribution in the measuring volume, i.e., interference fringes are formed at the intersection of the two beams. The spacing X between the fringe, i.e., the distance between two consecutive maxima or minima) is given by X = r / [ k o sin(a/2)]. If a flow tracing particle is passing through this system of fringes with velocity U parallel to the x coordinate direction, then the scattered light would appear to fluctuate with a frequency o = 27rU1/X = 2koU1sin(a/2). This result is identical to Eq. (1.1.59), so it is concluded that the photodetector output from the homodyne configura-
118
1. MEASUREMENT O F VELOCITY
tion can be given this fringe interpretation. In view of this result, the homodyne contribution to the photodetector output could be called the Doppler fringe current; correspondingly, the heterodyne contribution might be called the Doppler heterodyne current. It is not appropriate to give the heterodyne contribution the stationary fringe interpretation of the preceding paragraph. In this case the fringes are moving, and, moreover, they are formed as a result of the phase difference between the light scattered by the flow tracing particle and the light in the reference beam, as discussed at the beginning of this section. Furthermore, the fringes may be located in the measuring volume (dual incident beam heterodyne) or at the beam combiner (single incident beam heterodyne and Yeh and Cummins configurations). The same considerations also apply to the single incident beam homodyne arrangement, where the phase difference between light scattered from the flow tracing particle in two different directions interferes at the beam combiner placed in front of the photodetector. This phase difference changes with time as the particle moves, because the path difference along the two scattering directions changes with particle motion. In spite of this, there have been attempts to apply the stationary fringe model to this configuration by suggesting that so-called virtualfringes are formed whenever a flow tracing particle passes through the measuring volume. Clearly the explanation based on moving fringes formed at the photodetector is physically much more satisfying. 1.1.4.4.3. COHERENCE FUNCTIONS. As already pointed out, the preceding calculation of the photodetector output current has neglected to account for the finite sizes of the photodetector, the flow tracing particles, and the light source. The complete governing equation will therefore be obtained by integrating Eq. (1.1.56) over the respective areas AD, A , , and A , of these components of the velocimeter. We have, from Eq. (1.1.58), for the general information bearing term of Eq. (1.1.56), assuming, for simplicity, that the spatial distribution of the light across the effective area AP of the flow tracing particle is uniform* ( i = 0, 1, j = 1, 2) 2 9 ( E f E f ) = 2PJ'j) cos[ko(el - ej) Ut]% A ![D
I,,I,.
exp(Xiko6)dA,dA,dAD
where
* This is a reasonable assumption for particles that have dimensions much smaller than the characteristic dimensions of the illuminated measuring volume.
1.1.
I19
TRACER METHODS
and Pf is the monochromatic power of the light field over a defined area in a particular direction. If the flow tracing particle is spherical with radius u p , and can be treated as a circle of the same radius, and if the source and photodetector are both circular with apertures or radius as and a,, respectively, we have* from Eqs. (1.1.61)
~ ~ ( E ~ =E 2(PiPj)1'2W(t)ppp&, T) cos[k,Jef - el) * U t
+ k$],
(1.1.62a)
where pup= 2J,(a,kp sin a ) / ( a , k , sin a),p , = 2J1(a,k,sin B ) / ( a , k , sin B), 2Jl(aDkDy)/(aDkDy)(see Table XIX and Fig. 18 for definitions of the angles a,p, and y ) , and 8 is an optical path difference that depends on the location, relative to the photodector, of the center of the light source f , and/or the center of the illuminated measuring volume R0 (see Table XVII). The quantity W ( t )is a function that defines the light distribution along the path followed by the flow tracing particle. This depends on the light distribution in the measuring volume, which depends, in turn, on the method used to produce the light beams (see Table XVIII). The quantities pp,p , , and pD are the so-called coherence functions (or heterodyne efficiencies), and they can be defined as the ratio of the effective signal power to the total power incident on the photodetector. The quantity J l ( x ) / x is plotted in Fig. 16, where it can be seen that it is qualitatively similar to the sinc function of Fig. 15. From Fig. 16 it can be seen that J l ( x ) / x has its first zero at x = 1 . 2 2 0 ~ , so to obtain a satisfactory photodetector output signal the argument x must be a lot smaller than 1 . 2 2 0 ~ . Since for the three coherence functions shown in Eq. (1.1.62),the argument is proportional to the product of a radius a and an angle 8, a restriction is placed on the magnitudes of these two quantities. This is of great importance in understanding certain operational characteristics of the laser Doppler velocimeter. p D =
*34 235
238 ~
V. J . Corcoran, J . Appl. f h y s . 36, 1819 (1965).
V. J. Corcoran, J . Appl. f h y s . 37, 31 17 (1967). P. Buchhave, D i m Inf. No. 15, p. IS (1973).
~~
* The calculations of C o r ~ o r a n and ~ ~ B' ~u ~c h~ h a ~ illustrate e ~ ~ ~ the methods used to obtain this result. In the papers of Corcoran, it is assumed that the source and detector have rectangular openings, which is a form that has been used occasionallyz27in the laser Doppler velocimeter. However, if the apertures are very narrow slits (as would be required if a white light source were being used; see the end of the section) the measuring volume is extremely elongated in one direction. This gives poor spatial resolution (see Section 1.1.4.4.4).Also, for flow tracing particles having a trajectory that does not coincide closely with the longer axis of the measuring volume, it can give a signal of very short duration. This will affect the uncertainty of the velocity measurement (see Section 1.1.4.4.S), and the ability of certain types of signal processors to interpret the photodetector output (see Section 1 . I .4.4.7.6).
1.
120
MEASUREMENT OF VELOCITY
TABLEXVIII. Light Distribution Functions W(r) Method of forming light beams"
Light distribution function w(r)
Division of amplitude (unapertured laser exp source)
Division of wavefront (rectangular aperture)
sincz
Definition of optical parameters ( y - direction normal to plane of paper)
[- U p cosP(a/2)] b;
u:ig4 2b
(5u,r )
.t
f 0
o f
The radius of the laser light focus bo = 0.64 fX/b. At b , l / l o = e-l. See Table XXI for definitions of I and 1,. Apertures in screen are of finite width c normal to the plane of the paper.
i @I$ I
k
f
d
[Ji(x)/xI'
Division of wavefront (circular aperture)
2ar, where x = -U,r
Af
h
f
H
I, See latter part of Section 1.1.4.4.3 for methods of forming light beams.
-
0.6
-
0.4
-
x
\
x
7
v
(u
0.2 1.635lr 7
-
-0.2
X
FIG. 16. Coherence function J1 (x)/x. The remarks concerning negative values of the function made in the caption of Fig. 15 also apply to this function.
1.1.
121
TRACER METHODS
f
TIME, t
FIG.17. Ideal photodetector current as a function of time for a single flow tracing particle pedestal. Depth of modulation = (i,,,,, in the measuring volume. ---,envelope; -, im,,,)/(imax + in,") = fringe visibility. Heterodyne configuration: Depth of modulation = 2 ( a ) 1 ' 2 p p p s p ~I T+/ (a), a = P,/P;; pedestal level = aaP$W(r). Homodyne configuration: Depth of modulation = pppspT;pedestal level = 2 a P W ( t ) .
From Eq. (1.1.62a), combining the various terms according to Eq. (1.1.56) and using Table XVII, the photodetector current for the heterodyne configuration is i = aW(f)[Po+
+ 2pp/p&D(POfi)'IZcos(uDf + kO6)],
(1.1.62b)
where it is assumed that the terms corresponding to 2B(E&J) is small because both the receiving aperture and the scattered light power are small (see discussion below of detector size effects). For the homodyne arrangement,
i = aW(t)[P& + Pi2 + 2ppps(P&P~2)1'2 cos(o,t
+ k08)].
(1.1.62~)
From the preceding expressions it can be seen that the magnitude? of the information bearing component of the photodetector current depends on W ( t )and the quantities pu, p,, and F ~ .This is illustrated in Fig. 17, where the form of Eqs. (1.1.62b) and (1.1.62~)is shown. The amplitude, which is proportional to the so-called depth ofmodulation of the signal (see Fig. 17) of the information bearing component, is the vertical distance between the two envelope curves (shown dashed in Fig. 17). A decrease in the coherence functions can cause this amplitude to decrease t In addition, a further coherence function p T ,related to the spectral characteristics of the light source, can also affect the photodetector current, as discussed later in this section.
122
1.
MEASUREMENT OF VELOCITY
significantly, until there may be no information bearing component in the photodetector output. Before considering the effect on the laser Doppler velocimeter performance of the coherence functions, it is appropriate to note that the single and dual incident beam configurations have a property that allows us to avoid a lengthy and repetitive discussion of the coherence function effects. Thus, as noted earlier (Section 1.1.4.4.l), there is an inverse relation among the optical configurations, namely, any single incident beam configuration can be obtained from a dual incident beam arrangement, and vice versa, by interchanging the light source (S) and the photodetector (D). There is, likewise, a corresponding interchange among the coherence functions, so that restrictions on the source characteristics imposed by the coherence function in one case become restrictions on the photodetector in the inverse configuration, and vice versa. In view of this property of the optical configurations, the following discussion will be restricted to the dual incident beam case, since, as will be explained (Section 1.1.4.8), this configuration is the only one in use for fluid velocity measurements. However, if, for some reason, the following discussion is to be applied to the single incident beam arrangements, this can be done by replacing such words as source by detector, electric currents by light waves, and so on (see also Table XIX). Returning to the discussion of the effect of the coherence functions on the performance of the dual incident beam laser Doppler velocimeter configurations, we see that in these functions, according to the previously indicated limitation on the argument x, the radius a must be much smaller TABLEXIX. Restrictions on Particles and Source/Detector Characteristics ~~
Configuration" Item
Angle 8 in Eq. (1.1.63)
Single incident beam
Dual incident beam
Particle size effect
a
h a,, << 2 sin a
a p <<
h 2 sin a
Light field combinationb or divisionc method
P
aD<< -
a, <<
A 2 sin P
Particle location effect
Y
h as << 2 sin y
aD <<
A 2 sin y
A 2 sin P
Angles a,8, y are shown in Fig. 18. Single incident beam configurations (criterion only applies to combination of wavefront, i.e., mask arrangements). Dual incident beam configurations (criterion only applies to division of wavefront, i.e., mask arrangements). a
1.1. TRACER
(a)
METHODS
123
SOURCE
Y
FIG.18. Angles a,p, in Table XIX. (a) Dual incident beam heterodyne configuration (division of amplitude, p = 0). (b) Dual incident beam homodyne configuration (y = 0 for all locations of the flow tracing particle). (c) Single incident beam heterodyne configuration (combination of amplitude, p = 0). (d) Single incident beam homodyne configuration (y = 0 for all locations of the flow tracing particle).
than (1.220~)/(k sin 8). This can be written in terms of wavelength A as a << X0/2 sin 8,
(1.1.63)
where 8 is an angle defined by the particle size (angle a),the particle location p , and the method of light field division at the source y (see Fig. 18 for angles). Equation (1.1.63)is specialized for the various coherence functions and optical configurations in Table XIX. This shows that the realiza-
124
1.
MEASUREMENT OF VELOCITY
tion of a satisfactory photodetector output current depends on the size and location of the flow tracing particles, the size of the light source, and the method used to divide the light field. According to Table XIX, the flow tracing particles must be small enough to ensure that d, << A,,/ sin a. For a helium-neon laser (A = 623.8 nm) and a = 22.5". this requires the use of particles with a diameter smaller than 2 pm. However, in practice this requirement can probably be ignored, since Mason and BirchenoughZ3' have obtained experimental coherence function curves that have the form of a series of lobes, which are the same as in Fig. 17 except that each lobe is of approximately the same height. This suggests that the maximum particle size is not controlled by the theoretically determined coherence function, which is in any case limitedz3*to particles with a diameter much smaller than the light beam diameter in the measuring volume (say, 0.5 mm), but for the particle sizes of interest in flow tracing applications (0.1- 1000 pm), there is no upper limit on the size of particles that may be used. The second coherence function involves the source size a, and the angle /3 between the directions in which the light leaves the source. The relation of the photodetector output current to these quantities depends on the method used to divide the light from the source into the two beams that are incident in the measuring volume. This division may be carried out either by division of amplitude or division of wavefront. In the former case, the beams contain light that has left the source in the same direction and is separated by a beam splitter, so that the angle p is zero. If the beams contain light that has left the source in different directions, as would be the case with division of wavefront, then the angle /3 is different from zero. Division of amplitude is carried out by using a partially reflecting glass beam splitter, while division of wave front usually employs a mask, with two small openings, which is located between the source and the measuring volume (this is the classical experiment carried out by Thomas Young in 1807). According to the criterion developed above, for the case of division of amplitude the coherence function is unity for all values of the source size a,, since the zero value of the angle /3 ensures the criterion is satisfied. In the case of division of wave front, both the source size a, and the angle /3 affect the value of the coherence function. These conclusions have important practical consequences for the operation of a laser Doppler velocimeter.
*''J . S . Mason and A. Birchenough, in "The Engineering Uses of Coherent Optics" (E. R. Robertson, ed.), p. 541. Cambridge Univ. Press, London and New York, 1976. *" D. B. Brayton, Appl. Opt. 13, 2346 (1974).
1.1.
TRACER METHODS
I25
Since, as far as coherence considerations are concerned (see Section 1.1.4.4.5 for other limitations), there is no limit on the size of the light
source that may be used with the division of amplitude method for forming the two light beams, the use of this method will ensure that the maximum amount of light is projected into the measuring volume. With the maximum amount of incident light available, the scattered light will also be a maximum, thereby ensuring that the signal-to-noise ratio of the laser Doppler velocimeter is not limited by the strength of the scattered light. For this reason, practically all the velocimeters reported in the literature and all those produced commercially employ division of amplitude. In spite of the advantages of using division of amplitude, it does require comparatively expensive optical components and does introduce some limitations with regard to the temporal coherence of the light used in the velocimeter (see discussion below). Accordingly, from the point of view of simplicity (and cost), and considering requirements of temporal coherence, the division of wavefront method has some advantages, although its use has only been occasionally reported in the l i t e r a t ~ r e . ~ ~This ' * ~ is ~~ probably because its main disadvantage, the limitation on the amount of light available in the measuring volume, militates against its use in all but a very limited number of special situations. According to the criterion given in Table XIX, the light source radius us must be of such a size that u , << A/2 sin /3 or M u , << A/2 sin(a/2), which is the spacing between the fringes for a homodyne configuration [M is the magnification of the lens placed between the effective light source (see below) and the measuring volume]. This requires a very small source size, of the order of a few micrometers in diameter. When a laser is used, this condition is easily satisfied, because this acts as a virtual point source located within or behind the laser.* However, if a thermal light source, e.g., a mercury arc light, is employed, it is necessary to condense the image of C. Greated, Houille Blonche 6, 631 (1969).
* This characteristic of the laser is a manifestation of the very good lateral coherence properties of the emitted light. This means that the constancy of the phase difference between different points on the source wavefront is maintained very closely as the light propagates away from the wavefront. The lateral coherence of the light emitted by a source affects the ability to focus the light into a very small region of space, and the divergence of a beam of light emitted by the source. For example, typical beam divergences are for (a) a mercury arc source: 5 sr; and (b) a helium-neon laser: lO-'sr. Sources of light with a good lateral coherence, while not essential to the operation of the Doppler velocimeter, nevertheless ensure, with much less elaboration in the physical arrangements, and much less effort in adjusting the equipment to obtain satisfactory operation, that the performance is superior to that obtained with a nonlaser light source.
126
1.
MEASUREMENT OF VELOCITY
the source into a small spot. A mask is then introduced to limit the size of this virtual source so that the above discussed limitation on the source size is satisfied. It is found that this requires the aperture of the condensing lens to be considerably with a consequent severe reduction in the light that is available. In practice, larger apertures than that required by the theory would have to be used, leading to a decrease in the coherence function p , . Accordingly, a compromise would be required between a satisfactory light level and an adequate coherence function. To ensure a satisfactory output signal from the photodetector, the flow tracing particle must be so located* (angle y) relative to the photodetector that sin y << Hence, when using a helium-neon laser (A = 623.8 nm) with a receiver aperture diameter 2aDof 2 cm, y must be much smaller than 6 s of arc. With the photodetector located at a distance 9of 50 cm from the measuring volume, this would require the flow tracing particle to be located in the illuminating beam within 16 pm of the axis through the photodetector (see df in Fig. 19) if a satisfactory signal is to be obtained. The measuring volume may then be defined as a cylindrical region, the coherence volume, formed by the intersection of the cone of coherence (solid angle ODshown in Fig. 19) with the illuminating light beam. The distance df is then an indication of the measuring volume dimensionst and is directly related to the time AT required by the flow tracing particle to traverse the measuring volume. Intrinsic noise (the ambiguity noise) is associated with the passage of the flow tracing particle through the measuring volume, and this introduces a spectral spread Ao into the signal, which affects the uncertainty associated with the determination of the flow tracing particle velocity (see Section 1.1.4.4.7.1). Therefore a very small measuring volume (a small value of the angle y ) , although improving the spatial resolution of the velocity measurement, is not desirable from an uncertainty point of view. Fortunately, because of the nature of their relation, the angle y can be increased by decreasing the detector aperture diameter 2aD. This could be done by placing a pinhole immediately in front of the photodetector. For a pinhole of diameter 0.5 '40
C. A . Taylor and B. J . Thompson, J . Sci. Instrum. 34, 439 (1957). A. E. Siegman, Appl. Opr. 5, 1588 (1966).
* This result is known in communication theory as the antenna the~rem,"~ where it is usually expressed in the corresponding three-dimensional form A& << 'A (see Fig. 19). t In practice the coherence volume will only define the measuring volume if the former is smaller than the volume illuminated by the incident light beams, but usually the coherence volume is larger than the illuminated volume, which therefore defines the measuring volume (see Section 1.1.4.4.4).
127
1.1. TRACER METHODS -REFERENCE
BEAM
BEAM
I PINHOLE (DIAM=2
a
AREA DEFINES A, DETECTOR
,-
D
1
p
iJ"=.wa; :2aD I
PHOTODETECTOR
(a)
\k -%A?
ILLUMINATED
REG'oN
LENS
BEAM
;%TAD=rD2 1.
PINHOLE (DIAM =2aD ) m TO MINIMIZE STRAY LIGHT
wzzc
H
2 a D = 2 . 4 4X -f D
PHOTODETECTOR
(b) FIG.19. Detector optics for heterodyne configuration: (a) without lens nI, << A Z / 4 W ~ 2df- Az; and (b) with lens n, << A2/D2,2 4 - Az, In principle it is only necessary to focus the scattered light; however, the signal-to-noise ratio of the photodetector output is improved if the reference beam is also focused (see Section 1.1.4.4.6).
mm, the angle y is 8' of arc; so, for example, if the measuring volume is located 8.5 cm from the photodetector, the distance df (Fig. 19) is smaller than 100 p m . The suitability of this magnitude from the point of view of spatial resolution and the ambiguity noise would have to be considered further (see Sections 1.1.4.4.4 and 1.1.4.4.7.1). Another approach to decreasing the size of the effective area on the photodetector is to place a
128
1.
MEASUREMENT OF VELOCITY
lens in front of the photodetector; the scattered light collected by the lens is then concentrated into an area approximately equal to the size of the Airy disk. As an example, for a lens of 250 mm focal length and 67 mm diameter, the diameter 2aDof the Airy disk (A = 623.8 nm) is 2.84 pm, which gives df << 113 pm (see Fig. 19). This approach will project more of the scattered light onto the photodetector than will be the case with the pinhole. The allowable value of the angle y can be given a physical interpretation. The output current from the photodetector is made up of contributions from each element of the photocathode [hence the integration in Eq. (1.1.61) over the photocathode]. The currents from each element differ in phase because of the different lengths of the paths (from all elements of the flow tracing particles) followed by the light incident on the photocathode (an interference pattern is formed on the photocathode). The amplitude (depth of modulation) of the information bearing component of the photodetector current is a maximum when all these contributing currents are exactly in phase, and decreases as the phase difference between the elemental currents increases. Therefore the maximum permissible value of the angle y is a measure of the maximum permissible phase difference between the elemental currents originating at different points of the photocathode. The preceding discussion concerning the particle location effect only applies to the Doppler heterodyne current. The angle y is always zero in the homodyne contribution since this involves the superposition of two scattered light fields that have their origin at the same point on the flow tracing particle. This means that, as far as coherence requirements are concerned, the receiving aperture in a dual incident beam homodyne velocimeter is unlimited in size. It can therefore be made large enough to ensure that a satisfactory signal-to-noise ratio is attained. However, increasing the receiving aperture increases the uncertainty of the velocity determination (see Section 1.1.4.4.5)and the signal-to-noise ratio (due to effects of stray light; see Section 1.1.4.4.6);so there will be an optimum aperture size which is probably best determined empirically. On the other hand, when there is more than one flow tracing particle in the measuring volume, coherence considerations do affect the performance of velocimeters with the homodyne configurations (see Section 1.1.4.4.6). The treatment of the source, detector, and particle size effects, and the detector temporal characteristics given above involved only a single frequency component of the light. The light emitted by any practical source is not limited to a single, pure frequency but is spread over a range of frequencies. This is true even for the laser, although the width of a single line of the light spectrum of a laser is among the smallest encountered,
1.1.
129
TRACER METHODS
which has important practical consequences in relation to the Doppler velocimeter. * Integrating Eq. (1.1.62) over the wavenumber range k = 0 to k = CQ (subscript 0 is suppressed to simplify the notation), Z% (E&;”)
=
(PIPI)”2W(t)ppCLspDpT cos[kdet -
el)
Utl, (1.1.64)
where pT is a chromatic coherence function [for fundamental reasons (see below) it is better called a temporal coherence function] given by
and Pu is the flux incident at the photodetector given by Pu(k) = (Pl(k)Pj(k)).1’2Note that in deriving Eq. (1.1,64),use was made of the observation that the dependence of p p , p,, and pDon the wave number is so weak that it may be ignored,233and a similar assumption has been made for W(t) . If the light is propagated through a nondispersive medium, then 8 is independent of wavenumber k, and we can write
where k is the spectral mean wave number of the light. Thus it is seen that the temporal coherence function pTdepends on the Fourier transform of the energy distribution Pu of the light incident on the photodetector. The coherence function p T can be used to make an estimate of the importance of the effect of the spectral characteristics of the light on the performance of the velocimeter. To do this, it is necessary to assume that the spectrum is unaffected by the light scattering process. This is probably satisfactory for the homodyne case, where the signal strength is related to the visibility of the fringes formed in the measuring volume by light that is transmitted directly from the light source. However, such an assumption is probably not appropriate for the heterodyne contribution, where the scattered light has experienced a frequency shift relative to the * It must be emphasized that light with a very narrow spectral width is not essential to the operation of the Doppler velocimeter (see Section I . 1.4.4.5), although, as will be shown, it is a highly desirable feature (see below). The characteristic that makes the laser attractive for the Doppler velocimeter is the spatial energy density (W/pmZ)of the emitted light. As an example, for comparable situations, the energies incident in the illuminated measuring volume of the Doppler velocimeter using a 100 W (546. I nm) mercury arc source and a 5 mW helium-neon laser are (W/pm*) 8 X lo-@and 2 x lo-’, respectively.
1.
130
MEASUREMENT OF VELOCITY
incident light, but since only an estimate of the source spectral effect is being attempted this will be ignored. Assuming the light has a spectral profile of Gaussian form
Pu = (l/Ak) exp[-.rr(k - R)/AkI2,
(1.1.67)
- ~(AhkS)~l,
(1.1.68)
we have p T = C O S ( ~ )exp[
where Ak is the half-width at the half-maximum of the spectral line profile. Then, in view of Eqs. (1.1.64) and (1.1.68), Eqs. ( I . 1.62b) and ( I . 1 . 6 2 ~ ) become
+ Pi + 2 / . h p p ~ p D p T ( P &cos(wDt)] ))~’~ i = a W ( t ) [ B , + P& + 2pppspT(Pi1P~2)1’2 cos(wDt)], i
=
aW(t)[Po
(1.1.69a) (1.1.69b)
with Eqs. (1.1.69a) and (1.1.69b) applying to heterodyne and homodyne configurations, respectively. The coherence function pTplays the same role as the coherence functions p,, pp,and pD,namely, unless it is sufficiently large, the amplitude of the information bearing component of the photodetector current will be too small to provide an adequate signal. Assuming that a suitable signal is obtained when the exponential function in Eq. (1.1.68) is greater than 0.9, it can be shown that the velocimeter must satisfy the condition S < 0.18/Ak or 8 < 0.18/Aw [where the approximation exp( - x) = 1 - x has been used]. According to this criterion, the allowable optical path length difference 8 between the paths traversed by the light in the Doppler velocimeter is inversely proportional to the spectral line width Aw of the light emitted by the source. The inverse spectral line width has the dimensions of time and is known as the coherence time; hence the origin of the terminology temporal coherence for pT. The coherence time represents the time over which the relative phases between two points on a wave remain unchanged. However, for the purposes of this discussion, the allowable optical path difference 8 is better related to the coherence length I = c/Aw of the emitted light, the coherence length being a measure of the distance along a wave over which the relative phases remain constant. Alternatively -and this is the origin of the terminology temporal coherence for pT-the coherence length is related to a coherence time l/Am, representing the time over which the relative phases between two points on a wave remain unchanged. Typical estimated coherence lengths for light emitted by three sources that could be used in a Doppler velocimeter are shown in Table XX.
1.1.
TRACER METHODS
131
The importance to the Doppler velocimeter of the spectral characteristics of the light depends on the optical arrangement used to produce the two incident beams in the measuring volume. In the case of division of amplitude, the typical path length differences introduced by the optical components could be as large as several millimeters, but with division of wave front the path length differences can be limited to a few micrometers for points close to the axis of symmetry that passes through the source, the mask holes, and the point of observation. Accordingly, as can be seen from Table XX, to obtain a satisfactory signal with optical configurations that use division of amplitude, it is essential to use a laser. However, for division of wavefront, a nonlaser source of narrow band light, e.g., a high pressure mercury arc light, and, possibly, even a white light source, can be used; but, because of the limited amount of light available from a nonlaser source (see the earlier discussion on the need to spatially filter nonlaser light sources and the consequent effect on the available light), such sources have not been used very extensively.* As stated earlier, the preceding discussion of coherence effects in the laser Doppler velocimeter only applies to the two incident beam configurations. For the single incident beam arrangements, the limitations on the source and the detector are the inverse of those that apply to the two incident beam systems. Thus, for the heterodyne arrangement, the coherence volume is defined by the source aperture, which is consequently limited by the antenna theorem. The detector aperture is limited if combination of wavefront is used to direct the two scattered light beams onto the photodetector, and unlimited if combination of amplitude is used. In the homodyne systems, the detector aperture size also depends on the method of beam combination, but there is no restriction on the source size, so the maximum available light can be directed into the measuring volume. 1 . 1 . 4 . 4 . 4 . SPATIAL RESOLUTION.It is usual to assume that the measuring volume in a laser Doppler velocimeter is the illuminated volume formed at the intersection of the two incident beams. However, because 242 243
244
E. A . Ballik and J . H. C. Chan, A p p l . Opi. 12, 2607 (1973). A . Lohmann, Opr. Acta 9, 1 (1962). C. P. Wang, A p p l . Opt. 13, I193 (1974).
* Doppler velocimeters using nonlaser light sources have been demonstrated by S c h w a P and Ballik and Chan.2'z The most serious limitation is the amount of light that can be projected into the measuring volume, and Ballik and Chan have shown that a Ronchi grating has much better characteristics in this regard than the conventional double aperture arrangement. The theory of using Ronchi gratings for division of wavefront has been considered by L ~ h m a n n ~and ' ~ Wang.*"
1.
132
MEASUREMENT OF VELOCITY
TABLEXX. Typical, Estimated Coherence Lengths for the Light Emitted by Sources to Be Used with the Doppler Velocimeter Mean wavelength of light Source
Ad&
Filtered mercury high pressure arcb Helium-neon laser' White light'
5460
6328
5000
Estimated bandwidth" of light AA(A) 65
0.012 2000
Coherence length (0
Acceptable path difference (0. 180
20 p m
4 Pm
18 cm
6 cmd
1 Pm
0.2 pm
This is the half-width at the half-maximum of the spectral line profile. W. Elenbaas, Physica (The Hague) 3, 859 (1936). T. S. Durrani and C. A. Greated, "Lasers Systems in Flow Measurement." Plenum, New York, 1977. In many applications it is necessary to insert an optical flat in one of the paths in order to ensure that the path difference is within acceptable limits. H. H. Hopkins, in "Advanced Optical Techniques" (A.C.S. van Heel, ed.), p. 236. North-Holland Publ., Amsterdam, 1967. a
of the observed temporal variation of the signal amplitude associated with a single flow tracing particle (see Fig. 17), a more logical definition of the measuring volume would be to define it in terms of the spatial location of the particle as it affects the signal amplitude. In this case, in addition to the spatial distribution of light, the dimensions of the measuring volume are influenced by: (a) the light scattering characteristics of the flow tracing particle, (b) the requirement that a minimum number of signal cycles must be available to the signal processor in order to ensure that the determination of the Doppler frequency is sufficiently accurate, and (c) the heterodyne efficiencies (pP, ps, pD). One further complication is that the measuring volume dimensions may also depend on the aperture and focal length of the receiving optics.* R. M. Huffaker, Appl. Opt. 9, 1026 (1970).
* This would be the case when stops and apertures are introduced to improve the heterodyne efficiency in the heterodyne configuration or to control stray light in the homodyne configurations. H u f f a k e P has used such a basis [but not including items (a), (b), and (c) of the preceding list] for estimating the volume and dimensions of the measuring volume. This reference also includes useful graphs for quickly estimating the measuring volume dimensions.
1.1.
TRACER METHODS
133
A rather complex theory would be required to take all the preceding factors into account in the definition of the measuring volume dimensions. The effort involved is probably not worthwhile so it is usual to determine the measuring volume solely from the distribution of light in the incident beams. The definition of the measuring volume using only the illuminated volume concept is complicated by the nature of the spatial distribution of light in the incident beams. This depends on the method used to divide the light field (see Section 1.1.4.4.3), and Durrani and GreatedZz0deal with the various arrangements commonly used with laser Doppler velocimeters. The measuring volume is probably most conveniently defined in terms of its dimensions A x , A y , and Az along appropriate coordinate directions having their origin at the center point of the measuring volume(see Table XXI). Using this approach, the measuring volume dimensions for the systems considered by Durrani and Greated220are listed in Table XXI. From Table XXI it can be seen that decreasing the angle a will increase Az, so that the measuring volume becomes elongated in the z direction. This can have an undesirable effect on the measurement of the velocity of the flow tracing particles. If, for example, the flow is in the x direction and there is a variation in the z direction of the magnitude of this velocity, then a measuring volume that is elongated in the z direction will introduce an ambiguity into the velocity measurement. In addition to the z direction elongation, the rectangular aperture mask (division of wavefront) has the property that Ax/Ay = c / a , so that using narrow slits ( c / a >> 1) results in a measuring volume that is elongated in the x direction.* This is undesirable for flows in which the flow tracing particles have two or three components of velocity. The accuracy of the determination of the x component would then depend on the angle of the path of the particle relative to the x direction. In this case it is better to use a square aperture ( a = c) or a circular aperture, since this leads to a measuring volume in which A x = A y . 1.1.4.4.5. SPECTRAL BROADENING. To obtain useful measurements from the laser Doppler velocimeter, the following conditions must be met: (a) the optical portions of the instrument must be designed and adjusted in accordance with the principles discussed in Section 1.1.4.4.3, (b) the resolution o r precision of the frequency measurements must be appropriate for their intended application, and
* In these circumstances it would probably be acceptable to treat the measuring volume as an elongated cylinder of radius Ax = Ay and length Az (see HuffakeP'?
TABLEXXI. Method of forming light beams Division of amplitude (unapertured laser source)
Measuring Volume Dimensions
Definition
Definition of optical parameters ( y direction normal t o plane of paper)
of measuring volume" -I=
e-l
4
Ax
AY
2 b0 cos(a/2)
2b0 = 1.27-f A 2b
Az
260 sin(cr/2)
A V 4T -Ax 3
-~
2+?pJF:ou'
1.6 ({)'A
=
Radius b, of laser light beam at focus =
Ay Az
2b
X
b Division of wavefront (rectangular aperture)
Division of wavefront (circular aperture)
I = O(first lo zero of sinc function)
-I =
C
w
A x A y Az
ab
2Af a(&)
=-
1.22
If ra
Af 1.22 ra
2.44
AP bra
zero of Ji
a
o [first
10
2Af
2Af -
W/xI
I is the luminous intensity at point x, y , z ; l o ,luminous intensity at point (O,O,O). A V values for rectangular and circular apertures are only approximate.
4T -Ax 3
by Az
c
f
4
is dimension of aperture in y direction.
0.64 f-.A b At b , I/Io =
e-1.
1.1.
135
TRACER METHODS
(c) the effects of extraneous variables (noise) must be small compared to the magnitude of the information bearing signal. This section is concerned with item (b), except that the discussion of resolution will be limited to the optical portions of the instrument. The effect on the resolution of the spectroscopic methods used to determine the Doppler frequency shift has already been considered (see Section 1.1.4.3). Noise effects will be considered in Section 1.1.4.4.6. The observed frequency in the heterodyne and homodyne cases are given, respectively, by (see Table XVII):
wD = ko(e8 - eo) * U
(1.1.70a)
wD = ko(eoz - eol) U .
(1.1.70b)
and In terms of the angles shown in Fig. 20, Eqs. (1.1.70) can be written (putting ko = 2.rr/X0)
S $2
FIG.20. Angles for dual incident beam velocimeter: (a) homodyne configuration, and (b) heterodyne configuration.
I36
1. OD
MEASUREMENT OF VELOCITY =
(~T/A~)U,(COS+ cos $)
(1
.1 .71a)
and
where it is assumed that the flow tracing particle is moving parallel to the so U = ( V ,, 0, 0). From an inspection of Eqs. (1.1.71), it is clear that the resolution or precision with which uDcan be determined depends on the uncertainty in: (a) the wavelength A. of the incident light, (b) the particle velocity U 1 ,and (c) angles cp, $, and a. Items (b) and (c) can be further broken down. The particle velocity uncertainty can be due to the dynamic characteristics of the flow tracing particles (see Section 1.1.2.2) and to variations in the particle velocity as it passes through the finite size measuring volume. The angles cp, $, and a have uncertainties that are associated with the establishment of the magnitudes of these angles in the optical train of the receiving and transmitting optics. A further uncertainty in the magnitudes of the angles is introduced because a working velocimeter must have finite apertures for the light source and for the receiver. In consequence, light is incident in and is scattered from the measuring volume in a range of angles 6cp, 6$, and 6a and, as a result, a corresponding range of velocities is sensed by the velocimeter. With the exception of the uncertainty in the measurement of the angles a, cp, and JI, the uncertainties which have been identified above manifest themselves as an apparent increase 6wDin the width of the spectrum of the photodetector output signal, i.e., whereas in theory according to Eqs. (1.1.70) and (1.1.71),a single frequency wD is associated with a single flow tracing particle, in practice a range 6 0 of ~ frequencies distributed about wDis observed. It is usual to refer to these uncertainty effects as spectral
x axis,
broadening * (ambiguity noise).
The uncertainty in the measurement of angles a,cp, and $ can only be observed by replicating velocity measurements and resetting the optical train between each measurement replication. It can, of course, be estimated by the usual techniques that are used for random contributions to measurement uncertainty. Compared to the other contributions to the measurement uncertainty that have been mentioned, this is small and can be ignored. In order to estimate the relative magnitudes of the various contributions to the observed spectral broadening (ambiguity noise), it will be assumed
* There are other sources of spectral broadening; see the discussion of ambiguity noise in Section 1.1.4.4.7.1.
1.1. TRACER METHODS
I37
that each contributes additively to the overall broadening, i.e., Eqs. (1.1.71),
( I . 1.72) The first term is the contribution of the finite width Sho of the spectrum of the light emitted by the source to the spectral broadening. The second term represents only the effects of the finite size of the measuring volume, and the effects of particle dynamics are considered in Section 1.1.2.2. The final term in Eq. (1.1.72) is the uncertainty arising from the finite apertures, which has one or two terms, depending, respectively, on whether the homodyne (0 = a) or heterodyne (0, = cp, O2 = 4) cases are being considered. Since it is conventional to express the spectral width aho of the light emitted by a source in terms of the half-width of the spectrum at half the maximum value of the spectrum, the same convention will be adopted in estimating the second and the final terms in Eq. (1.1.72), i.e., the calculated values of the contributions to GoD/wD will be halved before they are compared with the contribution due to light source effects. The determination of the various contributions to the spectral broadening (ambiguity noise) will be considered in greater detail in the succeeding paragraphs. Light source efects. The contribution to the spectral broadening due to the finite width aho of the spectrum of the light emitted from the source is ~w,/oD
=
GXo/ho.
(1.1.73)
Using the data of Table XX, 6Xo/Xo is for a mercury arc 1.2%, for a helium-neon laser 0.0002%, and for a white light source 40%. Spatiul resolution. The contribution to the spectral broadening due to the finite magnitude of the measuring volume can be written as GOD/WD
= ( I / U , ) ( 6 U J 6 x )AX.
( 1.1.74)
A proper estimate would require the appropriate velocity distribution U,(x) to be used. For the purposes of making a general comparison of the various contributions to the spectral broadening of the photodetector output signal, it is convenient to assume U1 = U,[l - (XIR)~],which is the parabolic velocity distribution for laminar flow in a duct of circular cross section, with U , the center line velocity. With this assumption, Eq. (1.1.74) becomes ~wD/wD
= (UC/UJ(2x/R) Ax/R.
(1.1.75)
138
1.
MEASUREMENT O F VELOCITY
Where U , is small, i.e., adjacent to the duct walls, the effect of the spatial variation in velocity on the spectral broadening can be very large. At points distant from the duct wall, U , / U , and 2x/R are of order one and 6 0 D / w D depends on A r / R . To determine Ar, the dimensions of the measuring volume must be estimated. The usual practice in the literature is to assume that the illuminated volume where the two incident beams intersect is the measuring volume. This is an appropriate estimate for a homodyne configuration. However, for a heterodyne configuration, this would only be acceptable if the coherence volume (see Section 1.1.4.4.3) is larger than the illuminated volume. If the coherence volume defines the measuring volume, then the methods of Section 1.1.4.4.3 should be used to estimate the magnitude of the measuring volume. For the purposes of the present discussion, take Ax as 0.2 mm, which appears to be typical of many laser Doppler velocimeter applications. Then assuming R is 10 mm, 8oD/oDis 2%, or for the spectral width at the half-maximum, SOD/OD is about 1%. This is in agreement with the measurements of Morton,24s who has reported values of the spectral broadening due to the measuring volume size of 0.08%. Aperture size effects. From Eqs. (1.1.71), we have for the heterodyne configuration 8wD/oD
=
[sin
$/(COS
cp
+ cos $)] 6p
+ [sin $/(cos cp + cos $)I a$,
(1.1.76a)
and for the homodyne system 8wD/oD =
4 cot(a/2)
8a.
(1.1.76b)
In these equations, when applied to the double incident beam arrangement,* the quantities S$ and 8a are the angles subtended by the transmitting aperture at the measuring volume, while 6cp is the angle subtended by the receiving aperture at the measuring volume (see Fig. 20). Hence spectral broadening in the heterodyne case depends both on the size of the transmitting aperture and on the size of the receiving aperture, while with homodyne detection only the size of the transmitting aperture is significant. This has been demonstrated experimentdly by Mazumder and Wank~m.~~' To make a general estimate of the magnitude of the aperture size con*M
J. B. Morton, J . Phys. E 6, 346 (1973).
* For single incident beam arrangement, SJI and 6a depend on the receiving aperture; and &$, on the transmitting aperture.
1.1.
TRACER METHODS
I39
tribution to the spectral broadening, consider the following parameters of a homodyne system: (Y = 5" and 6a = 0.032 rad (250 mm focal length lens with an 8 mm diameter light beam). From Eq. (1.1.76b), we have 6 o D / o D = 18% or, for the width at the half-maximum, 6wD/oD = 9%. For a heterodyne system, SJi = 0.032 rad, Ji = W",6cp = 0.019 rad, and cp = 85". From Eq. (1.1.76a) we have 6oD/oD = 56% or, for the width at the half-maximum, ~ o D / o D = 28%. These estimates are in reasonable agreement with the measurements of spectral broadening reported by Mazumder and W a n k ~ m . ~ ~ ~ Comparison. The preceding demonstrates that the aperture size contribution to spectral broadening is by far the most important (say, 10-30%). This is followed by effects due to the finite size of the measuring volume, when there is a spatial variation in the fluid velocity (about 1%). The contribution due to the spectral width of the source is negligible when using a laser (0.002%). In addition to these sources of broadening, HansonZ4'has shown that curvature of the wavefronts can cause spectral broadening that is significant compared to that from the sources considered above, say, 5%. There is, in addition, a contribution due to the presence of more than one flow tracing particle in the measuring volume (see Section 1.1.4.4.7.1). There are two approaches to decreasing aperture broadening, viz., increasing the angles a, or cp and Ji, and/or decreasing the appropriate apertures. The practicality of the first method will depend on: (a) the nature of the flow system and its ability to accommodate light beams from various directions, (b) the need to provide a signal of sufficient duration to allow the velocity measurement to be made with the desired accuracy (see Section 1.1.4.4.7.6). (c) the effect of spectral broadening due to other velocity components248;and (d) light scattering properties of flow tracing particles. To decrease the aperture broadening by decreasing the aperture sizes requires the transmitting aperture to be decreased in the homodyne case. In the heterodyne configuration, both the transmitting and receiving apertures must be decreased. However, making the transmitting aperture smaller will decrease the signal strength, except that it may be compensated, at least in part, by increasing the power of the light source. Controlling the aperture broadening in the heterodyne case by decreasing the receiving aperture may not be feasible, because the receiving aperture size is limited by the need to meet the necessary coherence conditions (see Section 1.1.4.4.3). This requires a small aperture (0.5 mm or
*"
S. Hanson, J . Phys. D 6, 164 (1973). D. A. Jackson and D. M. Paul, J . Phys. E 4, 173 (1971)
I40
1.
MEASUREMENT OF VELOCITY
smaller), and this makes it difficult to align the light beams on the aperture and to maintain that alignment in the presence of vibration. A further decrease in the receiving aperture would further increase these operational difficulties. From a consideration of available information on turbulence, the spatial resolution Ax = Ay = Az required for the study of turbulence is 10-50 p m in liquids, and in gases 10-400 pm. However, if the index refraction of the observation port is to be matched in a liquid system (see Section 1.1.3.5),then a spatial resolution of about 60 pm may be the minimum attainable. 1.1.4.4.6.SIGNAL-TO-NOISE RATIO. Because the signal power incident on the photodetector of a laser Doppler velocimeter is very small (e.g., 10+ W),the magnitude of the noise in the velocimeter could be about the same as or greater than the signal, thereby making it difficult or even impossible to obtain useful information from the velocimeter. However, this situation can be alleviated to a greater or lesser extent by the choice of the velocimeter optical configuration and/or signal conditioning and signal processing techniques. This section will consider in quantitative terms the relative importance of the signal and of the noise. To do this, the conventional method of communication theory will be followed, and the ratio (SNR) of the power in the signal to the power in the noise will be evaluated. If the estimated signal-to-noise ratio of a laser Doppler velocimeter is substantially greater than unity, then no modification of the optical configuration will be required, nor will it be necessary to use special signal conditioning and signal processing methods. On the other hand, signal-to-noise ratios of unity or less will require special attention to be given to the photodetector output signal if the ratio cannot be improved by modifying the optical arrangements. The smaller the signal-to-noise ratio is relative to unity, the more sophisticated the signal handling methods must be. The signal-to-noise ratio will be discussed in this section relative to the type of photodetector (photomultiplier tube or photodiode) used, to the optical configuration, and to the number density of the particles in the measuring volume. The relation of the signal-to-noise ratio to signal conditioning and signal processing will be discussed in Section 1.1.4.4.7. There are three sources of noise in a laser Doppler velocimeter: (a) optical, (b) photodetection, and (c) electronic. This section will be concerned with the optical and photodetection sources of noise. Electronic noise, which has its source in the processing and analysis of the photodetector output, will be considered later (see Section 1.1.4.4.7). Of the optical and photodetection sources, the latter is by far the most important
1.1.
TRACER METHODS
141
because by careful design and adjustment of the velocimeter, it is usually possible to make the optical noise of negligible importance.* However, optical noise can arise from spatial and temporal variations in the refractive index of the medium along the velocimeter light paths, and from light scattered by the optical components of the velocimeter. These sources of optical-noise are not under the experimenter's control. Refractive index effects arise when velocity measurements are made in turbulent flows. They are a consequence of variations with location and time of the local density, temperature, or chemical composition of the fluid. Buchhave et al.3ahave shown that these effects can be very significant in liquids. They are of less importance in gases, and this is confirmed by observations, which appear to indicate that over the path lengths usually encountered in laboratory flames, refractive index variations do not affect the velocimeter signal. On the other hand, in full size furnaces, according to the reports of Baker et al., 2s1 refractive index variations can have a very significant effect. Observed effects of variations in the refractive index on propagating light beams are the following: (a) spreading of the averaged focused spot size relative to its size in a vacuum, (b) wandering and bending of the light beam, and (c) amplitude and phase fluctuations in the light waves. The frequency of the phenomena listed under (a) and (b), which have been investigated, among others, by Chiba,2s2Dowling and L i v i n g s t ~ n , ' ~ ~ and Buchhave et uf.,3ais low in comparison with the signal frequency, so the determination of the spectrum of the signal is not affected. However, the intermittency of the signal may increase as the light beam incident on the photodetector moves over its surface. In addition, variations in the relative positions of the light beams defining the measuring volume will suggest that cause the dimensions of the latter to vary. Buchhave et D. A. Jackson and D. M. Paul, J . Phys. E 2, 1077 (1969). B. M. Watrasiewicz, J . Phys. E 3, 823 (1970). 251 R . J. Baker, P. Hutchinson, and J . H. Whitelaw, J . Hear Transfer 96, 410 (1974). 252 T. Chiba. Appl. Opt. 10, 2456 (1971). 253 J . A . Dowling and P. M . Livingston, J . Opt. Sor. A m . 63, 846 (1973). 240
* Laser noise, which is particularly serious with argon ion lasers, can usually be decreased to a negligible level by either an appropriate choice of operating conditions,24eor by the use of the noise canceling scheme demonstrated by W a t r a s i e w i c ~ . ~ ~ ~
1.
142
MEASUREMENT O F VELOCITY
one method of correcting for beam wandering is to estimate the frequency spectrum of this phenomenon by heterodyning light scattered from refractive index homogeneities in the flow field with a reference beam that passes through the fluid. The effects of propagation through a turbulent flow field on the coherence of the light incident on the photodetector has been considered by G a ~ - d n e rFried,255,zs6 ,~~~ and Wall.z57 These authors show that degradation of the signal due to the decrease in the heterodyne efficiency arising from turbulence effects depends on the wavelength of the light, the length of the path over which the light is propagating, the aperture of the photodetector, and the intensity of the turbulence. For atmospheric turbulence, Friedzs5showed that if the diameter D of the photodetector aperture is much less than a characteristic dimension r o , which depends on the scale and intensity of the turbulence, the fluctuations in the index of refraction along the light beam are unimportant, and the signal-to-noise ratio can be improved by increasing the photodetector aperture. On the other hand, for D >> ro the effects of turbulence are dominant, so that no improvement in the signal-to-noise ratio results from increasing the receiver aperture. Since these calculations have been carried out for atmospheric turbulence, it is probably not advisable to apply them to the flow in furnaces and other technically important fluid systems. There is clearly a need to extend the calculations of Fried to these cases. The character of the noise that arises in the photodetection process depends on the type of photodetector used in the velocimeter. A number of photodetectors are available,216but the two that have found the widest use in laser fluid velocity measurements are the photomultiplier tube and the photodiode. The sensitivity of the latter is much greater than that of the photomultiplier tube; however, there is more internal noise associated with its operation, and in consequence in most cases, as will be shown, the photomultiplier tube has significantly higher SNR than the photodiode when used in a laser velocimeter. There are three sources of internal noise in a photodetector: (a) Johnson noise due to the thermal fluctuations of the electrons in the photodetector and associated electrical circuits, (b) dark current noise (this is usually much smaller than any other noise source and may be ignored), and (c) photon fluctuation or shot noise associated with the signal and zM S. 255
05'
Gardner, IEEE I n t . Conv. R e c . Part 6, p. 337 (1964). D. L. Fried, Proc. IEEE 55, 57 (1967). D. L. Fried, IEEE J . Quantum Electron. qe-3, 213 (1967). L. S. Wall, J . Opt. Soc. A m . 64, 1005 (1974).
1.1.
I43
TRACER METHODS
with any background radiation (external noise) that is also incident on the sensitive surface of the photodetector.* The Johnson noise power P , is given by Nyquist’s theorem260:
P J = kg7 Af,
(1.1.77)
where kB is the Boltzmann constant; r , the effective temperature of the photodetector; and Af,the bandwidth of the signal processing electronics, which is essentially the signal bandwidth. The shot noise power PSHgenerated in the resistive elements R of the photodetector is given by Schottky’s theoremZ6O PSH= 2qQkR Af
[lim T T-rm
-T/Z
i dr] ,
(1.1.78)
where G is the gain (if any) of the photodetector, and k is a factor which accounts for the increase in noise induced by the internal current gain process.261 In Eq. (1.1.78) the current i is, for a heterodyne configuration, from Eq. (1.1.69a), i = UW(t)[Po
+ P i + PB + ~ / L , / L ~ ~ / L T PCOS(W,~)], , ~ P ~ ” * (1.1.79)
where P B is background radiation power. t From Eqs. (1.1.77)-( 1.1.79), the noise power N of the photodetector is N = [ 2 q ~ f f K , k ( P+ , Pi
+ PB)R + kgT] Af,
(1.1.80)
258 S. A. Self, in “Combustion Measurement” (R. Goulard, ed.), p. 103. Academic Press, New York, 1976. 259 H. Bossel, W. J. Hiller, and G. E. A . Meier, J . Phys. E 5, 893 (1972). A. van der Ziel, “Noise.” Prentice-Hall, Englewood Cliffs, New Jersey, 1954. 281 H. Melchior, M. B. Fisher, and F. R. Abrams, Proc. IEEE 58, 1466 (1970). 262 L. Lading, Opto-electronics 4, 385 (1972).
* S e l P has used the coincidence between the pulses generated by two photomultipliers, both of which receive the signal, to discriminate against photon fluctuation and shot noise. A similar approach has also been demonstrated by Bossel et o/.p38 t The background radiation power Y s would be scattered light from the light source thermal radiation originating from objects in the field of view of the photodetector, and light scattered by flow tracing particles outside the measuring volume. The thermal radiation is usually concentrated in the infrared region of the spectrum, because most objects in the laboratory environment will be at the ambient temperature. Scattered laser light can be controlled by appropriate screens, baffles, and a pinhole in front of the photodetector (see Fig. 19). Ladingze2shows that Pe can be minimized by making the receiver aperture [lens diameter D in Fig. 19(b)] as large as possible and the pinhole [diameter 2aDin Fig. 19(b)] equal to the image of the measuring volume. In these circumstances, it is reasonable to assume that the background radiation PB may be ignored.
144
1.
MEASUREMENT OF VELOCITY
where K, comes from including the light distribution function W(t)in the integration of Eq. (1.1.78). The signal power S iszs0 S =R
[lim T lT'*it d t ] , T-m
-T/Z
(1.1.81)
where signal current is is given by the information bearing portion of the current i is = 2 a p p p ~ C L p T W ( t ) G ( P o Pcos(o,t). I)1i2
(1.1.82)
From Eqs. (1.1.81) and (1.1.82), S = 2aZpCL'pp~p~&K~GzP$~,
(1.1.83)
where KL comes from including the light distribution function W ( t ) in the integration of Eq. (1. I .81). The signal-to-noise ratio for a general photodetector with internal gain G , acting as an optical mixing detector, is obtained from Eqs. (1.1.80) and (1.1.83) (ignoring the background radiation PB):
S I N = 2',p~~floo/[2qaK,(P0+ Pi)k
+ (kB~/Rc2)]Af.
(1.1.84)
When the photodetector is a photomultiplier tube, the gain G is such ( lo6) that the Johnson noise term [the second term in the denominator of
Eq. (1.1.84)] is much smaller than the shot noise term [the first term in the denominator of Eq. (1.1.84)], so the signal-to-noise ratio becomes (replacing P8 by F ,the luminous power of the scattered light)
S I N = apZ$c$PoP"/qKw(Po+ F ) k Af
(1.1.85a)
or, using the quantum efficiency 7 = ahf/q of the photodetector,
S I N = r/&&P'/hfk
Af(1
+ P'/Po),
(1.1.85b)
where K, = K&/Kw. It can be seen that the photomultiplier tube gain does not enter explicitly into the calculation of the signal-to-noise ratio; it need only be large enough to make the Johnson noise negligible compared to the shot noise in the tube (sometimes called the shot noise dominated condition). The signal-to-noise ratio in the heterodyne configuration can be maximized by increasing the reference beam power Porelative to the power Ps in the signal (see Section 1.1.4.4.1), and Eq. (1.1.85b) becomes (1.1.86)
1.1.
I45
TRACER METHODS
In practice, the ratio P o / F is increased by increasing the reference beam power P o , but little is gained by doing this indefinitely because eventually the saturation limit of the photocathode is reached. Furthermore, laser noise may become significant at high laser power output. The usual practice is to make the reference beam power about five or ten times the signal power. By a calculation similar to that used to develop Eqs. (1.1.85) for the heterodyne configuration, it can be shown that the corresponding result for the homodyne system is S I N = ( q ~ ~ / 2 h&)Pi. fk
(1.1.87)
It can be shown that K~ (and hence the signal-to-noise ratio) reaches a maximum when the two incident beams, each of equal intensity, are aligned so that the normal to the photodetector bisects the angle between the incident beam. A cursory comparison of Eqs. (1.1.86) and (1.1.87) suggests that for a given bandwidth Af and signal power F ,the signal-to-noise ratio should be higher in the heterodyne configuration operating with the maximum reference beam power. However, in practice this is not found to be the case because the signal power Pi in the homodyne system can be increased indefinitely (up to the saturation point of the photodetector, although this is not likely to be a practical limit in view of the very low power of the light scattered by the flow tracing particles) by increasing the receiving aperture. In the heterodyne system, the maximum receiving aperture, and hence the maximum signal power (effectively p2Dp”),is limited by the need to satisfy the coherence conditions (see Section 1.1.4.4.3). As long as the receiving aperture in the heterodyne configuration is smaller than that required by coherence conditions (& = l), the signal p p and, consequently, the signal-to-noise ratio increase in proportion to the receiving aperture. For heterodyne systems with receiving apertures larger (pt << 1) than that dictated by coherence conditions, the light gathered by the additional aperture area makes a negligible contribution to the signal bearing portion of the photodetector output, and hence the signal-to-noise ratio is independent of the aperture size. Thus, for large receiving apertures the signal-to-noise ratio of the heterodyne configuration approaches an asymptotic value SIN
=
(qKw/hfk
Af)(/4PS)max
9
(1.1.88)
where (pp),,, depends on F and on the coherence characteristics of the optical system. Thus, as DrainZg3shows, with a single particle in the L. E. Drain, J . Phys. D 5, 481 (1972).
146
1. MEASUREMENT
OF VELOCITY
measuring volume, the signal-to-noise ratio of the homodyne system is usually superior to that of the heterodyne system. It is therefore generally accepted that the homodyne configuration is the appropriate choice for low concentrations of flow tracing particles. The use of a photodiode instead of a photomultiplier tube as the photodetector of a laser Doppler velocimeter is attractive because of its much lower cost compared to the latter. Photodiodes also require less complicated circuitry and are more robust than photomultiplier tubes. However, a photodiode has unity gain (G = l) and, as will be shown, this has a significant effect on its application to the laser Doppler velocimeter. The signal-to-noise ratio characteristics of the photodiode can be deduced from Eq. (1.1.84) by setting the gain G, and hence k , equal to one. As a result, and because at the signal frequencies (200 MHz) encountered in laser Doppler velocimeters, the resistance of the diode, which is frequency dependent, is extremely small [about 1 ohm (Melchior at ~ 1 . ~ ~ l the Johnson noise is the dominant noise constituent, and the signal-tonoise ratio is, from Eq. (1.1.84) 2 2 2 2K' _ s - 2a2P2&3 p p = r) 4 P " P p (1.1.89) N - kgr Af kgrf h2 Af Using typical magnitudes* encountered in laser Doppler velocimeter practice, a comparison of the signal-to-noise ratio of photomultiplier tubes and photodiodes, using Eqs. (1.1.85a) and (1.1.89), shows that the photodiode introduces much more noise onto the signal than does the photomultiplier tube. Thus, even though the radiant sensitivity a of photodiodes is very high (typically ten times that of a photomultiplier tube in the red portion of the visible spectrum), their poor noise characteristics limit their application to situations involving high light levels. Sufficiently high light levels are likely to be encountered with a homodyne configuration, and, for example, RuddZ2'has demonstrated the use of a photodiode in his combined homodyne-heterodyne configuration (see Section 1.1.4.4.2),where the signal is probably much higher than that encountered in either simple homodyne or heterodyne systems. With both the photomultiplier tube [Eqs. (1.1.85)] and the photodiode [Eq. (1.1.89)], the signal-to-noise ratio can be improved by decreasing the
* The following typical data for a laser Doppler velocimeter, for both types of photodetectors, were used: signal power p6 = 1.72 X 10-B W, P o / P = 34, and Af = 0.457 X 108 Hz. For the photomultiplier tube, a = 27 mA/W and k = 4, giving SNR = 1.5 x 1oJ = 50 dB. For the diode, a = 350 mA/W, T = 300 K , and R = 1.5 ohm, giving SNR = 2 x 10' = 20 dB.
) ] ,
1.1.
TRACER METHODS
I47
frequency bandwidth Afof the signal frequency that can be handled, but this will limit the maximum fluid velocity (or fluid velocity fluctuations) that can be measured. Both Eqs. (1.1.85) and (1.1.89) show that for a heterodyne velocimeter the signal-to-noise ratio depends on the coherence function pD.Since this depends on the location of the scattering particle relative to the photodetector (see Section 1.1.4.4.3), it follows that the signal-to-noise ratio depends on the particle position. The signal-to-noise ratio will also vary with particle position because of the spatial variation of the incident light at the particle (indicated by the quantities K,,,, KL, K ~ ) . In view of this it is possible to define the size of the measuring volume on the basis of an acceptable signal-to-noise ratio, rather than the illuminated volume approach used in Section 1.1.4.4.4. Up to this point, the explanation of the operation of the laser Doppler velocimeter has assumed that there is only one flow tracing particle in the measuring volume at a given time. In practice it is very unusual to find this condition, and in most flow systems the measuring volume contains a large number of particles which are randomly distributed in space. This affects the photodetector output in two ways: (a) the photodetector current has a random character even when the flow is steady and laminar, and (b) the signal-to-noise ratio is different from that observed with a single flow tracing particle and, as will be shown, this can affect the choice of velocimeter optical configuration. This section will concentrate on the signal-to-noise ratio, and the effect of the random character of the signal on the velocity measurement will be considered in Section 1.1.4.4.7. When there is more than one particle in the measuring volume, the light scattered by the particles may undergo optical mixing (heterodyning). If the particle number density in the measuring volume is sufficiently high, this can degrade the performance of a homodyne configuration laser Doppler velocimeter by decreasing the signal-to-noise ratio. On the other hand, the signal-to-noise ratio of the heterodyne systems is unaffected by the presence of multiple particles in the measuring volume. In view of this, it is probably desirable to limit the application of the homodyne systems to situations where the number density of the flow tracing particles in the flow field is low. These phenomena will be discussed in greater detail in the remainder of this section. With N particles in the measuring volume, the electric field at the photodetector will be [cf. Eq. (1.1.55a)l (1.1.90)
1.
I48
MEASUREMENT O F VELOCITY
where Eo is defined in Eq. (l.l.SSb), and Epl and Ep2are given by [cf. Eq. (1.1.55C)l (1.1.91) Epi = (IA;tl/R) exp[-j(o;r + c0 + k;RO)], where i (= 1, 2) indicates the beam illuminating the particle. From Eq. (1.1.44a), 0; = WO + (k; - ko) Up, (1.1.92)
so Epi = (IA&l/R) exp{-j[wot
+ (k;
- k),
*
UPt
+ c0 + k;RO]}.
Then the output current i from a point on the photodetector is
On carrying out the indicated mathematical operations and taking account of the various physical possibilities that could arise, we have2s4 i = a(lo
+ Z1 + Z2 + Z3 + l4 + Z5 + 16 + Z7),
(1.1.93a)
where 10
= e(EOm7
4
=
(1.1.93b)
N
e(EPiE;J
(i = 1, 21,
( 1.1.93~)
P=l
(1.1.93e)
(1.1.93g) and (1.1.93h) 284
F. Durst, 2. Angew. Math. Phys. 24, 619 (1973).
1.1.
TRACER METHODS
149
The terms I,,, I,, 12, I,, 16, and I, that appear in Eq. (1. I .93a) also appear in the expression [Eq. (1.1.56)] for the photodetector current when only one particle is in the measuring volume. However, in the multiple particle case these terms represent the summation of N terms of the type appearing in Eq. (1.1.56). The additional contributions, I , and 15,are associated with the presence of more than one particle in the measuring volume. Specifically, I , is due to the mixing of light scattered by different particles (k and I ) in the same light beam ( 1 or 2), and Z5 arises from the mixing of light scattered by different particles (k and I) in different light beams ( 1 and 2). To obtain the total photodetector output current, Eq. (1.1.93) must be integrated over the area of the photodetector AD, the area of the flow tracing particles A p , and the area of the light source As). This introduces the corresponding coherence functions pD,ppand p, (see Section 1.1.4.4.3). In the case of the terms 1, and Z5 , an additional coherence function* p; depending on the distance dPart between particles and the area AD of the photodetector is introduced. As a result. for these terms to contribute significantly to the photodetector current, the photodetector aperture must be limited to a certain maximum size, and the distance d,,,, between the particles must not exceed a certain maximum value. Since these terms depend on the heterodyne efficiency, Drain263calls them the coherent contribution to the photodetector signal. Correspondingly, the term I,, the homodyne contribution, does not depend on the heterodyne efficiency (see Section 1. I .4.4.3) and is therefore called the noncoherent contribution by Drain. This means that since I, and 1, contribute to the photodetector current in the homodyne configuration, heterodyne phenomena are present even in the homodyne systems. Although this somewhat contradicts the terminology used to describe this particular configuration, it will nevertheless be retained because of its simplicity. To estimate the signal-to-noise ratio, the signal power must be determined. Following the discussion earlier in this section, the signal power 285 B. M . Watrasiewicz and M. J . Rudd, “Laser Doppler Measurements.” Butterworth, London, 1976.
* Because of the random times at which the different flow tracing particles enter and leave the measuring volume, the coherence function p;) can only be given a probabilistic interpretation (with the prime introduced to distinguish it from the detector aperture coherence function pD associated with the terms /, and /,; po also has a probabilistic character in the multiple particle case). Space does not permit the details of this calculation, or any of the others dealing with the various heterodyne efficiencies to be given, but a more extensive discussion will be found in Drainzs3and Watrasiewicz and R ~ d d The . ~ coherence ~ ~ functionp;) has the same general form as the other coherence functions (pD,pS,and p p )introduced in Section I . 1.4.4.3.
1.
150
MEASUREMENT OF VELOCITY
S is, from Eqs. (1.1.93) (with P, and pup omitted for clarity, and assuming a photomultiplier of gain G is used as the photodetector),
zc N
N
S = 2a2G2RK$(2pb2
k=
kPf
Pip;
x N
+
1 I
P$
POP;),
(1.1.94)
P= 1
P=l
=
N
+ 2p;
where p; is the previously mentioned coherence function for optical mixing between the light scattered from different flow tracing particles. * To obtain the result shown in Eq. (1.1.94) the following assumptions must be made2? (a) the flow tracing particles are randomly distributed in the measuring volume, and (b) the averaging times are sufficiently long to allow many particles to pass through the measuring volume. Equation (1.1.94) is a general expression and can be specialized for the homodyne and heterodyne configurations. Thus, for the former, the last term in Eq. (1.1.94) is omitted. Then
z N
S
=
2a2G2KsNe
P"p'
( 1.1.95a)
P=l
where (1.1.95b) The quantity N, is the effective number of flow tracing particles in the measuring volume. If all the particles have the same scattering properties, then N, = (N - I), and for N >> 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.
MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
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 << oo. This means that useful information is carried by the signal frequency* and not by the signal amplitude, as in the case of chronophotographic observation of flow tracing particles or in hot wire anemometry . Another important characteristic of the signal which affects its processing is its temporal variation. Depending on the number of flow tracing particles, it can either be in the form of bursts separated by periods with no signal [see Fig. 21(a)], or it can consist of a continuous signal [Fig. 21(b)] which has random phase and amplitude and can therefore only be described statistically. It is important to note that the random nature of the continuous signal is quite independent of the nature of the fluid flow, so that a statistical description is required even with a laminar, steady * It is desirable at some points in the subsequent discussion for the reader to differentiate clearly between the carrier frequency ooand the Doppler frequency shift oD. Clarity is often aided by associating the symbol wD in the text with symbol U pfor the velocity of the flow tracing particle.
1.1. TRACER METHODS
I55
f
TIME, i
(b)
FIG.21. Typical photodetector output signals: (a) single burst associated with a single flow tracing particle, and (b) continuous signal resulting from the simultaneous presence in the measuring volume of more than one flow tracing particle.
flow, as well as with a turbulent flow. In the latter case, the statistics for the velocimeter signal would include both the statistics of the turbulent flow field and the intrinsic statistics of the signal. This intrinsically random nature of the continuous signal has its origin in the random arrival times of the flow tracing particle in the measuring volume, the variation in the light scattering properties of the flow tracing particles, and the variation in the paths of the particles through the measuring volume. This is discussed in greater detail below. One further important feature of the signal is that its signal-to-noise ratio may be very poor, in which case it is essential to use a signal processor with a narrow bandwidth. Since the passband of the processor will also affect the magnitude of the velocity fluctuations that can be handled by the processor, it is clear that the presence of noise in the signal can influence the measurement of fluid velocities in turbulent and other fluctuating flows. In order to discuss the theory of the various signal processors, which
156
1.
MEASUREMENT OF VELOCITY
are essentially FM demodulators, it is appropriate to devise a suitable mathematical expression for the signal. First consider the signal resulting from scattering by a single particle. It has a finite duration AT, which depends on the dimensions of the measuring volume and on the velocity of the flow tracing particle. The photodetector current i with a heterodyne system for a single particle p entering the measuring volume at a time t , can therefore be written [from Eq. (1.1.69a)l as In this expression it is assumed, following Durrani and Greated,220that the receiving aperture in the heterodyne configuration is small (in order to assume adequate coherence conditions, so that pD= l), so that the terms that contain no velocity information make a negligible contribution to the photodetector output current. It is usual to assume that the flow tracing particle size, source aperture, source spectral characteristics, and optical system adjustment are such that the corresponding coherence functions (p,, ps,pT,and pD)may be taken as unity. Then i = ~ U ( P ~ P : ) ~-’ r,)~ W cos[wD(t (~ - r,)].
(1.1.105)
For the homodyne system, Durrani and Greated220assume that the receiving aperture is so large that both those terms with velocity information and those without contribute to the photodetector output, so from Eq. (1.1.69b), assuming P& = Ps = P & , i
=
2uPsW(t - t,){l
+ p&,pT
cos[wD(t -
[,)I}.
(1.1.106a)
Again assuming that p,, p,, and pTare both close to unity,*
i
=
2uPsW(r - rp){l + cos[fJ&
- tp)]}
(1.1.106b)
The photodetector current has some particular features, which were alluded to earlier: (i) Since the measuring volume is not of uniform dimensions for all possible paths (see Section 1.1.4.4.4), the duration AT of the signal (which is related to the function W in the preceding equations) differs from particle to particle. (ii) The flow tracing particles enter the measuring volume at random times t,, so the corresponding signals are randomly distributed in time.
* The first term (low frequency component or pedestal) on the right-hand side of Eqs. (1.1.106b) carries no information about the particle velocity, and it is therefore filtered out either by a high pass filter at the processor input, or optically before the scattered light is incident on the photodetector (see Section 1.1.4.4.7.2).
1.1.
TRACER METHODS
157
(iii) The amplitude of the light scattered by the moving flow tracing particles will vary with time. The main cause of this variation is the spatial distribution of light in the measuring volume. The variation in the light scattering characteristics of the flow tracing particles and the passage of different particles through differently illuminated portions of the measuring volume can also affect the amplitude. In addition, there are a number of other reasons for amplitude variation which are classified as signal dropout (so-called because amplitude can drop to zero), and these are discussed in Section 1.1.4.4.7.4. Equations (1.1.105) and (1.1.106b) represent the photodetector signal when there is only one flow tracing particle in the measuring volume, which would be the case when the concentration of flow tracing particles is very low. If the flow tracing particle concentration is high so that there is always more than one particle in the measuring volume at any time, then the signals from each of the particles will overlap in time and the photodetector output will be continuous. However, as can be seen in Section 1.1.4.4.6, a simple summation of Eqs. (1.1.105) and (1.1.106b) over the number N of particles is not appropriate, because optical mixing will occur between light scattered by different particles. The frequency of these signals represents the difference in velocity between the particles in the measuring volume, and their spectrum is distributed about zero frequency (the majority of the flow tracing particles have zero relative velocity), so they contribute to the low frequency portion of the spectrum (the pedestal). Expressions for the photodetector output current in the heterodyne and the homodyne cases, respectively, can be written in the following somewhat inexact, but nevertheless useful, form: N
N
and
The low frequency terms are filtered out, so the expressions for the
158
1. MEASUREMENT OF VELOCITY
photodetector currents in both heterodyne and homodyne configurations can be written as identical expressions. N
i =u
2 Ppw(r - rp) cos[oD(t - rp>],
(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.
MEASUREMENT OF VELOCITY
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.
MEASUREMENT OF VELOCITY
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
1. MEASUREMENT OF VELOCITY
"
~
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, << t v ) ,where t , is the period of signal fluctuations, and t, is the period of sweep.
swept oscillator wave analyzer can be improved by making (w2 - wl)/Aw as close to unity as possible, and the values of the ratios t,/t, and tSA/rs as small as possible. Although the experimenter has considerable latitude in the selection of w2 - w1 and t S A , the sweep time t, of commercially available wave analyzers is fixed, typically & s. This can introduce errors when the statistical characteristics of the signal are not stationary with time (the most difficult statistical characteristic to control in practice is the mean number and size of flow tracing particles passing through the measuring volume). Accordingly the conventional, commercially available swept oscillator wave analyzer, with its comparatively slow, fixed sweep rate, is normally only used for qualitative measurements. However, Iten and Dandliker275 275
P. D. Iten and R. Dandliker, Proc. lEEE 60, 1470 (1972).
168
1. MEASUREMENT OF VELOCITY
have developed a method for significantly reducing the sweep time t , so that t s / t , can be reduced. If the sweep rate can be made much smaller than the period t , of the temporal variations in the signal frequency,* then the output from the swept oscillator wave analyzer is a voltage, proportional to the instantaneous frequency7 w, of the photodetector output (see Fig. 24). Strictly this converts the swept oscillator wave analyzer into a tracking bandpass filter, which is considered in detail in the following section. The noise characteristics of the swept oscillator wave analyzer have been considered by Cummins and SwinneyzlQ(see also Wang276*277). They show that
‘1
I&[
(1.1.1 13) = (1 + T AwSA)~”, N out 1 + S/NIi” where T is the time constant of the smoothing circuit. For commercially available wave analyzers, T AosA is much larger than one, and if the input signal-to-noise ratio is also substantially greater than one, we can write
(s/Mlmt = ( T AwsA)~’~,
( 1.1.1 14)
i.e., the output signal-to-noise ratio of the swept oscillator wave analyzer is independent of the input signal-to-noise ratio. Accordingly, the noise characteristics of the swept oscillator wave analyzer are satisfactory for the processing of laser Doppler velocimeter signals, including cases where the input signal-to-noise ratio is less than one. In spite of its limitations, the swept oscillator wave analyzer does have a number of advantages: (a) the mean Doppler frequency shift oDcan be estimated even with very poor quality and infrequent signals, because errors are not introduced by signal dropout (see Section 1.1.4.4.7.4), whereas signal processing by the tracking bandpass filter is very sensitive to signal dropout (see Section 1.1.4.4.7.4); (b) the method is not affected by the level of the turbulence, i.e., very rapid changes in the fluid velocity, and hence in the Doppler frequency oD,can be accurately measured;
‘11
c. P. Wang, J . Phys. E 5, 763 (1972). c. P. Wang, Appl. Phys. Lett. 20, 339 (1972).
* It should be noted that since this high speed sweep technique samples the signal, it is necessary, by Nyquist’s criterion, for the sweep frequency to be at least twice the most probable signal fluctuation frequency, if a satisfactory reconstruction of the signal is to be obtained. t This sounds like a contradiction in terms, but it is a widely used terminology. See Section 1.1.4.4.7.4, in particular Eq. (1.1.116).
1.1.
TRACER METHODS
169
however this is not true for the tracking bandpass filter, in which there is an upper limit to the frequency of the velocity fluctuations that can be measured (see Section 1.1.4.4.7.4; (c) commercial spectrum analyzers are readily available at modest cost as oscilloscope plug-ins; and (d) it is a very useful diagnostic tool for setting up a laser Doppler velocimeter, for monitoring its performance, and making preliminary evaluations of the flow. 1.1 A.4.7.4. Tracking Bandpass Filter. The inefficiency of the swept oscillator could be avoided if the center frequency w, of the filter, instead of changing independently of the signal frequency o,were to change by following the latter (see Fig. 25). The output would then be a real time signal proportional to the instantaneous frequency* wi of the photodetector output, given by wi(r) = wD(t)- dp(r)/dt.
(1.1.115)
Such a technique would result in perfect utilization of the data in the signal. This is the essential principle of the tracking bandpass filter. The elements of a practical tracking bandpass filter are shown schematically in Fig. 26. Its operation is very similar to that of the swept oscillator wave analyzer (Fig. 23) in that the output from a voltage controlled oscillator (VCO) is mixed with the photodetector signal w which has been stripped of the low frequency pedestal and low frequency noise in the high pass filter (Fig. 26). The mixer output is narrow band filtered in the intermediate frequency (IF) filtert to remove as much small amplitude broadband noise as possible (this makes the tracking bandpass filter particularly insensitive to broadband small amplitude noise of electronic or optical origin). Instead of rectifying the signal, as in the swept oscillator wave analyzer (Fig. 23), the signal passes through a frequency-to-voltage converter. The output v from this device is used to control the voltage controlled oscillator so that the output from the mixer is always at the fixed center frequency w, of the intermediate frequency filter. The voltage controlled oscillator driving voltage v , as controlled by the frequencyto-voltage converter, is thus related to the photodetector output signal, so
* Actually, the filter must have a small but finite time constant, so it is not really true that the output is proportional to the instantaneous frequency. However, the time constant is usually so small, particularly when compared to the time required to obtain an adequate sample with, say, the swept oscillator wave analyzer, that the output may be treated as real time for all practical purposes. t This bandwidth can be made very small because the IF signal leaving the mixer has a frequency that is constant regardless of the variations in the frequency o,of the photodetector output.
1.
170
MEASUREMENT OF VELOCITY
FREQUENCY, w
f
I
TIME, t
FIG.25. Relation between signal frequency wD and filter center frequency dashed line) in a tracking bandpass filter.
wc
(shown by
any change in the photodetector signal is followed by the voltage controlled oscillator controlling voltage, which is therefore the output from the tracking bandpass filter. Further details on the principles and design of tracking bandpass filters will be found in the monograph by Klapper and F ~ a n k l e . ~ ~ ~ Greated and D ~ r r a n i , ~ W’ i~l m s h ~ r s tand , ~ ~Adrian280 ~ have investigated the resolution AUIV of signal processors that operate in the time domain (i-e.,tracking bandpass filters and frequency counters). In the absence of noise (other than the intrinsic ambiguity noise), they have shown that
A U / U = K ( 2 A T / T ) ~ / ~ ( A ~ o / ~ o ~( T ) -
m)
(1.1.116a)
and
AU/U
=
K Ao/oD
( T small),
(1.1.116b)
where K is a constant depending on the optical parameters of the system, AT is the average time of passage of a flow tracing particle through the measuring volume, and T is the time constant of the output smoothing circuit. For very large time constants ( T + m), the resolution of the tracking bandpass filter is superior to that of the swept oscillator wave analyzer, because (AT/T)”* is very much smaller than one. With small time constants, the resolution approaches that of the swept oscillator wave analyzer. Greated and D ~ r r a n i discuss ~ ~ @ this in numerical terms. z78 J. Klapper and J . T. Frankle, “Phase-Locked and Frequency Feedback Systems.’’ Academic Press, New York, 1972. IT@ C. Greated and T. S. Durrani, J . Phys. E 4, 24 (1971). 280 R . J. Adrian, J . Phys. E 5, 91 (1972).
1.1.
171
TRACER METHODS
T O VOLTAGE
wvco
t
tv VOLTAGE
d
FIG.26. Tracking bandpass filter.
The superior resolution of the tracking bandpass filter compared with the swept oscillator wave analyzer is a consequence of the superior efficiency of the former instrument. The output from the tracking bandpass filter is a continuous real-time signal proportional to the instantaneous frequency wi = WD + wb + wN of the photodetector signal. This output can be exhibited on a cathode ray oscilloscope, galvanometer recorder, or XY recorder, with X equal to the time t, and Y equal to the instantaneous frequency wi. For subsequent analysis, the processor output can be treated like the output from a linearized hot wire (see Section 1.1.4.7). Since the tracking bandpass filter processes the photodetector output in the frequency domain, it may process noise with an amplitude comparable with that of the signal rather than processing the signal. This means that the photodetector signal-to-noise ratio must be high enough to ensure that the processor will only respond to the information carrying signal. The manufacturers of tracking bandpass filters quote acceptable input signal-to-noise ratios that lie between -6 and 4 dB, but a perusal of the scientific literaturezs1 suggests that higher values (e.g., 15 dB) are more realistic. The sensitivity to noise of the tracking bandpass filter can be varied, either at the design stage or, within limits, by the experimenter, by adjusting the passband (sometimes known as the capture frequency range) of the intermediate frequency filter. However, decreasing the capture frequency range decreases the so-called frequency slew rate (hertz per second) at which the central frequency of the filter can change. This, in turn, limits the rate at which the tracking bandpass filter can follow variations do,/dt = k du/dr in the velocity u of the flow tracing particles. T. S . Durrani and C. A . Greated, IEEE J . Aerosp. Elecrron. Sysr. AES-10, 418 (1974).
172
1.
MEASUREMENT OF VELOCITY
Hence, decreasing the width of the passband presented to noise narrows the bandwidth for fluctuations in the frequency of the photodetector output signal. Limiting the ability of the signal processor to handle velocity fluctuations can lead to serious inaccuracies in the output from the laser velocimeter. Furthermore, the inability of the tracking bandpass filter to track high frequency velocity fluctuations may cause it to lose lock, i.e., cease to provide processed data. It is clear that the adjustment of a tracking bandpass filter in any measurement situation must always be a compromise between speed of response and good noise rejection. Commercially available tracking bandpass filters have frequency slew rates that lie between 100 kHz/ms and 500 MHz/ms. For this reason, it is best to restrict the use of the tracking bandpass filter to liquid flows, where the frequencies are usually lower than those encountered in gas flows. The limiting input signal-to-noise ratio of the tracking bandpass filter also introduces other problems. In order to minimize the effect of noise on the measurements, the tracking bandpass filter is designed to discriminate against those signals that do not exceed some minimum preset amplitude. Any signal which has a lower amplitude is ignored by the processor, so the apparent output from the photodetector is zero. The tracking bandpass filter output must be held constant under these conditions; otherwise, it will attempt to locate the signal by driving the voltage controlled oscillator output to an extreme value, so that on the reappearance of the photodetector output signal the processor will be unable to locate the input signal. The system will then become inoperative until the signal is manually relocated by varying the voltage controlled oscillator output. The constant nonzero output represents erroneous data, and a large number of such events could lead to substantial error. The occurrence of signals below the chosen minimum amplitude is known as dropout. This condition has a number of causes. A very low concentration of flow tracing particles can result in a situation in which there are periods when no particles are in the measuring volume. This is particularly likely to arise when particles, e.g., dust, that occur naturally in the fluid are used as flow tracing particles. If the flow tracing particles are large compared to the fringe spacing in the measuring volume, this can lead to a low output signal from the photodetector (see Section 1.1.4.4.3). When there are multiple particles in the measuring volume, destructive interference may occur at the photodetector between light scattered by the particles. Spatial variations in the light distribution can cause the photodetector signals to have apparent dropout. Thus, the signal resulting from light
1.1.
TRACER METHODS
I73
scattered by a flow tracing particle which does not pass through the geometrical center of the measuring volume will have a twin-peak temporal variation, each peak originating in one of the intersecting beams. Similar effects also appear when using high power lasers operating in the TEMol mode,* where the intensity of the light, rather than reaching the peak associated with the Gaussian spatial light distribution of the TEMoomode, decreases at the beam center line. The light distribution in the measuring volume then has a twin peak distribution, and even though a particle passes through the center of the measuring volume, there could be a signal dropout, which is apparently the same as that described in the preceding paragraph. t The errors due to the effect of signal dropout in the tracking bandpass filter data can only be minimized$ by ensuring that none of the above causes of dropout are active in a given experimental situation. In particular, it is important to restrict the use of the tracking bandpass filter to situations where the concentration of flow tracing particles is high enough to ensure that there is always at least one particle in the measuring volume. In practice, this is usually found only in flowing liquids. A number of practical tracking bandpass filters suitable for analyzing J. W.Foreman, Jr., Appl. Opt. 6, 821 (1967). P. J. Titterton, Appl. O p f . 7, 206 (1968). 284 S. J. Barker, J . Fluid Mech. 60, 721 (1973). z8z 283
* The optical modes of lasers are discussed in de Lange.217 The fundamental mode, TEMm (TEM is the transverse electromagnetic mode), has the appearance of a single spot of light, and the intensity distribution is Gaussian, with the maximum intensity at the center of the beam. In addition-and this is important in connection with heterodyning-the phase is uniform over the whole wavefront. This is the mode usually assumed in theoretical discussion of the laser Doppler velocimeter, e.g., in Sections 1.1.4.4.3 and 1.1.4.4.4). The next two modes are the TEMol and the TEMII, which appear as two spots and four spots of light, respectively. This means there is a non-Gaussian distribution of light in the measuring volume, with the results indicated above. Furthermore, there is a phase reversal at the center of the beam. This makes optical heterodyning extremely difficult because of the difficulty of ensuring that only light of the same phase is mixed at the photodetector. Consequently, the presence of the modes higher than the fundamental can cause a serious decrease in the magnitude of the coherence function pD. These effects are discussed in detail by Foremanz" and T i t t e r t ~ n . ~ ~ ~ t In turbulent jet flow there is an apparent dropout if flow tracing particles are only present in the jet. This is clearly connected with the fluid mechanics of the situation, and could be avoided by introducing flow tracing particles into the external flow as well as thejet flow. $ Dropout errors can be estimated by comparing the peak frequencies obtained with a tracking bandpass filter with the corresponding frequency obtained with a swept oscillator wave analyzer.2M
174
1. MEASUREMENT
O F VELOCITY
laser Doppler velocimeter signals have been described in the literat ~ r e , and ~ ~tracking ~ - ~ bandpass ~ ~ filters are also available commercially. 1.1.4.4.7.5. Frequency Counting. With high turbulence intensities (say, >lo%) the tracking bandpass filter introduces errors because of its inability to follow large fluctuations in the fluid velocity. Since this error has its source in the feedback feature of the tracking bandpass filter, measurements under these conditions require a signal processor that does not use feedback. This objective can be achieved by carrying out the signal processing in the time domain. In principle, this can be done in two ways. In the first method, which we will callfrequency counting, the signal frequency is determined by measuring the number N of zero crossings in the signal between any two points of equivalent phase in the signal separated by some preselected time TN. The other method measures the period of the signal by determining the time TN between the two points of equivalent phase separated by a chosen number N of cycles. Since the time TN is proportional to the period TN/ N of the signal, this second method will be called period counting. In either case the signal frequency o can be determined from the relation o = ~ T N / T N Frequency . counting is used when there is more than one flow tracing particle at a time in the measuring volume, and the photodetector current is, in consequence, almost continuous. For the single burst photodetector signal associated with very low flow tracing particle number densities, the period counting method is used. This section will deal with frequency counting, and the next section will be concerned with period counting. A standard laboratory timer-counter can be used as a frequency counting signal processor for the photodetector output. However, to allow further analysis of the signal (see Section 1.1.4.7), an electrical signal proportional to the number of zero crossings is more useful. This can be obtained by passing the output from the timer-counter into a frequency-to-voltage converter. The output from this device, because of its continuous nature, is then a real-time signal proportional to the instantaneous frequency miof the photodetector output, and it can be processed and analyzed in the same way as the output from the tracking bandpass filter. 21u E. Rolfe, J. K . Silk, S. Booth, K. Meister, and R. M. Young, Laser doppler velocity instrument. NASA Conrracr Rep. CR-1199(1968). J. D. Fridman, K. F. Kinnard, and K. Meister, in “Proceedings of the Technical Program-Electro-Optical Systems Design Conference” (K. A. Kopetzky, ed.), p. 128. Ind. Sci. Conf. Manage., Chicago, Illinois, 1970. M. 0. Deighton and E. A. Sayle, Disa Inf. No. 12, p. 5 (1971). T. H. Wilmshurst and J . E. Rizzo, J . Phys. E 7,924 (1974). P. D. Iten and J. Mastner, in “Flow-Its Measurement and Control” (R. B. Dowdell, ed.), p. 1007. Instrum. SOC.Am., Pittsburgh, Pennsylvania, 1974.
1.1. TRACER METHODS
175
The resolution of time domain signal processors, including frequency counters, has been considered in Section 1.1.4.4.7.4. If the photodetector output exhibits dropout, this will appear to the frequency counter as an excessively long interval between zero crossings, and the resulting incorporation of this portion of the signal in the zero crossing count will lead to an erroneously low velocity. Frequency counting signal processors are sensitive to noise, other than ambiguity noise, because the good small amplitude broadband noise rejection characteristics of the tracking bandpass filter are not available, i.e., the frequency counter is a wide band signal processor with a bandwidth limited only by the signal conditioning electronics (see Section 1.1.4.4.7.2). Noise effects can be decreased by increasing the intensity Ff of the light incident on the flow tracing particles by using a more powerful laser, and/or by improving the light scattering characteristics of the flow tracing particles and/or the scattering angle so that the observation system collects the scattered light as close to the forward scattering direction as possible. Noise at the input to the counter-timer can be decreased by setting the signal level (the discrimination level) at which a zero crossing is registered sufficiently high so as to ensure that broadband noise with an amplitude less than the discrimination level does not register as a signal. However, this may result in small amplitude signals being ignored. Wilmshurst et a/.289ahas devised a frequency counter with a bandpass filter at the photodetector output and having a center frequency that can be adjusted automatically to follow the instantaneous frequency at the frequency counter output. Signal processing by frequency counting has been reported by a number of i n v e s t i g a t ~ r s . ~Jerskey ~ * ~ ~ ~and * ~Penne1392 ~~ used a frequency counter in the form of a commercially available frequency meter. Frequency counting based on a programmable hybrid computer is described by Garon and G o l d ~ t e i n . ~ ~ ~ 1.1.4.4.7.6 Period Counting. Tracking bandpass filters and frequency counters cannot be used with concentration of flow tracing particles which are so low that there is, on the average, less than one particle in the measuring volume at a given instant. The photodetector output signal then consists of a succession of single wave packets (or Doppler bursts). Under these conditions, the susceptibility of the tracking bandpass filter and the frequency counter to dropout errors introduces inaccuracies into T. H . Wilmshurst, C. A. Greated, and R. Manning, J . Phys. E , 4, 81 (1971). J . F. Meyers and W. V. Feller, in ICIASF ’73; I n t . Cong. Instrumentation in Aerospace Simulation Facilities,” p. 1%. Inst. Electron. Elec. Eng., New York, 1973. m 1 K . A. Blake, Opto-electronics 5, 27 (1973). 2g2 T. Jerskey and S. S. Penner, Phys. Fluids 16, 769 (1973). A. M. Garon and R. J. Goldstein, Phys. FIuids 16, 1818 (1973). 289a
zeo
I76
1. MEASUREMENT OF VELOCITY
the signal processor output. A signal processing technique is therefore required which avoids the tendency of the tracker to lock onto the last measured frequency (see Section 1.1.4.4.7.4). In addition, since the frequency counter will, incorrectly, count periods when no signal is present, the frequency must be measured in a time that is shorter than the time occupied by the signal. The first of these requirements can be met by discarding the feedback feature of the tracking bandpass filter. This means, as was pointed out in the preceding section, that some degradation in the quality of the processor output will have to be accepted, since the processor no longer has the narrow pass band of the tracking band pass filter to filter some of the low amplitude broadband noise on the signal. To minimize the effects of input noise, it may be necessary to adjust the discrimination level and/or use a more powerful laser, as was suggested in the previous section when discussing the frequency counter. The very short duration of the photodetector signal can be accommodated by measuring the frequency over a specified number of cycles of the signal, i.e., by period counting. In addition to permitting satisfactory measurement of short duration signals, period counting is also more accurate than frequency counting, which is an important consideration with a limited measurement time, in which small uncertainties could represent a very large percentage of the measured quantity. In practice, the period counting is restricted to about 10% of the maximum number of cycles in the Doppler burst. This allows for the varying duration of signals which arise from the different paths followed by the flow tracing particles through the measuring volume, and it also allows for variations in the speed of the particles. Further, by limiting the signal duration in this way, the effective spatial resolution of the laser Doppler velocimeter is improved. Velocity measurements using period counters are sensitive to errors introduced by the determination of the first and last zero crossing points. This depends on the setting of the discriminator used to minimize noise effects. From Eq. (1.1.48), we have the measured velocity Urn Urn= N X / T h ,
( 1 . 1 . 1 17)
where N is the number of points at which the signal (“positive going”) crosses the discrimination level (these points are called the zero crossings although this is only a correct terminology when the discriminator level is set at zero photodetector current), X is the fringe spacing, and the superscript prime on TNindicates the measured value. The measured time Th is related to the true time TN by TN = Th Z , where Z is a random variable because of the random character of the “burst” signal. Hence the error
+
1.1. E, =
177
TRACER METHODS
U = 6Um/Umof the velocity measurement is given by EU
=
( 1.1.1 18)
ZIT,.
There are two contributions ({1 and {2) to the quantity 2 [see Fig. 27(a)], so that Z = The minimum value of Z is zero = &), and the maximum value is the maximum of [{1 and 52 correspond to lines of different slope, as shown in Fig. 27(a)]. From Fig. 27(b) we see that the maximum (1 is T N / 4 N . Accordingly, 0 S Z S T N / 4 N , and hence the maximum error is ~.ij = -
1/4N.
(1.1.119a)
IMINATOR ING ME, t
f
FIG. 27. Time measurement in period counting. (a) Time measurement for a single Doppler burst. (b) Time of one cycle of a Doppler burst.
178
1.
MEASUREMENT OF VELOCITY
This shows that the discriminator setting tends to introduce bias toward higher velocity estimates, and that the error can be reduced by increasing the number N of zero crossings, i.e., by increasing the number of fringes in the measuring volume (when using a homodyne system). The preceding estimate has been for a single flow tracing particle. In order to minimize this error, and others, it is appropriate to average a number of measurements made on successive bursts. Wangzn4shows that the worst case error in the measurement of the mean flow tracing particle velocity is EU ? -
(1/12N
+
1/96W).
(1. I . 119b)
Therefore, as expected, the error is reduced by averaging over a number of burst measurements. Wang also shows that the mean square velocity fluctuations fi’ have an error given by ?
-
(1/6N
+ 1/32W).
(1.1.1 19c)
Other sources of error associated with period counting can be introduced by the signal processing equipment. Although Wang shows that these can introduce errors much larger than those arising from the discriminator setting, they will not be considered here because of their hardware related nature. It is sometimes claimed that the output from a period counting signal processor is not degraded by ambiguity noise. This is not strictly true, and it would be better to say that of all the signal processors available, the output is affected least by ambiguity noise. Because it processes signals in the time domain, ambiguity noise can only be introduced by variations in the signal duration and the time at which particles enter the measuring volume (see Section 1.1.4.4.7.1). However, the period counter only processes signal bursts and is therefore not sensitive to the latter source of ambiguity noise; also, the duration of the signal to be processed is fixed by the number of signal cycles to be processed and, unless the temporal variations in fluid velocity are very large, this will be almost constant, so ambiguity noise from variations in signal duration should be quite small. The output from the period counter consists of a succession of pulses of height TN (see Section 1.1.4.4.7.5) proportional to the transit time A7 of a particle passing through the measuring volume. Although, in principle, a single pulse contains all the information necessary to determine the fluid velocity in steady, laminar flow, a number of such pulses should be used m4 J. C. F. Wang, “Measurement Accuracy of Flow Velocity via a Digital-FrequencyCounter Laser Velocimeter Processor” Report 75CRD196. General Electric Co., Schenectady, N.Y.,1975.
1.1.
TRACER METHODS
I79
to form an average which will minimize the effects of broadband and ambiguity noise. In turbulent flow such statistical analysis is, of course, essential. Commercially available pulse height analyzers have been used to analyze the output from the period counter, by determining the probability distribution of particle transit times TN. A pulse height analyzer classifies the pulse signals according to their height TNby assigning each pulse to a “bin” which is reserved for pulses of a certain height. The number of pulses in each bin is added and the totals displayed as a function of pulse height TN, i.e., as the probability distribution of pulse heights. A sample-and-hold can be used to reconstruct, within the limits of uncertainty imposed by the intermittent nature of the processor output, the temporal variation of particle transit times TN (see footnote f in Table XXV). After computer processing to extract the Doppler frequency, the signal provided by the sample-and-hold can be analyzed as if it were the continuous output signal from, say, a tracking bandpass filter. Two further advantages of the period counter are (a) the dynamic range [(oDMAX/ODMIN) without changing the instrument setting] is much greater than that of the tracking bandpass filter because it has a much wider bandwidth than the latter type of processor, and (b) rapid changes in the signal frequency can be accommodated because there is no feedback in the processor. It should be noted that the period counter is a sampling device. This has its origin in two features of the processor: (a) a finite time is required to process the photodetector output signal, and no other signal can be handled during that time; and (b) the fluid velocity is measured intermittently, i.e., only when a particle is present in the measuring volume. The latter introduces sample biasing which can affect the accuracy of the data. This aspect of particle tracking methods of fluid velocity measurement is considered in Section 1.1.1.2. The finite processing time is required because a finite time TN is needed to count the specified number N of cycles in the signal, and a further time Tc is necessary to carry out data validation (see below) and “housekeeping” functions, such as passing reset commands to the digital components of the processor. This means that the data rate is limited to l/(TN + Tc), allowing typically 5 x 104-1 x lo6 data points to be handled per second (0.1 MHz wD 25 MHz). Although it has been stated earlier that ambiguity noise effects are less important in period counting signal processors than in any other type of processor, there is, the possibility of ambiguity in data interpretation. One source of this is sample biasing, mentioned above; the other is data dropout. With the latter, the photodetector output signal may provide some of the N cycles which have to be counted, and the remaining cycles
180
1.
MEASUREMENT OF VELOCITY
will probably be contributed by the signal from the next particle to pass through the measuring volume.* The elapsed time for N cycles would then be inaccurate (errors can be as large as 10% of the counts, because N is usually small). To eliminate this possibility, AsherZo5required overlapping series of cycles (say, 5 cycles and 8 cycles) to be counted and the counting times compared. If the ratio of the times are in agreement, i.e., the ratio indicates that the two period counts are from the same Doppler burst, the signal is accepted; if not, it is rejected. t Such data validation circuits, although they do have some inherent are essential to the satisfactory operation of a period counting signal processor. Period counting processors were independently developed by AsherZo5 and Brayton et af.2wand are now commercially available. The application of period counting to very low light level signals is discussed by Kalb and Cline.207 Although the period counting technique is intended to operate with signals consisting of a single Doppler burst, with suitable modification it can be used where the particle number density is high enough so that there is occasionally more than one flow tracing particle in the measuring volume. In addition to measuring the frequency over a specified number of cycles of the signal, all the cycles that exceed the threshold limit are counted. The frequency of this second, corroborative signal is compared with the measured frequency. If the two measurements agree, the measured frequency is accepted. If there is no agreement, it is assumed that the signal is a hybrid, associated with more than one flow tracing particle, and it is therefore rejected. 1.1.4.4.7.7. Photon Counting Correlation. Period counting has a poor immunity to noise if the photodetector output has a low amplitude H, because the minimum sensitivity of the processor must be decreased so as to ensure that valid signals are not ignored. This means that period counting cannot be used where the signal-to-noise ratio is one or less. Low amplitude signals also introduce other difficulties into signal processing. Thus it is possible that with very low scattered light levels there *01 286 97
J. A. Asher, Prog. Astronaut. Aeronaut. 34, 141 (1974). D. B. Brayton, H. T. Kalb, and F. L. Crosswy, Appl. Opt. 12, 1145 (1973). H. T. Kalb and V. A. Cline, Rev. Sci. Instrum. 47, 708 (1976).
* Noise of sufficient amplitude could also introduce invalid data. This could occur if the period counter were set to trigger with a signal amplitude that was not great enough to discriminate between the signal and the noise. t This could result in the rejection of data where the velocity of the flow tracing particle vanes as it passes through the measuring volume. See the preceding footnote on possible, sources of ambiguity noise in period counting.
181
1.1. TRACER METHODS
c
z W
(a) 2 V
c
3
a
c
3 0
n c
0
(b)
TIME
V W I-
; 0
c
0
I
a
TIME
(c)
n
TIME
FIG. 28. Photodetector output current in photon counting correlation. (a) Current that would be observed at high light levels. (b) Current corresponding to (a) but at low light levels (pulses are shown after the current has passed through the discriminator, so that all the pulses have the same amplitude). (c) Number n of individual pulses (corresponding to photons) in the time interval of duration T .
will be insufficient photons incident at the photodetector to allow meaningful interpretation of the signal (see Fig. 28). Durst and WhitelawZg8 imply that at least one hundred photons per signal cycle are required to provide a valid signal. Ladingzg9states that for a satisfactory signal, the number of photons detected per signal cycle should be larger than the allowable variance on the determination of one cycle. Very low light levels, together with low concentrations of flow tracing particles, is a situation usually encountered where naturally occurring particles are used as flow tracers, and where, in addition, the signal is carried by light that is backscattered into the photodetector. Typically, this is a feature of the atmospheric velocity measurements or measurements in high velocity wind tunnels. In the latter case, the addition of flow tracing particles to the air flow usually requires an impractically large mass flow rate of particles; in addition, the dynamic characteristics of naturally occurring particles are much better suited for high speed flows than artificially added particles. Since measurements in the atmosphere and in wind tunnels constitute important applications of the laser velocimeter, a method is required for the analysis of the signals in such situations.
zDB
F. Durst and J. H. Whitelaw, Am. SOC. Mech. Eng. [Pup.] No. 72-HT-7. L. Lading, Disa fnf. No. 19, p. 12 (1976).
182
1.
MEASUREMENT OF VELOCITY
When the intensity of the light incident on the photocathode is low, the photodetector output is made up of discrete pulses,* each pulse being associated with a photon that is incident at the photodetector. A careful examination of this signal will show that it has periods of higher or lower pulse density in a given time interval. These correspond to periodic variations, at the Doppler frequency, in the intensity of the scattered light (see Fig. 28), i.e., as a flow tracing particle passes through the bright portions of the measuring volume, it scatters more photons into the photodetector; correspondingly for dark portions, fewer photons arrive at the photodetector. In principle, the signal could be processed by a swept oscillator wave analyzer (but not by a tracking bandpass filter because of the signal dropout in the output from the photodetector at low incident light levels); however, as shown earlier (Sections 1,1.4.4.7.3), this device is an inefficient signal processor. This is an important consideration when the signal is very weak, because to overcome the limitations of the swept oscillator wave analyzer, it is necessary to extend the duration of the measurements if results of high statistical accuracy are desired. The efficiency could be improved by using a filter bank, but this has the previously noted disadvantage of being very inflexible. The analog in the time domain of the filter bank is the autocorrelator. Besides its efficiency, this device is attractive as a signal processor because practical autocorrelators do not have the inflexibility of the filter bank. The autocon-elator has a further advantage relative to the processing of the photodetector output at low light levels, in that it is easily realized as a digital device. As noted, the photodetector output in this situation has a natural pulse character, i.e., within a short time increment the number of photons is either zero or one. The conversion of such signals to analog form so that they may be processed in an analog correlator must inevitably degrade the information content of an already very weak signal. Any method that avoids this is desirable, because it will ensure the highest possible accuracy for the data.t The autocorrelation function for a signal with random amplitude and phase describes the dependence of the signal at a given time on its value at later times. For the laser Doppler velocimeter the photodetector output current has a periodic variation with time, related to the Doppler shift fre* In fact, the photodetector current has this pulse character at high scattered light intensities, but it is not evident from an examination of the signal because the pulses are superposed and give rise to an apparently continuous current. t At high incident light levels (i.e., with photon count rates in the range from 108 to 108 counts/s), it would also be possible to process the photodetector output signal digitally. However, fast digital electronics are expensive, and since analog signal processors are both comparatively much less costly and capable of processing the continuous output from the photodetector, it is appropriate to use analog methods at high light levels.
1.1.
TRACER METHODS
I83
quency wD and a constant value which represents the noise in the velocimeter. The separation of the signal and noise is consequently assured. Correlation of the photodetector output requires the signal to be sampled. It is important that the sampling procedure itself not introduce correlations. Thus the samples must be independent. The best way to ensure this is to integrate all the pulses in the photodetector output in some time interval T, either by counting the pulses or integrating the charge.* At the end of the sample time T, the data can be transferred to the correlator, and the integrator or counter reset for the next sample. The autocorrelation function R is found by counting the number n of photons incident on the photodetector in successive time intervals k, centered at time f k , with each interval of duration T (see Fig. 28). The successive counts are stored, and an estimate (indicated by the caret) of the autocorrelation function is formed by taking the sums of products
In Eq. (1.1.120), M is the number of autocorrelation channels, and N, the number of intervals in a run. The quantity m is an index for the autocorrelation lag time 7,such that T = mT. To simplify the circuitry of the correlator, it is usual to employ a “clipped” rather than a full correlation. This gives an undistorted correlation; however, a slightly longer measuring time is required in order to obtain a correlation coefficient that is equal in accuracy to that given by the full counting technique. Clipped correlation is described in h s e y et a/.,301 Foord et a/.,302and Jal~eman.~O~ The choice of the sample time T depends on the duration of the signal associated with the passage of a single particle through the measuring volume, because the correlation of the samples from the photon counter is intended to determine the period of the signal coming from any one particle. Lading has proposed that the optimum sample time T is one slightly longer than the anticipated time of transit of a flow tracing particle through the measuring volume. 3oo C. J . Oliver, in “Photon Correlation and Light Beating Spectroscopy” (H. Z. Cummins and E. R. Pike, eds.), p. 151. Plenum, New York, 1973. 301 P. N . Pusey, D. E. Koppel, D. W. Schaefer, R. D. Camerini-Otero, and S. H . Koenig, Biochemistry 13, 952 (1974). 302 R . Foord, E. Jakeman, C. J. Oliver, E. R. Pike, R. J. Blagrove, E. Wood, and A. R. Peacocke, Nature (London) 227, 242, (1970). 305 E . Jakeman, J . Phys. A 3, 201 (1970). ~
* Oliver has shown that photon counting is a more accurate way than RC integration for providing the input to the correlator.
184
1.
MEASUREMENT OF VELOCITY
She304,She and L ~ c e r o , ~Durrani O~ and Greated30s, Birch et Smart and Moore308,Createdws, and Mayo310have shown how the count correlation can be related to the Doppler frequency shift wD of the scattered light. For a steady fluid flow it can be shown, on ignoring the pedestal and any broadband noise that may be present (but including the ambiguity noise), that
k(7)= u1 e x p ( - ~ ~ ~ i / 4 a 2 )+[ 2cos(wD~)],
(1.1.121)
where a = [2(7#%,/Afl cos(a/2). This shows that the dependence of the correlation function on the delay time 7 is a damped cosine wave with a period directly related to the Doppler shift frequency wD. The velocity of the flow tracing particles can therefore be determined directly from the correlation function. With a turbulent flow it is necessary to make assumptions about the statistical nature of the velocity fluctuations in order to relate the observed correlation function to the fluid flow. If it is assumed that the amplitude of the turbulent velocity fluctuations are represented by Gaussian distribution, then
k(7)= al exp(-u272) + u3 exp(-u472) cos(u57),
(1.1.122)
where the constants a 2 , u 4 ,and us are related to statistical parameters that describe the turbulent flow field. To determine the constants, the observed correlation function must be fitted to Eq. (1.1.122). This can be a lengthy computation and, in consequence, Durrani and Greated311have suggested that it is better to Fourier transform the correlation coefficient and thereby obtain the frequency spectrum of the photodetector output signal. This requires special procedures if the spectral analysis is to have an appropriate resolution, and these are discussed in the original reference. At very high turbulence levels, the exponential damping term in Eq. (1.1.122), associated with the cosine function, becomes very strong and effectively damps out the periodic variations in the correlation function. The correlation function is then essentially useless for turbulence measurements. However, the situation can be improved if the apparent freC. Y. She, Appl. Opr. 12, 2415 (1973). C. Y. She and J . A. Lucero, Opr. Commun. 9, 300 (1973). T. S. Durrani and C. Greated, IEEE Trans. Aerosp. Electron. Sysr.AES-l0,17(1974). ~ o ’ A D. . Birch, D. R. Brown, J. R. Thomas, and E. R. Pike, J . Phys. D 6, L71 (1973). A. E. Smart and C. T. Moore, AIAA J . 14, 363 (1976). C. Greated, in “The Engineering Uses of Coherent Optics” (E. R. Robertson, ed.), p. 481. Cambridge Univ. Press, London and New York, 1976. ‘Io W. T. Mayo, Jr., Appl. Opr. 16, 1157 (1977). J1l T. S. Durrani and C. A. Greated, Appl. Opr. 14, 778 (1975). JM 305
1.1. TRACER METHODS
185
quency of the signal is increased by frequency shifting. The duration of the sampling intervals can then be decreased, and the total number M of lag times can be confined to a much shorter total time M7. Correlation will then be restricted to early portions of the exponential function, where the effect on the cosine function will be much weaker. Consequently, the periodic character of the autocorrelation function and its exponential decay will be clearly exhibited, and desired turbulence data can be extracted. A major disadvantage of applying photon counting correlations to turbulent flows is the necessity of making assumptions about the character of the flow. Smart and Moore30s have investigated the effect of nonGaussian (skewed) velocity amplitude distributions on the correlation function, and have found the substantial errors can arise if it is assumed that the distribution is Gaussian. Birch et have shown that in this case a Gram-Charlier model for representing the probability density function of the velocity fluctuations is much more satisfactory than the Gaussian model. Further discussions of the theory of photon counting correlation have been given by Jakeman,303Pike and J a k e m a ~and ~ , several ~ ~ ~ authors in the volume edited by Cummins and Pike.313 It is worth noting that the velocity biasing conventionally associated with period counting (see Sections 1.1.1.2 and 1.1.4.4.7.6) is circumvented by the photon counting correlator, because although there are more particles to be observed, there are fewer collected photons from each. Because of the weakness of the signals encountered in photon counting, it is essential to keep the system broadband noise to a minimum. This requires careful choice of photomultiplier tubes, and Foord ef and have reviewed the requirements of tubes for photon counting. For the same reason, it is not usual to use the reference beam optical configuration with photon counting, when the intensity of the light scattered from the flow tracing particle is very low, because the shot noise originating in the photomultiplier tube could become unacceptably large (see Section 1.1.4.4.6), leading to greatly increased times for signal processing (see below). Although the photon counting correlation method is a very sensitive signal processing method, there are limits to the weakness of the signal E. R . Pike and E. Jakeman, Adv. Quantum Elecrron. 2, 1 (1974). H . 2.Cummins and E. R. Pike, eds., “Photon Correlation and Light Beating Spectroscopy.” Plenum, New York. 1974. 314 R . Foord. R. Jones, C. J. Oliver, and E. R. Pike, Appl. Opr. 8, 1975 (1969). ’I5 G. A. Morton, Appl. Opr. 7, 1 (1968). 31*
’I3
186
1.
MEASUREMENT OF VELOCITY
that can be correlated due to the impossibility of increasing the collection efficiency of the photodetector optics, the size of the photodetector, or its quantum efficiency. This means that the times over which the signal is acquired must be increased as the strength of the signal decreases, and experience has shown that the time to obtain a good correlation can range from seconds to minutes. The photon counting correlator was first discussed by J a k e m a r ~ and ,~~~ subsequently demonstrated by Pike,318Abbiss et al. ,317 Meneely et and Kalb and Cline.297 An excellent She and L u c e 1 - 0 , Birch ~ ~ ~ et discussion of the theory of photon counting correlation has been given by Pike and Jakeman.312 The practical realization of a photon counting correlator for fluid velocity measurements has been described by Pike and Jakeman,312and Oliver.300 1.1.4.4.8. REMOVALOF DIRECTIONAL AMBIGUITY. The conventional laser Doppler velocimeter is not sensitive to the direction of motion of the flow tracing particles. This is a serious limitation in the application of the velocimeter to certain flow situations, e.g., oscillatory fluid flows, recirculating flow regimes, and turbulent flow. A number of techniques are available for eliminating this problem, which will be briefly reviewed in this section. A comprehensive discussion of this topic has been published by Durst and Zar15,~~’ and extensive use has been made of that reference in preparing this section. The main methods used to overcome the directional ambiguity may be classified as: (a) frequency shifting [or single sideband suppressed-carrier (SSBSC) modulation], (b) polarized light beams, or (c) multicolor lasers. A number of other methods which have either only been proposed or used very rarely are described by Durst and This section will briefly consider the techniques of items (a), (b), and (c) above. The Doppler frequency shift oDis related to the particle velocity U,in a one-dimensional flow by Eq. (1.1.48): wD = 2k01U11sin(a/2).
(1.1.123)
Since negative frequencies are not defined, an absolute value sign has been introduced. This equation is plotted in Fig. 29, and it can be seen that for a given angle a,it is impossible to determine the sign of the particle velocity without some modification to the velocimeter. E. R . Pike, J . Phys. D 5 , L23 (1972). J. B. Abbiss, T. W. Chubb, A. R. G. Mundell, P. R. Sharpe, C. J. Oliver, and E. R. Pike, J . Phys. D 5 , LlOO (1972). s18 C. T. Meneely, C. Y. She, and D. F. Edwards, Opr. Commun. 6,380 (1972). slB C. Fog, in “The Accuracy of Flow Measurements by Laser Doppler Methods” Proc. LDA Symp., Copenhagen, Denmark, Augusr 25-28, I975 (P. Buchhave et al., eds.), p. 336. Disa Elektronik A/S, Kovlunde, Denmark, 1976. 316 317
1.1. TRACER METHODS
I87
WITHOUT FREQUENCY SHIFTING [EQ. (l,I.l23)]
U’
D U
FIG. 29. Doppler frequency shift oDas a function of particle velocity U for constant a. U’ must be the largest anticipated value of U to ensure that directional ambiguity is eliminated from the signal.
In the frequency shifting technique,* a constant term Am, of typical magnitude between 5 kHz and 50 MHz, is added to the right-hand side of Eq. (1.1.123), so the curve relating the Doppler frequency shift wD and the particle velocity U is thereby shifted so that for U > U’(see Fig. 29), it is possible to determine the direction of particle motion. The frequency shift Aw is applied to one of the incident beams (in the two incident beam configurations), i.e., the frequency of the light in one beam is changed by Ao relative to the frequency in the other beam. Three methods are available for producing the required shift: (a) mechanical, (b) acousto-optical, and (c) electro-optical. In the mechanical and acousto-optical methods, the frequency shift is imposed on the incident light by placing a suitable time dependent diffracting device between the laser and the measuring volume. The electro-optical frequency shifting method employs a crystal which acts as a rotating half-wave plate on light which has been circularly polarized before passing through the crystal. The light leaving the electro-optic cell has its frequency shifted relative to the value of the incoming light. In the diffracting devices (mechanical or acousto-optical), the diffracted beams leave at different angles 8, and at different frequencies + n o A . These are related by n(wA/mo)= sin On, where n indicates the so-called order of the beam. The principle of this method of optical frequency shifting is to take the first order beams (which contain most of the diffracted energy) and transmit them to the measuring volume. This can be done by the use of appropriate combinations of optical components and * Other applications of frequency shifting are discussed in Sections 1.1.4.4.7.2 and 1.1.4.4.7.7.
188
1. MEASUREMENT OF VELOCITY
ma~king.~"If the positive and negative first order diffracted frequencies + wA and - wA are used, the two incident beams at the measuring volume will differ in frequency by 2wA, i.e., Aw = 2wA. Diffraction type frequency shifters are commercially available, and this makes the technique attractive to many users of the laser Doppler velocimeter. Electro-optical frequency shifters can take various forms320-323 but the principle of the technique can be conveyed most simply by considering the system described by Peters. In this device circularly polarized laser light is passed through two electro-optic (Pockels) cells (circular polarization of the laser output, which is usually plane polarized, can be obtained by passing the light through a h/4 retardation plate). Circularly polarized light may be considered as having two components, one in the x direction and one in the y direction. The electro-optical cells are placed in series along the light path, and one is arranged so that it only modulates the x component and the other only modulates the y component. The modulating frequencies applied to these two cells are 90" out of phase with respect to one another. The output from the two cells in series consists of a fundamental (carrier) wave and an infinity of waves separated by equal steps in frequency (sidebands); there are upper and lower sidebands disposed symmetrically about the carrier frequency. Each of the sidebands has an additional property in that it is polarized in a particular direction. This means that on passing the output from the two cells through a circular polarizer, the carrier (linearly polarized) and all linearly polarized sidebands are suppressed. The remaining sidebands are circularly polarized. It is possible, by an appropriate choice of the amplitude of the modulating voltage applied to the electro-optical cells, to suppress all but the desired sideband. The output from the frequency shifter is therefore light with its frequency shifted relative to the carrier (laser) frequency. A different technique of electro-optical modulation has been described by Foord et ul. ,324 and an appropriate modulator for laser Doppler velocimeter applications is commercially available. In this device, the modulator consists of two Pockels cell, each located in the path of one of the beams of a dual incident beam homodyne velocimeter. By suitable adjustment of the modulating voltages, the light at the output side of the modulators can have a relative difference in phase that varies with time. This causes the fringes in the measuring volume to move in space. If the L. E. Drain and B. C. Moss, Opto-electronics 4, 429 (1972). C. F. Buhrer, V. J . Fower, and L. R. Bloom, Proc. IRE 50, 1827 (1962). C. F. Buhrer, L. R. Bloom, and D. H. Baird, Appl. Opt. 2, 839 (1967). 529 C. J. Peters, Appl. Opt. 4, 857 (1965). R. Foord, A. F. Harvey, R. Jones, E. R. Pike, and J . M. Vaughan, J . Phys. D 7, L36 (1974).
1.1.
I89
TRACER METHODS
amplitude of the modulating voltage is a saw-tooth which causes the phase shift of the light to increase linearly with time from 0 to 180" and then return instantly to zero phase shift in a repetitive pattern, the fringes will be observed to move steadily in one direction. If the fringes move against the flow, the apparent Doppler frequency shift is moved to higher frequencies, and vice versa for fringes moving with the flow. The use of polarized light to obtain frequency shifting has been demonstrated by Iten and Dandliker.325 The two incident beams* are orthogonally polarized with respect to one another. t On the assumption that the polarization of the light is not changed by scattering from the flow tracing particles (Iten and Dandliker demonstrate that this is justified for small scattering angles, s loo),the scattered light is divided between two optical channels, and the light is directed onto two separate photodetectors. A phase shift, of a magnitude which can be controlled by the experimenter, between the signals from the two channels is introduced by appropriate optical components (see Iten and D a n d l i k e ~for - ~ ~details). ~ The sign of this phase shift depends on the flow direction. By introducing a phase sensitive tracking receiver at the output from the two photodetectors, the phase, and hence the particle velocity, can be determined continuously and automatically. The remaining technique for eliminating directional ambiguity that will be described uses multicolored lasers. If a number n of different colored i laser beams are incident at the measuring volume, the corresponding Doppler frequency shift wm will be given by wm = (27r/Ai)2U1 sin(a/2)
( i = 1 , 2,
.
..
,n).
(1.1.124)
The scattered light from the flow tracing particle can be divided into single colors either by narrow bandpass optical filters or by a dispersion prism. Each separated light beam is then introduced to its particular photodetector. If the incident light beams are arranged so that they overlap, but are not quite coincident, at the measuring volume, then the temporal sequence of the signals from the different photodetectors will indicate the 3*5
P. D. Iten and R. Dandliker, Appl. Opt. 13, 286 (1974).
* The technique could presumably be applied to a single incident beam system with the functions of all the various components interchanged (see Section 1.1.4.4.1). t Because of their relative polarization, the two beams incident at the measuring volume do not interfere. Hence, by the introduction of suitable polarizing elements in each receiving channel, the light scattered out of each of the incident light beams will interfere at the photodetector. This means that the results of Section 1.1.4.4.2 will apply, and the magnitude of the velocity can be determined from the Doppler frequency shift. However, the signals from the photodetector will have a relative phase difference that is related directly to the direction of motion of the flow tracing particles.
I90
1.
MEASUREMENT OF VELOCITY
order in which a flow tracing particle has passed through the light beams, and hence the direction of motion of the particle.* The use of multicolor lasers for the elimination of directional ambiguity is limited to flow fields in which the number density of particles is low enough to ensure that the photodetector output signal is in the form of a Doppler burst (see Section 1.1.4.4.7.6),because if the particle concentration is high enough so as to cause the signals from individual particles to overlap, it would be difficult to determine the order in which the different beams (fringe patterns in the case of the dual incident beam configurations) are crossed. The advantages and disadvantages of the various techniques for overcoming velocity ambiguity are summarized in Table XXIII. 1.1.4.4.9. SIMULTANEOUS MEASUREMENTOF SEVERAL VELOCITY COMPONENTS. It is often desirable to make simultaneous measurements of more than one velocity component in a fluid flow field. Such measurements are particularly important in turbulent, flows, but there are also laminar flow situations in which the measurement of two, or even three velocity components leads to a more complete picture of the flow field. Equation (1.1.46) is the appropriate governing equation for measurements of two or more fluid velocity components. If 6 and es (or ei and 4) are arranged for each velocity component as shown in Fig. 13, then the simplest form of Eqs. (1.1.46) results. In practice this would require two or three velocimeters, each lying in one coordinate plane. However, this direct approach has a number of disadvantages: (a) Each photodetector collects scattered light from the other velocity component measuring channels, so optical mixing can occur between this “spurious” light and the “valid” light; consequently there is “cross talk” between the channels. (b) It is expensive to duplicate velocimeter optical trains and signal processing equipment. (c) It is difficult to adjust the alignment of the separate velocimeters with respect to each other. D. H . Thompson, J . Sci. fnstrum. 1, 929 (1968). 317
=’
W.Mathes, W. Riebold, and E. de Looman, Rev. Sci. fnstrum. 41, 843 (1970). L. H. Tanner, Opt. Laser Techno/. 5, 108 (1973). A. Quick, 2. Flugwiss. 24, 17 (1976).
L. Lading, in “The Engineering Uses of Coherent Optics” (E. R. Robertson, ed.), p. 493. Cambridge Univ. Press, London and New York, 1976.
* This idea is also used in multidimensional velocity measurements (see Section 1.1.4.4.9). A number of velocimeters using two beams, both of the same color, have been reported; see Thompson?“ Mathes et al.,3Z7Tanner,’*’ Durrani and Greated,’” Greated,908 Quick:*@ Lading.’”
TABLEXXIII. Characteristics of the Techniques for the Removal of Directional Ambiguity Technique Mechanical frequency shifting
Acousto-optical frequency shifting Electro-optical frequency shifting Polarized light
A6J
0- 106 Hz
107
HZ
101-105 Hz
-
Advantages Cheapest and simplest technique.
Luminous energy loss is small.
Spectral broadening effects are minimized because there are moving parts. Simple and cheap technique. Spectral broadening effects are minimized.
Multicolored light sources
a
References
Luminous energy losses significant because some light is lost by diffraction. Spectral broadening of the signal may be significant due to vibrations of, and speed variations in the rotating diffraction grating. Expensive. Accurate alignment required.
a -d
Expensive. High voltage power supply required. Accurate alignment required. Zero velocity cannot be detected because there is a step change in frequency at zero velocity. Limit on scattering angle due to depolarization effects. Number density of flow tracing particles must be low. Luminous energy losses significant because of the use of filters. Measuring volume relatively large compared to LDVs using single color light sources.
h
T. Suzuki and R. Hioki, J. Opt. SOC.A m . 57, 1551 (1%7).
* E. B. Denison and W.H. Stevenson, Rev. Sci. Instrum. 41,
J
Disadvantages
1475 (1970). W. H. Stevenson, Appl. Opt. 9, 649 (1970). E. B. Denison, W. H. Stevenson, and R. W.Fox, AIChE J. 17, 781 (1971). 0. Lanz, C. C. Johnson, and S. Morikawa, Appl. Opt. 10, 884 (1971). L. F. Jernqvist and T. G. Johansson, J . Phys. E 7, 246 (1974). P. Buchhave, Disu In$ No. 18, p. 15 (1975). L. E. Drain and B. C. Moss, Opto-electronics 4, 429 (1972). R. Diindliker and P. D. Iten, Appl. Opt. 13, 286 (1974). F. Durst and M. a r e , Appl. Opt. 13, 2562 (1974).
e-g
i
192
1.
MEASUREMENT OF VELOCITY
This situation has led to the development of a number of techniques to overcome these limitations. To minimize cross talk, the signal is filtered so that the respective channels respond only to scattered light associated with a particular velocity component. This is achieved in practice either optically or electronically. Optically, polarized or colored light is used, with particular planes of polarization or colors assigned to particular velocity components. In the electronic approach, the output from the photodetector (in this case only a single photodetector would be needed) is filtered so as to separate the different velocity component signals. Special optical configurations have been developed which avoid the duplication of optical components. This assists both in minimizing the cost of the equipment and in making the alignment of the velocimeter easier (the same objectives can also be achieved using the electronic filtering described in the preceding paragraph). Such special optical arrangements result in the illuniinating and receiving systems being located331so that Eqs. (1.1.46) are complicated functions of the optical configuration angles. Examples of such expressions are given in H ~ f f a k e rWelch , ~ ~ ~et al.332Farmer,333and Blake.334 The remaining portions of this section will consider in more detail the various techniques for the simultaneous measurement of several velocity components. Polarized light beams. Since light beams that are polarized at 90” with respect to each other do not interfere, it is possible for two laser Doppler velocimeters to be applied to the same measuring volume without mutual interaction. This is accomplished in practice by appropriately polarizing the incident light beams and inserting polarizing elements between the measuring volume and the photodetector. Each direction of polarization, i.e., each photodetector, is associated with a particular direction in which a fluid velocity component is to be measured. In their turn, each photodetector is connected to its own signal processor. In addition to minimizing the cross talk between measuring channels, this technique has the added advantage that only a single light source need be used. This is attractive from the point of view of cost and, because it decreases the number of optical components, it simplifies the alignment of the velocimeter. Polarized light multidimensional velocimeters have been demonstrated H. H. Bossel, W. J. Hiller, and G. E. A. Meier, J . Phys. E 5, 893 (1972). N . E. Welch, R. H. Hines, F. L. Worley, and E. C. Gaddis in “Proceedings of the Technical Program-Electro-Optical Systems Design Conference” (K.A. Kapetzky, ed.), p. 147. Ind. Sci. C o d . Manage., Chicago, Illinois, 1970. sJJ W. M. Farmer, Appl. Opr. 11, 770 (1972). w . K. A. -Blake, J . Phys. E 5, 623 (1972). s91
1.1.
TRACER METHODS
193
by Blakem4for the dual incident beam configuration, and by Bossel et al.331 for the single incident light beam heterodyne and homodyne systems. Multicolor light beams. A very simple technique for minimizing cross talk is to illuminate the measuring volume with two or three different colored light beams* of frequencies wgi, i = 1, 2, 3. Then equation (1.1.44a) becomes 0 s
=
~
g
+ (kf - kot) * U. i
( 1.1.125)
In this expression, i = 1 , 2 , 3 indicates the frequency of the incident light. The spatial orientation of the vectors kf and k,,, will decide the particular velocity components that are to be associated with a particular color i. The light scattered from a flow tracing particle can be optically filtered according to the frequency wgi of the incident light and collected at two or three photodetectors (depending on the number of velocity components to be measured). Each photodetector would respond, because of the optical filtering, to scattered light of one color, and hence to a single velocity component. Individual photodetectors would be connected to their own signal processing circuits. Clearly the principle of this type of velocimeter is identical to that using polarized light, except that color rather than direction of polarization is used to minimize cross talk effects. Two color laser Doppler velocimeters for the simultaneous measurement of two velocity components have been demonstrated by Grant and O r l ~ f fand , ~ by ~ A b b i ~ s .Components ~~~ for constructing such a velocimeter are available commercially. Instead of using different colored light sources, Farmer and Hornkohlw7 and Crosswy and HornkohF3* have demonstrated a velocimeter that employs incident light beams of different frequency which are generated by modulating at two frequencies the light emitted by a single source. The modulation is accomplished by a two-dimensional Bragg cell. Because only two modulating frequencies are used, only two velocity components can be measured, but a system for measuring three nonorthogonal velocity components has been described, but not demonstrated, by Hallermeier.33e ass G . R. Grant and K . L. Orloff, Appl. Opt. 12, 2913 (1973). J. B . Abbiss, Elecrro-Opt. Sysf. Des. 6 (No. 7), 29 (1974).
W. M. Farmer and J. 0. Hornkohl, Appl. Opt. 12, 2636 (1973). F. L. Crosswy and J. 0. Hornkohl, Rev. Sci. fnstrum. 44, 1324 (1973). sBD R. J. Hallermeier, Appl. Opt. 12, 924 (1973).
* The argon ion laser provides two strong lines at 4880 A (blue) and 5150 A (green), and is therefore a very convenient source if two velocity components are to be measured.
194
1. MEASUREMENT
OF VELOCITY
The two-dimensional Bragg cell of Farmer and Hornkohl, and Crosswy and Hornkohl uses two ultrasonic traveling waves which are directed along mutually orthogonal directions. The input to the cell is the single beam from a laser, and the output consists of four equally intense beams. Three of the output beams are diffracted, and one is a portion of the original, undiffracted input beam. The frequency of the diffracted beams can be chosen by appropriately orienting the cell relative to the incoming laser beams. Polarizers located between the Bragg cell and the measuring volume polarize the illuminating beams so as to minimize cross talk between the different velocity component measuring channels. The scattered light is directed onto a single photodetector, and the output signals associated with the two velocity components are separated by bandpass filters (see below). Two advantageous features of the Hornkohl et al. system are (a) it is inherently self-aligning because of the use of a single Bragg cell to generate the light beams and a single photodetector to receive the scattered light, and (b) directional ambiguity can be avoided because of the use of frequency shifting (see Section 1.1.4.4.8). Electrical filtering of the photodetector output. The separation of the signals associated with the different velocity components can be accomplished electrically, so that a single photodetector may be used for all the signals. This avoids the duplication of receiver optical components, and hence the problems associated with aligning the receiving system. Where the frequency spectra of each signal incident on a single photodetector are individually distinguishable, the output from a swept oscillator wave analyzer may be interpreted in terms of the velocity components. This method, of course, is subject to all the limitations of the swept oscillator wave analyzer (see Section 1.1.4.4.7.3). However, the sampling techniques described in Section 1.1.4.4.7.3 that have been devised to improve the efficiency of the swept oscillator wave analyzer can still be used when measurements of two or more velocity components are being made (see, in particular, Sullivan and Ezekie13'O). Crosswy and H ~ r n k o h Phave ~ ~ used bandpass filters to separate the velocity components signals where different carrier frequencies ooare associated with different velocity components (see above for details).
1.1.4.5. Fabry-Perot Spectrometer 1.1.4.5.1. INTRODUCTION. The advantage of using direct spectral analysis of the scattered light by means of a Fabry-Perot spectrometer has already been considered briefly in Section 1.1.4.3. This section is con340
J. P. Sullivan and S . Ezekiel, J . Phys. E 7, 272 (1974).
1 . 1 . TRACER METHODS
195
cerned with the principles and practical applications of laser Doppler velocimeters (LDVs) with direct spectrum analysis. It discusses the design principles of scanning and servo laser Doppler velocimeter techniques, describes some types of multiple-beam interferometers incorporated in laser Doppler velocimeters, analyzes the performance of specific velocimeters, and reports some results of their application in aero- and gasdynamic experiments. A number of fast-response laser Doppler velocimeter systems with high time and space resolution utilizing electro-optical devices are also proposed and discussed in this section. Methods of optical data processing as well as some engineering problems, e.g., design of different types of beam splitters, and focusing and polarizing arrangements, which apply to all types of laser Doppler velocimeters, are touched on briefly, since they are discussed at length in several references .341-345 1.1.4.5.2. PERFORMANCE PRINCIPLES OF LDVs WITH DIRECT SPECTRUM A N A L Y S I S .MULTIPLE BEAMINTERFEROMETERS.In addition to the optical mixing techniques discussed in Section 1.1.4.4, direct spectrum analysis of a scattered signal may be used for extracting the Doppler frequency shift, if an appropriate spectrum resolution is ensured. The most suitable methods for this purpose are those of high performance interference ~ p e c t r o s c o p y ~ that ~ ~ - ~incorporates ~~ multiple-beam interferometer-spectronieters having much higher resolving power (up to 10’- lo8) than other spectroscopic devices. The following paragraphs consider some specific features related to the design of laser Doppler velocimeters of this type. A schematic drawing of an LDV system with direct spectrum analysis is shown in Fig. 30a. It comprises a frequency-stabilized helium-neon laser ( l ) , a multiple-beam interferometer (2), a signal recording system (3), and automatic control (4) of signals as well as the usual optical elements (e.g., lenses, mirrors, beam splitters, diaphragms, used in LDV H. 2. Cummins, N . Knable, and Y. Yeh, Phys. Rev. Lett. 12, 150 (1964). m B . S. Rinkevicius, Usp. Fiz. Nauk 111,305 (1973)-Sov. Phys.-Usp. (Engl. Trans/.)16, 712 (1974). 3w F. Durst, A. Melling, and J . H. Whitelaw, “Principles and Practice of Laser-Doppler Velocimetry.” Academic Press, New York, 1976. 3u Yu. G. Vasilenko et a/.. “Laser Doppler Velocimeters.” Nauka, Siberian Section,
1975. 3u G. L. Grodzovsky, ed.), “Laser Doppler Measurement of Gas Flow Velocities.” Proc. N. E. Zhukovsky C.A.H. Institute Moscow, 1976. A. N. Zaidel, G. V. Ostrovskaya, and Yu. I. Ostrovsky, “Techniques and Practices of Spectroscopy.” Nauka, 1976. M7 P. Connes, Rev. Opt. 35, 37 (1956). M0 M. Hercher, Appl. Opt. I, 95 (1%8).
1.
196
1
MEASUREMENT OF VELOCITY
Al“
FIG.30. LDV with direct spectrum analysis: (a) block diagram of experimental setup; 1, single frequency laser; 2, multiple-beam interferometer; 3, signal recording system; 4, automatic control system; and 5-7, optical elements for beam separation, focusing, and collection of laser radiation. (b) Schematic of a typical spectrum record.
systems for the guiding, filtering, and focusing of a laser beam and collection of scattered radiation (5-7). If scattered radiation AS = A. - A D , is brought to an interferometer (for instance, to a Fabry-Perot etalon, plane or confocal) together with a reference beam Ao, then two systems of interference fringes corresponding to AS and A. are formed in a focal plane of a lens placed at the interferometer outlet (the corresponding spectrum is shown in Fig. 30b). By.measuring AD = As - Ao, the velocity of light scattering particles may be found from Eq. (1.1.4.4~). It is evident that unlike a photomixing method, the major problem in direct spectral diagnostics is the measurement of small Doppler shifts and, therefore, small velocities. It is usually assumed in interference that two spectral lines registered on one channel are resolved by an interferometer, provided their spacing equals a half-width of for the present method the lines SA; then a typical minimum velocity Urnin may be estimated from the condition AD B
( 1.1.126)
SAo = AIR,
where SAo and R are the instrumental half-width and resolving power of the interferometer, respectively. For an ideal etalon, SXo is defined only by the reflection coefficient of the mirror plates K’, and the plate spacing = r)A2(1 - K’)/2tT(K‘)”2.
With a plane-parallel etalon,
r)
= 1; with spherical, r ) =
(1.1.127)
4.
For the
1.1.
TRACER METHODS
197
system geometry shown in Fig. 30a, where cp is assumed to be 90”for the sake of simplicity, AhD is related to U as AD = (U/C)A sin a.
(1.1.128)
For typical values of A = 6328 A, a = 30°, and a resolving power of an inA), Urninis about 12 terferometer R = 5 x lo’ (aho = h / R = 1.2 x m/s. On the other hand, for convenience of interpretation and processing of experimental data, AD should not exceed the free dispersion range of the etalon, i.e., AD AX0 = F 6ho, ( 1.1.129) where F is the finesse (the number of reflections inside an interferometer). In modern instruments, F attains -50-100, to give Umax/Umin b 100. In addition to the effect of the resolving power R on the range and accuracy of measured velocities, the illumination of an interferometer has a strong influence on LDV sensitivity. Therefore, to characterize the interferometer efficiency in a LDV system, it is appropriate to use a parameter P which is a product of the illumination and the resolving power. For a spherical eta10n349*350 (Fig. 31a),
Ps
=
4.rrTt2,
( 1.1.130)
where T is the interferometer transmittance. On comparing348this expression with Pp = 2.rrTA for a plane Fabry-Perot interferometer (Fig. 3 1 b), we arrive at F s / P p = 8t2/DZ,
(1.1.131)
where the plate area is taken as A = .rrP/4, and the transmission coefficients for both interferometers are assumed equal. Letting the plate diameters be D = 3 cm, the ratio P s / P p at t = 10 cm and t = 1 cm is found to be 90 and 0.9, respectively. To provide measurement of Umin 10 m/s, high-resolutions etalons are required at t = 5-10 cm. In this case, the spherical interferometer efficiency, as it is exemplified above, is much higher compared to a plane one, and with increasing t a parameter P s / P p grows as - t 2 . It should also be noted that with the use of a spherical interferometer, the requirements for adjustment accuracy and surface quality are considerably reduced .351 When designing an LDV with a direct spectrum analysis (in particular, in servo LDVs; see below), it is important to provide constant linear dis-
-
348 350
H. G . Heard, “Laser Parameter Measurements Handbook.” Wiley, New York, 1968. D. J. Bradley and C. J . Mitchell, Philos. Trans. R . Soc. London, Ser. A 263,209(1968). A. Persin and D. Vukicevik, A p p l . Opr. 12, 275 (1973).
1.
198
MEASUREMENT OF VELOCITY
(C)
L ;
*/: I
z(e)
3
4 r
(dl
FIG.31. Schematic of a typical multiple-beam interferometer: (a) spherical interferometer (SIFP), (b) plane-parallel Fabry-Perot interferometer (PIFP), (c) interferometer with a wedge position of plane mirrors ( I n ) , and (d) defocused spherical interferometer. The free dispersion region versus the radius of the rings is shown for a defocused spherical etalon .s5’
persion of an interferometer system in passing from one order of interference to the other. This condition is satisfied with an interferometer having wedge-type plane mirrors, i.e., with a Fizeau-Tolansky interferometer (IFT)230,351as well as with a defocused spherical etalon (DSIFT)352*353 (see Figs. 31c and 31d, respectively). S. Tolansky, “High-Resolution Spectroscopy.” Methuen, London, 1947. 3~
I. V. Skokov, “Multiple-Beam Interferometers.” Mashinostr., Moscow, 1969.
1.1.
TRACER METHODS
I99
With the use of a Fizeau-Tolansky interferometer, there are important requirements for incident beam parallelism. To provide the necessary finesse F = AAo/GAo,an angular dimension of a source must not exceed a y = (A/2tF)1’2.
(1.1.132)
With characteristic parameters A = 6328 A, r = 10 cm, and F = 50, y = 3.5 x lo4. This imposes limitations on the collimating system: d/f3.5 x where d is the light source dimension (i.e., a size of a test region) andfthe focal distance of a lens, collecting the scattered light. A defocused spherical ktalon (DSIFP) is a modified spherical interferometer. A parameter h can be introduced to characterize the departure of such a system from confocality. It is shown in ref^.^^^*^^^ that for small h , within several orders of interference, the dispersion dvdr will be linear (Fig. 31e). A more thorough analysis of the operation of these interferometers is given in ref^.^^^*^^-^^^ Certainly, a proper choice of a particular LDV and of spectrum analyzer systems depends on the test flow characteristics and requirements of the experiment. Several types of Doppler anemometers have been reported with direct optical spectrum analysis intended for diagnostics of various aero- and gasdynamic processes, e.g., two-phase flows354-356 and high temperature and hypersonic ~ ~ o w sfor, measurements ~ ~ ~ * ~ ~ of turbu~ l e n ~ eand ~ ~boundary ~ . ~ ~layer ~ profiles. All these systems differ in the type of lasers and interferometers that are employed (plane,357conby the number of measured flow foca1,248*355 or defocused velocity components, by a direction of scattering back?, by scanning techniques, and by many other specific features of the optical systems. However, all these LDV schemes may be subdivided into two main groups depending on the resolved spectrum display principle (discrete or continious in time) and data recording: (1) spectrum-scanned LDVs and (2) servo LDVs. A. P. Alkhimov, A. N. Papyrin, and A. L. Predein, in “Gasdynamics Problems,” p. 265. Pure Appl. Mech., USSR Academy of Sciences, Novosibirsk, 1975. 3s5 A. P. Alkhimov, V. M. Boiko, A. N. Papyrin, and R. I. Soloukhin, Z h . Prikl. Mekh. Tekh. Fiz. No. 2, p. 36 (1978);J. Appl. Mech. Tech. Phys. ( E n g l . Trans!.) 19, 173 (1978). A. P. Alkhimov, V. M. Bolko, and A. N . Papyrin, in “Physical Gasdynamics,” p. 24. Pure and Appl. Mech., USSR Academy of Sciences, Novosibirsk, 1976. ~ 5 ’B. S. Rinkevicius, A. V. Tolkachev, and A. V. Kharchenko, in “Laser Doppler Measurements of Gas Flow Velocities” ( G . L. Grodzovsky, ed.), p. 155. Proc. N. E. Zhukovsky C. A. H. Institute, Moscow, 1976. H. L. Morse, B. J. Tullis, H. S. Seifert, and W. Babcock, J. Spacecr. Rockers 6 , 264 (1%8). D. M. Paul and D. A. Jackson, J. Phys. E 4, 170 (1971). sm J. M. Avidor, Appl. Opt. 13, 28 (1974).
2 00
1.
MEASUREMENT OF VELOCITY
Figure 32 illustrates a classification of LDVs with a direct spectrum analysis based on the spectrum recording method. Before considering these systematically, their main specific features should be emphasized. The scanning schemes rely upon the time dependent frequency characteristics of multiple-beam interferometers. This may be achieved either by optical length variation of an etalon or by electro-optical scanning methods. The whole contour of the scattered radiation line is recorded in this case (see Fig. 32); therefore, a readout is discrete in time. However, the measured line center and the line shape make it possible not only to determine B and AU, but also to obtain information on particle size and velocity distribution functions n(r) and n ( U ) ,since the scattered light intensity depends on sizes and number densities of light scattering particles.
LDV s with direct spectrum analysis (based on multibeam interferometers)
L5 Systems based on scanning interferometers
Systems with image converter scanned spectrum
7
Servo L D V systems
Spectrum-xanned L D V s
Systems with linear transmittance optical filters
Streak recorded interference fringe systems
'"I
U
FIG.32. Classification of LDVs with direct spectrum analysis.
1.1.
TRACER METHODS
20 1
Servo LDV schemes provide a continuous measurement of each moving particle velocity in the measuring volume by means of optical filters whose transmission I / I o is a linear function of a Doppler frequency shift I / I , = K A and, ~ therefore, the particle velocity U . In this case, the detector output signal, whose amplitude is directly proportional to U at each instant, allows the determination of the mean U as well as the particle velocity variation, AU (Fig. 32). 1.1.4.5.3. SPECTRUM-SCANNED LDVs 1.1.4.5.3.1. Scanning Principles. Requirements f o r an LDV System Spectrometer. This section will consider some design principles of LDV systems which employ scanning interferometers and photodetectors for spectrum recording. As already mentioned, this laser Doppler measuring technique is based on the frequency dependence of the transmittance in a conventional etalon. A multiple-beam interferometer was to operate as a narrow band filter whose frequency characteristic w is defined as
where A = a 2 C / 2 ~ ?and B = (1 - g),C/2$t are coefficients depending on interferometer parameters, the transmittance a,the reflectivity K' of mirrors, and the distance between them t. The following intrinsic frequencies of this filter exist: w,
= q.rrmc/t,
( 1 .1 . 1 34)
where m is the fringe order, and q = 1 for a plane-parallel etalon and 4 for a spherical one. An absolute half-width of each band (a half-width of an instrumental function of an ideal interferometer) is defined by &Jo
2 :
qC(1 -
Kr)/f.
(1.1.135)
The frequency characteristic H(w) of an interferometer may be changed through the optical length variation nt, by changing either the refractive index n or the plate spacing t . With decreasing t at a small value E , the resonance frequency omincreases, and its new value wk is found from the relationship3s1: w& = q.rrmc/(t -
E)
= qwm(l - E / t ) .
(1.1.136)
Here, if E = t / m , then wk = &,+I. Let us assume that radiation incident onto an interferometer correA . Papoulis, "Systems and Transforms with Application in Optics." McGraw-Hill, New York. 1968.
202
1.
MEASUREMENT OF VELOCITY
sponds to some time dependent signal with a power spectrum S(w). If the and varies rather spectrum S(w) is band-limited by frequencies (w, ,om+J smoothly within an interval 6wo (an instrumental half-width), then the mean output intensity Z will be proportional to S ( O & ) , ~ where ~’ a& is the instantaneous resonance frequency defined from Eq. (1. I . 136). Varying E from zero up to r / m results in increasing wk from wm up to om+l. This allows the whole spectrum of a process to be recorded from the photodetector output signal. In practice, a spectrum is usually scanned either by means of pressure variation in an interferometer chamber or by a mechanical displacement of one of its mirrors. Figure 33 depicts schemes of spectrometers incorporating the scanning Fabry-Perot etalons in which optical length variations are achieved via (a) chamber pressure variation Ap or (b) displacement of one of the mirrors with the use of a piezoelectric cell. The common principal components of these schemes are as follows: an interferometer with a variable free dispersion region (3), a photodetector (6), an automatic control mechanism (4), a recording system (7), and conventional optical elements ( 2 ) . A diaphragm ( 5 ) is set in the focal plane of an objective ( 2 )to select a required spectral region when using a photodetector to record the etalon output intensity. Its dimension d is defined from the condition for an optimal combination of the resolving power and illumination of the e t a l ~ n ~ ~ *
d
=
(2j2/R)”*,
(1.1.137)
where f i s the focal distance of an objective imaging an interference pattern, and R is the interferometer resolving power. The interference fringe order numbers Am recorded by a photodetector (6) positioned behind a diaphragm ( 5 ) depend upon the optical length variation A(nt) of the interferometer:
Am
=
2 A(nt)/A.
(1.1.138)
In the case of pressure scanning, Am is defined as (1.1.139)
where the known relationship relating the refractive index n to pressure p , i.e., n - 1 = (no - l)p/pO, has been used. For instance, in air (no - 1 = 3 x lo4) at t = 10 cm, A = 0.63 um, Ap = 1 atm, and Am = 95. In designing scanning spectrometers for the purpose of determining the Doppler shift, in addition to good spectral resolution and sensitivity, rapid time response is also of significance. In some experimental aerophysical studies it is desirable to be able to change the spectrum recording time
1.1.
TRACER METHODS
203
(C) 3
r
FIG.33. Block diagrams of spectrometers involving scanning interferometers with optical length variation attained by: (a) pressure variation, (b) mechanical piezoelectric displacement of one of the mirrors, (c) image converter scanned spectrometer with a double converter tube, and (d) image converter scanned spectrometer with a TV image tube.
within wide limits, T, = 10-10-4 s.355 The requirements for lasers are also critical, i.e., single-frequency lasing and high frequency-stability as well as sufficiently high radiation intensity. 1.1.4.5.3.2. Practical LDV Systems Based on Scanning Interferometers. At present LDV systems involving confocal piezoelectrically ~ , ~ ~ ~ scanned interferometers are the most widely ~ ~ e d . ~ ~These LDVs provide higher sensitivity and temporal resolution compared with those using a plane pressure-scanned Fabry -Perot i n t e r f e r ~ m e t e r . ~ ~ ' Besides, they are simpler, and for ease in operation an electrically con%*
P. L. Eggins and D. A . Jackson, J . Phys. D 7, 1894 (1974).
204
1. MEASUREMENT OF VELOCITY
trolled scanning circuit may be employed. Scanning rates of about lo3 orders per second can be attained with the aid of piezoelectric cells; this corresponds to a recording time of an instrument line contour t, =s lop4s. The latter can be easily varied from up to 10 s and more. of this type developed for supersonic A typical LDV two-phase flow investigations is presented in Fig. 34. A single-frequency helium-neon laser (1) was used as a light source. The output power was 5 mW at A = 6328 A. The laser beam passed through a diaphragm (2) and a beam splitter (3) to yield reference and illuminating beams, which were focused with a lens (4), via an inlet window, at a common point in the test flow (8). The light scattered by particles and the reference beam were collected and registered at an angle LY = 31'44'. The Doppler frequency shift wD for this system configuration was WD
(Hz) = 8.7 x lo5 U (m/s).
( 1.1.140)
The Doppler frequency shift was recorded by using a piezoelectrically scanned interferometer (6). With an ac scanning voltage applied to a piezoceramic element of an etalon, its plate spacing varies within several wavelengths, yielding a corresponding change in the resonance frequency of the interferometer, and it consequently follows the recorded frequency spectrum of the collected radiation. This spectrum was registered by a photomultiplier (7), an amplifier (1 l ) , and an oscilloscope (12) synchronously swept by a scanning voltage signal. The time scale of the oscilloscope screen corresponds to the frequency scale of the spectrum investigated in this case. In the experiments, the scanning interferometer was a confocal resonator with a plate spacing o f t = 10 cm formed by mirrors with multiple dielectric coatings and with a radii of curvature p = 10 cm and reflectivity K = 95% at A = 6328 A. The main spectroscopic characU
I
FIG.34. An LDV scheme involving a confocal piezoelectrically scanned interferometer: 1, single frequency laser; 2-4, projecting optical elements; 5, optical filter; 6, confocal interferometer; 7, photomultiplier; 8, test flow; 9, scanning and 10, synchronizationblocks; 1 1 , amplifier; and 12, oscilloscope.
1.1.
TRACER METHODS
205
13
FIG.35. A scanning LDV scheme based on the synchronous detection method: 1 laser; 2, modulator; 3-7, 9, 13, and 14, optical elements; 8, confocal interferometer; 10, photomultipliers; 11, narrow band amplifier; 12, scanning system; 15, synchronous detector; and 16, recording system.
tenstics of this system are as follows: a free dispersion range A& = 750 MHz and an instrumental half-width ti& = 10 MHz, which, according to Eqs. (1.1.126) and (1.1.140), corresponds to a minimum measured velocity of about 10 m/s. A scanning pulse generator (9) allows a wide variation of spectrum scanning rates, thus permiting 7: to be varied from 10 to lo4 s. A further improvement of this should be aimed at an increase of LDV sensitivity since for fine particles ( d , = 1- 10 pm) at low concentrations it is difficult to obtain reliable results for processing, since the signal-to-noise ratio was less than one under these conditions. It is evident that in this case the scheme may be improved either by the use of more powerful lasers or by increasing the recording system sensitivity. The use of a synchronous detection p r i n ~ i p l e ~can ~ ~ *increase ~~' the signal-to-noise ratio. One such LDV scheme designed3Json the basis of the principle illustrated in Fig. 34 is shown in Fig. 35. A laser beam amplitude-modulated by means of an optical or a mechanical (2) modulator, at a modulation frequency of 18 kHz, was passed through a beam splitter (3), and a portion of the light was directed onto a photomultiplier (10) whose signal was used as the reference for a synchronous detector (15). The main signal, amplitude-modulated with the same frequency, arrived at the synchronous detector after passing a selective amplifier (1 1). The synchronous detector output was recorded with an oscilloscope or other recording device (16). The experiments performed on an aerodynamic setup have shown that this scheme may register a scattered light signal with a signal-to-noise ratio of about 0.1 at a signal level sufficient for data processing at particle concentrations naturally present in air. Another signal-selecting LDV system was developed for high tempera-
206
1. MEASUREMENT OF VELOCITY
ture flows investigation^.^^^ To separate a useful signal from a background jet radiation, the system employes an electronic scheme composed of a discriminator and a coincidence circuit. Finally, a scanning LDV development based on a single frequency 300 mW l a ~ e T 2should ~ ~ be mentioned. With radiation scattered by fine particles produced by the condensation in a moist gas, a signal-to-noise ratio of about one hundred was attained in this system. The reader will recall that one of the major problems encountered in the application of LDV schemes with direct spectrum analysis is the measurement of small velocities. When registering two signals, references and scattered or both scattered, on one channel, the minimum measured velocity Umlnis limited by the Rayleigh criterion A D = 6hD. In this case, Umlnmay be reduced only by decreasing the instrumental half-width 6Ao of the system, and this requires an increase of the interferometer spectral resolution. However, if the signals are recorded on different channels, Doppler shifts A D < SADmay be measured. In the simplest case, to avoid overlapping the two contours corresponding, for instance, to reference and scattered signals, at A D < S A D , two independent recording channels may be utilized simultaneously, i.e., two interferometers and photodetectors are required. Unfortunately, this considerably complicates the measurement system. With the aid of specific optical and/or electronics methods and techniques, a separate recording of two contours may be provided with a single spectral instrument. The schematic shown in Fig. 36 illustrates separation of two independent registering channels by optical means.363*364 Laser radiation 1, scattered by particles, was confined by diaphragms (6, 6 ’ )to propagate along two paths k I s and k”s and then, with the help of tilting mirrors, was directed onto the mirror faces of a prism (9). On being reflected from the prism, the radiation, corresponding to k’s and k”s, passed along two nonintersecting channels and is incident at an interferometer (1 1). A face of the prism (9) was sharply imaged by the objective (10) in the plane of registration of an interference pattern whose dimension was preset by the focal distance of the objective (12). Thus, each of the two beams of scattered light occupied half the interferometer and, therefore, illuminated the corresponding half of interference fringe rings (Fig. 36b). The total frequency difference for radiation scattered in Jb9
B. S. Rinkevicius and A. V. Tolkachev, Zh. Prikl. Spektrosk. 9, 748 (1968); J . Appl.
Sprcfrosc. (Engl. Trans/.)9, 1171 (1968).
A. P. Alkhimov, A. N . Papyrin, R. I. Soloukhin, and M. S. Stein, in “Laser Doppler Measurement of Gas Flow Velocities” (G.L. Grodzovsky, ed.), p. 142. Proc. N. E. Zhukovsky C. A. H. Institute, Moscow, 1976. am A . P. Alkhimov ef al., in “Proceedings of the All-Union Symposium on the Methods of Aerophysical Studies,” p. 46. Inst. Pure Appl. Mech., Novosibirsk, 1976. W. L. Kuriger, Proc. IEEE 57, 2161 (1969).
1.1.
207
TRACER METHODS
7 8
U Diaphragms to photomultiplieri
(bl
7 --7~\
Frmge shift direction
FIG. 36. LDV with recording of two scattered radiation beams on separate channels: (a) block diagram of experimental setup; 1 , single frequency laser; 2-4, 6 , 7 , 10, 12, optical, and elements; 5, test flow, 9, 14, rotatable prisms: 1 1 , interferometer; 15, IS’, photomultipliers: 16, control and synchronization scheme; and 17, oscilloscope. (b) schematic of interference fringes for plane-parallel and wedge-type interferometers.
k J Sand k J J directions S was defined for the present scheme geometry as oD = (2U/A) sin(a/2) sin cp
(1.1.141)
Fig. 36b gives a schematic drawing of the structure of interference fringes obtained by the plane-parallel (PIFP) and wedge-type (IFT) interferometers employed in the present scheme. When scanning a spectrum, the diaphragmed radiations are brought by the prism (14) to two photomultipliers (15, 15’), whose signals enter two recording channels (e.g., two oscilloscope traces). This allows measurement of small A D , which are less than the instrumental width 8ADof an LDV system. For decreasing the lower limit of a measured velocity, it is to use an electronic technique based on modulation of two reference beams with different frequencies. If the output from the photodetector is passed through a narrow band selective amplifier tuned to these frequencies, it is possible to separate the signals that correspond t o the spectra of scattered light from the two illuminating beams. It is shown that this method provides measurement of a Doppler shift which is approximately twenty times smaller than AD defined by a Rayleigh crite-
1.
208
MEASUREMENT OF VELOCITY
-
rion. The measured minimum velocity in this system was - 5 m/s at U,,, 5 x lo3 m/s. 1.1.4.5.3.3. LDV Systems with an Image-Converter-Scanned Spectrum. Another type of LDV employs a Fabry-Perot etalon as a dispersion element, but with a constant free dispersion range. The spectrum is then scanned and recorded by image-converter devices, permitting an increase in LDV time resolutions by several orders. In addition, this permits the use of a steady state interferometer, which has been found both easier to design and to use compared to the scanning type, particularly in gas dynamic measurements. One LDV system is that incorporates a scanning spectrometer consisting of a “static” interferometer and a so-called double converter tube (DCT) composed of an electron multiplier system united in a single tube with an image converter system. Figure 33c schematically shows this spectrometer system. It includes a multiple-beam interferometer (3), a DCT (6), a power supply and control block (4), a recorder (7), as well as projection optics (2). To increase the resolution of and illumination in an instrument, it is more expedient to use in combination with the DCT an etalon with tilted mirrors (IFT), whose interference pattern represents a system of straight parallel fringes in the form of lines of the same thickness. Figure 37 demonstrates an LDV system based upon this principle. A test radiation spectrum produced by the interferometer (6) was projected on the photocathode (1) of the double converter tube (7), which is a combination of an image converter (IC) and an electron multiplier (EM). An electron image of the interference pattern, analogous to the light one, is converted by an electrostatic lens (3) into the diaphragm plane. The moving image is recorded by means of a slit ( 5 ) having a width of 80 pm. When adjusting the instrument, the interference fringes were made parallel to the slit (Fig. 37). Behind the slit, a dynode system of an electron multiplier (6) and a collector (7) were located, and the spectrum was scanned by a sawtooth voltage applied to the electron beam deflection plates (4). An electric signal from the collector was registered by an oscilloscope (10). A scanning generator (8) ensured a wide time scale in spectrum recording (10-10-6 s). The major parameters of the double converter tube used in the present system, e.g., integral sensitivity, temporal and spatial resolutions, as well as their dependences upon supply voltages have been i n v e ~ t i g a t e d . ~ ~ ~ , ~ ~ ’ 365 A. P. Alkhimov, V. A. Arbuzov, A. N . Papyrin, R. 1. Soloukhin. and M. S. Stein, Fiz. Goreniya Vzryvu 9,585 (1973); Comhusr. Explos., Shock Wuves (Engl. 7 r u n s . ) 9, 507
(1975).
E. P. Kruglyakov, in “Plasma Diagnostics.” Atomizdat, 1968. L. M. Diamant, A. M. Iskol’dskii, M. I. Kudryashov, and Yu. E. Nesterikhin, TeploJiz. V y s . Temp. 8, 163 (1970); High 7 e m p . [Engl. Trunsl.) 8, 151 (1970). 3BB 367
1.1. TRACER METHODS
209 7
Sweep dwxtmn t
FIG.37. LDV system involving a static wedge-type interferometer and a double image converter tube (DICT): 1, moving object; 2-5, optical elements; 6, interferometer; 7, DICT; 8, sweep generator; 9, power supply system; 10, recorder; 11, laser; and 12, mirror. Locations of the interference fringes and the DICT internal slit are shown schematically at the bottom.
A proper choice of optimal relationship between geometric dimensions of a slit and a contour width of a test signal should be provided.3m In particular, if a Lorentz line profile is analyzed, the Fabry -Perot interferometer contour may be approximated as I(x) = Zo[l
+ (x/x,)”]-’.
( 1.1.142)
The constants are chosen so that at x = ?xo and I = 0.5Z0. The time profiles of the electron current behind the slit can be obtained for different ratios of a line xo to a transverse dimension of a registering slit in the double converter tube ( K = xo/d). It is shown that there is no need for K =z 0.5, since in this case the resolution gain is insignificant, while the instrument illumination falls markedly. It is likely that an image-converter Fabry-Perot spectrometer based on a vidicon (a television image tube),368as shown in Fig. 33d, might be useful for LDV systems. This spectrometer allows both visualization of an interference pattern and photoelectric recording of a test spectrum (the recording time of an interferometer fringe contour is of about s). Finally, let us consider briefly one important circumstance associated with the practical use of multiple-beam interferometers. It is a known fact that interferometers with mirror spacing t 3 10 cm require a particu-
368
J . M. Gagne, J . Bures, and N . Laberge, Appl. Opt. 12, 1894 (1973).
2 10
1. MEASUREMENT OF VELOCITY
larly fine alignment which, to a great extent, specifies the real resolving power of an instrument. That is why for LDV systems operating in gasdynamic experiments, the protection of the interferometer from vibrations and acoustic noise is very important. Figure 38 and 39 present two types of multiple-beam interferometers.355*365 A static interferometer with plane mirrors, which can be set in parallel (PIFP) or at a small angle to each other (IFP) is shown in Fig. 38. The main advantage of this construction is the rigid support of the mirrors by threaded rings (3) permitting elimination of mirror vibrations at the points of contact with the interferometer body. The necessary alignment was done by rollers, incorporated in the body (2), which ensured sufficient rigidity of the instrument as a whole. The layout of a scanning spherical interferometer is shown in Fig. 39. Mirrors were rigidly mounted on rollers (one of the mirrors is fastened via a piezoelectric cylinder), which provided the instrument adjustment. One of the rollers could be displaced along three guides (2) made of invar rods that allowed a smooth variation of the plate spacing
FIG.38. General view of a static interferometer with plane mirrors: 1, roller; 2, body; and 3, clamping rings.
1.1.
TRACER METHODS
21 I
FIG.39. Layout and a general view of a scanning confocal interferometer: 1 , roller; and 2, guides.
within t = 5-20 cm. A spectrum was scanned by means of a ceramic piezoelement in response to a sawtooth voltage, V = 400 V. To protect the interferometer from acoustic disturbances, it was placed in a noiseabsorbing housing. Note that the above interferometers ensured LDV stability so that they could operate without additional alignment between experimental runs with a noise level at the measuring equipment site of about 120 dB. 1.1.4.5.4. SERVOLDV SYSTEMS. The LDV systems incorporating scanning interferometers suffer a common drawback, i.e., a presence of dead time in measurement processes (a discrete character of reading) since the test spectrum region, equal to that of the free dispersion of an interferometer, is always chosen to be greater than the scattered radiation contour width. In processes in which the fluid velocities vary rapidly, LDV systems that can ensure continuous tracking of time-varying particle velocities in a given flow region have received increased attention. One of the ways of realizing “instantaneous” velocimeters relies upon the use
212
1. MEASUREMENT OF VELOCITY
of multiple-beam interferometers operating in an optical filter regime that provides a linear dependence of transmission Illo upon a Doppler frequency shift oD,and, therefore, particle velocity U. A schematic of the detecting arrangement of an LDV of this type is presented in Fig. 40. Let scattered laser radiation with intensity lo and wavelength A = A. + A D , where AD = K ~ Ufall , on an interferometer filter ( 4 ) ( ~is~specified only by the LDV geometry). Then at the filter outlet (Illo = K Z A D , with I being the constant coefficient), the light intensity is I = KZAdo = K ~ K ~ I ~ U .
(1.1.143)
Directing the radiation to a photodetector ( 5 ) that has a linear sensitivity characteristic V1 = K$, an electric signal may be obtained with an amplitude v1
=
K$
= K1KZkSIoU.
(1.1.144)
To eliminate the dependence of the output signal amplitude upon fluctuations of the scattered radiation intensity I o , use can be made of an additional channel consisting of another photodetector (7) and electronics unit (8) separating signals from the outputs of both photomultipliers ( 5 , 7). The photomultiplier (7) receives some portion of the incident scattered radiation directed to the optical filter (4). In this case, a signal amplitude at the photomultiplier output (7) equals
vz = d
o ,
(1.1.145)
where K~ is the coefficient depending upon reflectivities of plates (3) and (6) and photomultiplier sensitivity. Then a signal from the separator (8) is of the form
FIG.40. Block diagram of a servo LDV: I , source of scattered radiation, 4 + AD; 2, 3, and 6, optical elements; 4, multiple-beam interferometer in an optical filter regime with linear transmission, I l l , = AD; 5, 7, photodetectors; and 8, signal separation system.
1.1. TRACER METHODS
Interference fringes
213
Slit
\ I
FIG.41. Realization principle of an optical filter with linear transmission I / I , = K A ~ . (a) Displacement of an individual interference fringe is recorded within a free dispersion region of an interferometer. (b) Displacement of a spectrum part within a linear region of the frequency characteristic of an interferometer (the optical discrimination method). AX, to the fringe shift; AD, the Doppler shift; and AA,, the free dispersion region of an interferometer.
v = v,/vz
=
KlKZKsU/K4 = KU.
(1.1.146)
Thus, the output signal amplitude in the present instrument is a linear function of particle velocity. The value of the coefficient K is defined only by the velocimeter parameters and does not depend on the scattered radiation intensity and, therefore, on the properties of the scattering particles. A numerical value of K may be found via LDV calibration, and upon defining V by the present detector system, velocity U may be found at any instant of time. Here, a high time response is specified only by the temporal resolution of the photodetector and the signal separation unit. Figure 41 illustrates the principles of realizing an optical filter with Z/Zo = K A D . Variations in wavelength A of the radiation incident on an etalon yields a spatial shift of the interference fringes. If in the plane of observation of the interference pattern, a diaphragm of a certain configuration is located, a relationship Z AD may be provided for the diaphragmed radiation intensity. This may be most easily obtained with
-
2 14
1.
MEASUREMENT OF VELOCITY
the help of interferometers whose linear dispersion dl/dA does not change with transmission from one order of interference to the other, since a fringe shift in this case is a linear function of A D . This condition is satisfied, as mentioned above, for example, with a defocused spherical Fabry -Perot etalon (DSIFP) or with wedge-type plane mirrors (IFT). Linearity of transmission may be achieved either by (a) visualization of an individual fringe shift by a special diaphragm within a range less than the free dispersion of an etalon Aho (Fig. 41a), or (b) by selection and recording of a spectrum region with 6h d aho, i.e., much less than the instrumental width of an etalon, so that a linear region of the frequency characteristic of the interferometer is selected as shown in Fig. 41b (the method of optical discrimination of Doppler frequency shifts369). With an interferometer pattern produced by a Fizeau -Tolansky etalon (Fig. 41a), the first principle can be readily used by setting a triangular diaphragm positioned at the etalon outlet. Then the transmitted light intensity I will be a linear function of A D . This method allows measurement of Doppler shifts AD G Aho, however, to accomplish this, a high sharpness of fringes (AXo/6ho)should be ensured for the given Aho. A schematic drawing of the LDV system based on this principle is presented in Fig. 42a. It comprises a single-frequency 5 mW helium-neon laser, a beam splitter (2) for the system calibration, a focusing lens (3), receiving optics (4-6), an interferometer with wedge-type plane mirrors (IFT), a diaphragm (8), an optical divider (9), two photodetectors (10, ll), an electronic unit for signal processing (12), and a recorder (13). If the scattered light incident on the interferometer has experienced a Doppler-frequency shift, interference fringe is shifted by AX
= (?h/QC)hD =
uXD
= K'U,
(1.1.147)
where cp and t are the angle and distance between mirrors, C is the velocity of light, and K is a constant dependent only on the instrument parameters. To detect an interference fringe displacement, use was made of a rectangular diaphragm (Fig. 42a) divided along a diagonal line with rotatable prisms placed together, base to base, so that the part of the radiation passing, for example, through the upper triangle, enters the first photomultiplier while the other reaches the second one, both operating in the same regime. The interference pattern displacement yields a redistribution of radiation fluxes incident on the photodetectors. Here, as is seen in Fig. 42a, the amplitudes of the photomultiplier output signals may be preP. Ya. Belousov, Yu. N. Dubnishchev, and V. A. Pavlov, Opr. Spektrosk. 43, 775 (1977); Opt. Specrrosc. (Engl. Transl.) 43,457 (1977).
1.1.
215
TRACER METHODS
. kl )
13
12
FIG.42. Servo LDV scheme involving a wedge-type interferometer. (a) 1, laser; 2, beam splitter; 3, focusing lens; 4-6, receiver optics; 7, wedge-type interferometer; 8, diaphragm; 9, optical divider; 10, 1 1 , photodetectors; 12, signal processing system; and 13, recorder. (b) A general view of an installation based on a wedge-type interferometer.
sented as V1 =
KL(U/2
- AX)Zo,
(1.1.148)
where u is the square diaphragm width; Zo , the scattered radiation intensity; and K; , the coefficient depending upon the photomultiplier sensitiv-
216
1. MEASUREMENT
O F VELOCITY
ity. Then expressions for total and difference signals are of the form
v+ = v1 4-
v2
=
UK2lo
V- = Vi - Vz
=
2 AX
(1.1.149) KJo.
( 1.1.150)
Taking the ratio of these signals, V-/V+, we arrive at
v = v-/v+ = 2 h x / U
= KU.
To calibrate the system, i.e., to determine K , with the aid of a stationary radiation source, a mechanical arrangement (Fig. 42b) was used which ensured a smooth diaphragm displacement (to a precision of 10 pm) parallel to a registered fringe in the plane of observation of the interference pattern. With a knowledge of the geometrical distance xocorresponding to a free dispersion region Aho, a value of h / x 0 could be found for each output signal V. On the other hand, for the present system geometry it appeared possible to calculate A x / x = f(hD) and, therefore, to define K relating the output signal amplitude with a Doppler frequency shift, i.e., K = VIA,. It is evident that a moving object, for instance, a rotating disk, with its speed being found independently, can also be used for calibration. Operation of the present recording system is illustrated by oscilloscope traces (Fig. 43a) which are obtained using a frequency shift simulation
U (rn/sl
(b) FIG.43. Experimental data obtained by a servo LDV using a wedge-type interferometer. (a) Signals from a tunable laser: the upper trace, V = V , - Vr; the lower trace, V = V , + V , . (b) Measured linear velocity of a rotating disk: solid curve, predicted relationship AX/Xo = f ( U ) ;circles, experimental data. cos 0 = 0.68, I = 5 cm.
1.1.
TRACER METHODS
217
with a tunable laser, one of its cavity mirrors being fixed on a piezoelectric cell displaced at a prescribed frequency. The upper trace shows the difference signal V = Vl - V,, while the lower trace shows the sum V+ = Vl + V,. Small variations of the lower signal (this channel sensitivity is ten times less than that of the upper) are due to the amplitude modulation of radiation observed together with the frequency modulation. Figure 43b gives a plot of A x / x = f(v), obtained in laboratory experiments with a rotating disk. Here, the solid curve is the predicted relationship, while the circles, the experimental data obtained with the help of the present LDV. The interferometer was operated at the plate spacing t = 5 cm and finesse F x 20. As is seen, this system provides velocity measurement with an error of no more than 1-2% starting with U = 15 m/s, its time response with the oscilloscope recording of the two signals V- and V+ being restricted only by temporal resolutions of the photodetectors. Similar LDV systems incorporating a defocused spherical interferometer were developed for turbulence parameter measurements in high speed gas f l o ~ ~ The . ~interferometer ~ ~ * ~ ~parameters, i.e., free dispersion range Afo = 500 MHz and finesse F 30, allowed a velocity resolution of about 3 m/s to be attained.360 Consider now the essential features of the second method of obtaining an outlet beam intensity, which is a linear function of the Doppler frequency shift. This method makes use of a linear region of the frequency characteristic of the interferometer. To do this, a slit [see Fig. 41(b)] is located at a point in the plane of observation of the spectrum corresponding to a frequency ol. If the interferometer transmission function is represented by a Lorentz profile,360then o1 = oof A/3lI2, where A is a constant defined by the reflection and transmission coefficients of the mirrors. For frequency shifts that do not exceed ? 20% of the half-width of the Lorentz line, the error due to assuming that the transmission function of the spectrometer is linear will not exceed 3%. This technique is especially suitable for measuring small velocities of about 1- 10 m/s, since these measurements are performed within the limits of a conventional instrumental contour. The measured velocity ranges may be varied by changing the profile width of the interferometer instrument function. An LDV with a spherical etalon used as an optical discriminator allows velocity measurements in the range 0.1- lo3 Its limitation is a low time response T = 0.25 x lo-, s due to the transmission band of the automatic frequency-control system. In addition to servo LDVs based on optical filters with linear transmission I/&, AD, another approach is based on the continuous observations of a scattered radiation spectrum by streak recording of an interference image. It should be noted that single-cascade electro-optical image con-
-
218
1.
MEASUREMENT OF VELOCITY
verters or opto-mechanical streak cameras are not applicable under real conditions of gasdynamic processes because of their small luminosity. Thus, in this case multicascade image converter amplifiers, with a gain of lo7, appear to be most p r o m i ~ i n g . Figure ~ ~ ~ ~ 44 ~ ~illustrates ~ the operating principle of this LDV. The reference and scattered laser beams formed by optical systems (2-7) are brought to a Fabry-Perot interferometer (8). An interference pattern is projected by the objective (9) on the photocathode of a five-cascade electron optical image converter system (11). A vertical slit (10) cuts a narrow band of the interference fringes. Scanning in time [through the sweep generator (12)] is carried out in the horizontal direction. Figure 44b depicts schematically the pattern obtained (a streak record) for two orders of interference positioned vertically. Nondisplaced horizontal lines refer to a reference radiation, while the displaced lines refer to a frequency shifted scattered radiation. By photometric evaluation of the light intensity of the interference pattern, the Doppler shift AD may be found. In addition, unlike the servo LDVs considered above, the scattered radiation spectrum profile may be obtained. Furthermore, there is no need for a velocimeter calibration, since the scattered and reference radiations are recorded simultaneously. It is well known that Doppler spectrum broadening may be attributed to certain physical processes in a test flow (e.g., presence of velocity gradients and pulsations, polydisperse particle velocity distributions) and to instrumental broadening in the optical system (see Section 1.1.4.4.5). The latter depends on the Fabry -Perot interferometer and the angular scatter of the observed radiation, and on the spatial resolution of the image converter system. The instrumental profile of an image converter device can be approximated by a Gaussian curve372whose half-width SAI is related to the resolving power of the image converter tube as
-
6x1 = (S/xo) A&,
(1.1.151)
where xo is the geometric distance corresponding to a free dispersion range AAo. A profile preset by an interferometer is described by Airy’s function and may be approximated by a Lorentz curve with a half-width S&, = Aho/F, where F is the interferometer finesse. With characteristic parameters S = 0.1 mm, x = 10 mm (the working section dimension of a photocathode), and F = 50, 6h1can be approximated to be of -0.5 aho. 3’oA. M. Iskol’dsky, Yu. E. Nesterikhin, A. N. Papyrin, and A. G. Romanenko, in “Plasma Diagnostics.” Atomizdat, 1968. J71 Yu. E. Nesterikhin and R. I. Soloukhin, “Rapid Measurement Methods in Gasdynamics and Plasma Physics.” Nauka, 1967. 372 S. P. Zagorodnikov, G. E. Smolkin, and G. V. Sholin, in “Plasma Diagnostics.” Atomizdat, 1968.
1.1.
219
TRACER METHODS
0 14
FIG.44. A continuous LDV scheme based on interference pattern scanning: 1, laser; 2-7, 9, 10, optical elements; 8, interferometer; 11, five-cascade light amplifier; 12, scanning system; 13, power supply; 14, recorder.
FIG.45. General view of an arrangement with a five-cascade image converter system.
220
1.
MEASUREMENT OF VELOCITY
A general view of an installation tested in laboratory conditions is shown in Fig. 45. The parameters of the Fabry-Perot interferometer are as follows: the plate spacing t = 10 cm, and the finesse F = 50. A five-cascade image converter tube with light gain of lo7 was utilized. The image sweep can be varied from several seconds per centimeter of the screen up to 1 &cm. The linear velocity of a rotating disk was measured within the range 50-200 m/s. A measurement accuracy less than 5% was obtained, being mainly associated with the photometric evaluation of a spectrum record. It should be noted that for continuous observation of the scattered radiation spectrum, use can also be made of multiple-channel detection techniques, e.g., with the aid of a system of mirrors or fiberoptic guides that bring radiation from separate sections of a line contour to several photomultipliers. At present these methods have been successfully applied, e.g., in plasma diagnostics for the measurement of scattered laser radiation*373.374 1.1.4.5.5. DIRECTSPECTRUM ANALYSIS LDVs I N GASDYNAMIC ExPERIMENTS. CONCLUDING REMARKS.Having reviewed LDV data and Doppler shift processing methods, it is appropriate to consider the type of studies that can be carried out in gasdynamic flows. These may be subdivided into two characteristic groups. The first group comprises typical aero- and gasdynamics problems aimed at determining the velocity profiles in some particular flow region, and measuring the velocity fluctuations. In this case, naturally occurring particles or those introduced into a flow act as scattering centers. In an ideal case, the particles must be very small and of the same size to be completely in equilibrium with a gas flow. The main task of these experiments is measurement of a mean velocity fr and velocity fluctuations u as well as a turbulence level3"
-
E
=
(ii2)1'*/0
= KSOf/WO,
(1.1.152)
where oois the frequency appropriate for I / , and 6of is the spectrum broadening due to velocity fluctuations. In high speed flow measurements of E and CJ, both schemes may be successfully applied, i.e., servo LDVs which relate the output signal amplitude fluctuations to the velocity fluctuations and scanning systems that provide 6wf through measurements of the broadened spectrum width 6oD. Investigations of these problems have revealed that Doppler spectrum broadening is attributed mainly to two factors: broadening due to the flow A. N. Zaidel and G . V. Ostrovskaya, "Laser Methods in Plasma Investigations." Nauka, L, 1977. w4 J. Katzenstein, Appl. Opt. 4, 263 (1%5).
1.1.
TRACER METHODS
22 1
velocity fluctuations and instrumental broadening. The total spectrum of a measured signal from a turbulent flow may be given as375
where F ( v) is the particle velocity distributions function, and the expression in brackets is the instrument function of the measurement system, whose width is directly proportional to velocity. Then the determination of the real profile, whose width is defined by velocity fluctuations, is reduced to the elimination of the instrument function from the measured spectrum. A procedure based on the statistical regularization method37s is elaborated to permit calculation of F ( U ) when its width is comparable The instrumental width of an with the instrument function of a interferometer as well as of the whole LDV may be determined separately, e.g., with the use of a signal spectrum in a laminar flow. When the range of particle sizes is very wide and the spectral broadening is correspondingly large, servo LDVs show greater promise, since their signal amplitude does not depend on the scattered radiation intensity. Moreover, the output signal of a servo LDV is more suitable for computer data processing. This is of significance in aerodynamic experiments when measurements must be made at several positions in each operation cycle of an experimental arrangement. The other class of problems deals with multiphase flow investigation^^^^ associated first and foremost with the physical processes of unsteady (gasparticle interactions), namely, the effect of velocity lag of particles and its dependence on various factors, e.g., density, size, and concentration, and particle interaction effects yielding, for instance, collisions, coagulations, and breaking, and as a result, a change in their size distribution function
44. Among the important problems arising in two-phase flow investigations, e.g., in the flow in rockets that use a solid propellant with metallic components, is378the determination of the size and velocity of particles that differ greatly in their disperse compositions. It is evident that scanning LDVs are the most promising schemes for solution of these problems, since they record a spectral function S(o) of the scattered radiation intensity fluctuations which exhibits information on particle sizes 375 N . G . Preobrazhensky, A. L. Rudnitsky, and B. G . Tamboutser, in “Gas Dynamics and Physical Kinetics,” p. 162. Inst. Pure Appl. Mech., Novosibirsk, 1974. s‘ N. G. Preobrazhensky and V. V. Pikalov, in “Optical Spectroscopy Techniques.” Moscow Univ. Press, Moscow, 1977. 377 S. Saw, “Hydrodynamics of Multiphase Systems,” Mir, Moscow, 1971. ‘“R. Hogland, Raker. Tekh. Kosrn. 32 (51, 513 (1962).
222
1. MEASUREMENT OF VELOCITY
and velocities. With polydisperse particle systems characterized by considerable particle velocity lag, the Doppler spectrum broadening will be attributed not only to the velocity fluctuations and instrumental broadening, but to the particle velocity distributions. In particular, a problem of importance is the extraction of the particle size distribution function n(r) from the observed Doppler spectrum data S(w). With no particle interaction, a relationship is obtained that relates S(o) to n(r)379:
where p(r, k) is the light scattering indicatrix and p( U / r )is the probability of a particle of radius r having a velocity u . The problems related to the determination of the particle size distribution function for a laminar flow within the Stokes drag regime,379and turbulent jets3@’are treated in the literature. It should be noted that further improvements of the Doppler spectrum processing procedure and, in particular, elaboration of the solution to the so-called inverse as it applies to Doppler spectroscopy are expected to extend significantly the LDVs’ capabilities in gasdynamic experiments. Finally, some experimental results obtained by LDVs with a direct spectrum analysis will be described. Consideration is given in the literat ~ r eto ~the~gas~-particle , ~ ~relaxation effects and particle velocity lags in a steady supersonic wedge-type nozzle flow (M = 2.8). Two-phase flows of different particle sizes and densities were investigated in the experiments. Gas and particle velocities were measured by an LDV involving a confocal piezoelectrically scanned interferometer (Fig. 34). Gas flow velocities were evaluated from the light scattered by smoke particles simultaneously introduced into the flow with the test particles. Figure 46 gives typical oscillograms to illustrate the Doppler frequency shift oDand the scattered radiation spectrum width versus parameters of particles introduced into the flow, i.e., their mean diameter dpand material density pp, as well as particle size distribution. Figure 47 shows the velocity distributions of the gas and different particles along a nozzle
[email protected]. Ivano, A. Ya. Khairullina, and A. P. Chaikovskii, Zh. Tekh. Fiz. 44, 429 Phys., (Engl. Transl.) 19, 267 (1974). A. Ya. Khairullina and A. P. Chaikovskii, Zh. Tekh. Fiz. 45, 689 (1975); Sov. Phys. --Tech. Phys. (Engl. Transl.) 20,435 (1975). s*l A. P. Alkhimov, A. N. Papyrin, and A. L. Predein, in “Physical Gasdynamics,” p. 83. Inst. Pure Appl. Mech., Novosibirsk, 1976. A. P. Alkhimov, A. N. Papyrin, A. L. Predein, and R. I. Soloukhin, Zh. Prikl. Mech. Tekh. Fiz. No. 4, p. 80 (19771, J . Appl. Mech. Tech. Phys. (Engl. Transl.) 18,4% (1974); Sov. Phys. -Tech.
(1977).
1.1.
223
TRACER METHODS
800
400
0
FIG. 46. Typical LDV spectra obtained for different particles accelerated in a supersonic nozzle flow, M = 2.3. Measurements refer to one distance from the nozzle throat, x = 15 mm. Particle characteristics are as follows:
Density, g/cm3 Diameter range, p m Average diameter, p m
1
2
3
4
5
6
0.5 22-28 25
2.7 1-25 15
1.2 50-350 200
8.6 5-40 25
8.6 40-120 80
8.6 160-240 190
(on-axis). Measurement at each point gave the Mach number M = ( U Ur)usand the Reynolds number Re = pd(U - Ur)/p as well as the particles' drag coefficients (1.1.155) With M = 0.6-2.1 and Re = 4 x 102-2 x lo4, the obtained Cd values were within 0.6-2.1 increasing with M. Experimental data3= on the effect of particle concentrations on their 585 N . N. Yanenko, R. 1. Soloukhin, A. P. Alkhimov, A. P. Vorozhtsov, E. V. Papyrin, and V. M. Formin, Reprint No. 2. Inst. Pure Appl. Mech., Novosibirsk, 1978.
224
1.
600-
MEASUREMENT OF VELOCITY
CI
-7
+ -8
0
10
20
30
40
X (mml
FIG.47. Gas and particle velocities versus length along a nozzle: 1-6 correspond to particle characteristics given in Fig. 46; 7 and 8, gas flow velocity; the solid curve, the predicted “pure” gas velocity distribution along a nozzle; 7, velocity of smoke particles in a dusty flow (LDV method); and 8, gas velocity in a “pure” flow (Pitot tube).
velocity show that with the attainment of volumetric concentrations in the nozzle throat of Q 3 0.5%, obstruction effects arise in the two-phase flow, which involve significant deviations of the gas parameters and particle velocities compared to that under flow conditions typical for single particle flow regimes. A comparison of numerical calculations with the experimental data allowed the domain of applicability to be established for some mathematical models describing flows of a medium that consists of a suspension of powdered material in a gas. It should be mentioned that a laser Doppler technique based on scanning spectrometers may be successfully used for investigation of moving particle interaction effects in a flow with different particle velocities at high disperse phase concentrations. To illustrate this, Fig. 48 presents oscilloscope records ( 1 -6) obtained when light is scattered by particles of two kinds (bronze particles, for which dp = 80 pm and pr 2 8.6 g/cm3, and lycopodium powder particles, for which d,, = 25 pm and pr = 0.5 g/cm3 simultaneously introduced into a flow. Here, it can be clearly seen that the variation of Doppler frequency shifts and scattered signal spectrum profiles depends on the concentration of the introduced particles. At Q C 0.1% (oscilloscope traces 1 and 2), two signals are distinctly detected, corresponding to the velocities of the two different types of particle: U’ = 410 m/s and U” = 190 m/s. With increasing concentration of both kinds of particles, in addition to a drop of each component mean velocity, one observes a considerable spectrum broadening of velocities for the light lycopodium particles as a result of their collisions with slower
1.1.
225
TRACER METHODS
FIG.48. Oscilloscope traces illustrating variation of a scattered radiation spectrum and Doppler frequency shift with increasing particle concentration for particles of two kinds, simultaneously introduced into a gas flow; bronze particles (ifp 80 p m , p = 9.6 g/cmJ), and iycopodium particles (d, 25 p m , p = 0.5 g/cm3).
-
-
bronze particles. With volumetric concentrations of cp B 2% in the nozzle throat, as is seen from oscilloscope traces 5 and 6, this spectrum becomes continuous and occupies the whole region from U’ to U ” . Figure 49 shows the experimental data on variations of different particle velocities with their passage across a standing plane shock wave ( M = 2.8) generated in a supersonic nozzle section in an expanded flow regime.382.384It is easy to see that in the laboratory coordinate system, the velocity of light particles, e.g., smoke or polystyrene, behind a shock wave sharply decreases, while that of bronze particles having the largest d and pr is practically unchanged. From the deceleration of the particles in the relaxation the particle drag coefficient was determined to be Cd = 0.9 at M = 0.6 and Re = lo3. In addition, sizes of the smallest smoke particles were found to be less than 0.2 pm. Measurements of the gas velocity profiles in a turbulent boundary layer, in the presence of the strorlg velocity profile distortions occurring in 3u A. P. Alkhimov, A. N. Papyrin, and A. L. Predein in “Physical Gasdynamics,” p. 40. Inst. Pure Appl. Mech., Novosibirsk, 1976. sBs G. Rudinger, in “Nonequilibrium Flows” (P. P. Wegener, ed.), Vol. I , p. 119. Dekker, New York, 1%9.
226
1.
MEASUREMENT OF VELOCITY
c
+-I 0-
2
0-3 0 - 4 x -5
X
0 10
x
x
20
15
x
X
25
X Imm)
FIG.49. Velocity variation of different particles with transition through a standing plane shock wave (M = 2.8; Po = 0.81 MPa; To = 260°K): 1, gas flow (prediction); 2, smoke particles; 3, polystyrol particles, d,, = 1-3 p m ,p = 1.2g/cm3;4, licopodium particles, d , = 25 pm, p = 0.5 g/cm3; and 5, bronze particles, d = 50 p m ,p = 8.6 g/cm3. A thickness of the shock wave oscillations zone is noted.
a separation region, as well as observations of the circulation flow development were performed in a series of supersonic wind tunnel experiments conducted with the use of an LDV involving a pressure-scanned Fabry-Perot i n t e r f e r ~ m e t e r .The ~ ~ ~literature describes the use of LDVs for the investigation of supersonic high temperature two-phase flows in a rocket nozzle that uses a solid propellant with metallic components. The velocity of the particles was measured at a section of the nozzle, and their size was evaluated from the predicted relationship U, =f(d,) to be of 3-6 pm for different experimental conditions. Finally, the turbulence parameters in a subsonic jet were measured by using a defocused spherical interferometer based servo LDV Measurements were made of the turbulence intensity distribution along the jet axis, and the results obtained were shown to be in good agreement with the data obtained by a hot wire anemometer. Thus, the experimental results discussed testify to the many uses and capabilities of laser Doppler velocimeters with direct spectrum analysis in various aero- and gasdynamic experiments. As for its technical use, the present method does not require complex electronic equipment; it is rather simple and may successfully supplement the classical photomixing based LDVs when measuring relatively high flow velocities, the latter still being applicable at velocities of the order of 1- 10 m/s. Further progress of velocimeters with direct spectrum analysis will involve the development of the theory and technique of the method and a J.
M.Avidor, AIAA J . 13, 713 (1975).
227
1.1. TRACER METHODS
data processing procedures. In particular, this involves an analysis of the limitations of the method with respect to sensitivity and accuracy of measurement, data handling by electronic computer, and the improvement of servo LDVs and of those with photoelectric spectrum registration on the basis of the image converter scanning systems providing high time responses. Solution of these problems will promote further elaboration and improvement of the laser Doppler measurement techniques. 1.1.4.6. Comparison of Signal Processing Methods. The selection of a signal processing method among the available optical mixing methods and the Fabry-Perot technique depends on a number of factors, which are, in order of importance (a) concentration of flow tracing particles, (b) signal quality (noise, dropout, level of scattered fight), (c) anticipated range and rate of variation of signal frequency (i.e., fluid velocity), (d) character of fluid flow (steady or turbulent), (e) statistical stationarity of the signal, (f) required precision of the data, (g) required character of the processor output (digital, analog, real time), and (h) cost. The relation of the preceding to the different signal analysis methods is summarized in Table XXIV. Some aspects of the problem of choosing a signal processing method are considered in the practical example given in Section 1.1.4.9. TABLEXXIV. Signal Processing Methods Technique
Advantages
Disadvantages
Swept oscillator 1. Modest cost. 1 . Inefficient signal processor, unless modified. wave analyzer 2. Can obtain satisfactory data 2. Output is not in real time. with noisy signals and in the 3. Measurements subject to presence of signal dropout. 3. Useful diagnostic tool for setting observer error because instruup a velocity measuring system. mental techniques not avail4. Permissible range of Doppler able for data interpretation. frequencies oD:10 Hz-500 MHz. 4. Ambiguity noise decreases precision of measurement. 1. Perfectly efficient data I. Maximum velocity fluctuation Tracking bandpass filter processor. limited by maximum frequency slew rate. 2. Real-time output proportional to instantaneous particle velocity, 2. Requires high concentration of flow tracing particles because allowing rapid accumulation of sensitive to signal dropout data and analog methods of (this, together with item 1, analysis. tends to restrict the tracking 3. Good broadband low level noise bandpass filter to measurerejection characteristics within ments in liquids). limitations imposed by maxi3. Ambiguity noise decreases mum frequency slew rate. precision of measurement. 4. Permissible range of Doppler 4. Limited dynamic range. frequencies oD:1 kHz-50 Hz. (continued)
TABLEXXIV (continued) 5 . Input must have a high SNR (15 dB) so that errors due to
Frequency counter
1. Perfectly efficent data processor. 1. 2. Real-time output proportional to instantaneous particle velocity, allowing rapid accumulation of data and use of analog 2. methods of analysis. 3. No dynamic range limitations. 4. Permissible range of Doppler 3. frequencies oD:dc-5 0 0 MHz.
Period counter
1. Used with low concentrations of
2. 3. 4. 5.
Photon counting 1. correlator
2. 3.
Fabry -Perot interferometer
1.
2. 3.
flow tracing particles, e.g., gas flows, because insensitive to signal dropout. No dynamic range limitations. Minimum signal degradation due to ambiguity noise. Output suitable for digital computer manipulation. Permissible range of Doppler frequencies oD:1 kHz200 MHz. Useful for very low scattered light levels, e.g., where naturally occumng particles are used for flow tracing, where light is backscattered from the flow tracing particles. Output suitable for digital computer manipulation. Wide frequency response (few Hz- 10 MHz). Can process signals from very high velocity flows (OD up to 300 MHz). Excellent spatial resolution. No diredtional ambiguity.
1.
2.
3. 4. 5.
large amplitude narrow band noise are avoided. Requires high concentration of flow tracing particles because sensitive to signal dropout (tends to restrict processor to measurements in liquids). Ambiguity noise decreases the precision of the measurement. Input must have a high SNR so that errors due to large amplitude noise are avoided. Poor broadband noise rejection chaacteristics. Not suitable for high particle concentrations because there is an upper limit on the allowable data rate. Data validation is essential. Data can be biased. Processor more expensive than tracking bandpass filter.
1. Interpretation of turbulent
flow data subject to considerable uncertainty. 2. Careful selection of photomultiplier tube for maximum sensitivity and minimum noise. 3. Long processing time. 4. Output not in real time. 1. Unable to resolve small
changes in particle velocity. 2. Output is not in real time, unless interferometer is modified. 3. Inefficient signal processor. 4. Measurements subject to observer error because instrumental techniques not available for data interpretation. 5 . Ambiguity noise decreases precision of measurement.
1.1.
TRACER METHODS
229
1.1.4.7. Signal Analysis. The output from the signal processor must be subject to further manipulation to obtain the parameters that are of interest in fluid mechanics. For laminar flow, the parameter of interest would be the fluid velocity. Under turbulent flow conditions (wD= WD + w;), a number of different statistical quantities are usually required. These are listed in Table XXV,which summarizes the techniques of signal analysis; the definitions of these quantities are written assuming that the output from each processor is of finite duration T. The processor output is in the form of either (a) a real-time signal proportional to the instantaneous frequency w , of the photodetector output, or (b) spectral information about the photodetector signal (indicated in Table X X V by an asterisk in the first row of the table). Processors of the second type can supply only a limited amount of information about the fluid velocity. Thus they cannot be used to measure the various turbulent flow correlations, or to determine the power spectral density of the fluid velocity fluctuations. Accordingly, an asterisk has also been inserted in the table at entries corresponding to those quantities. In addition, such processors cannot be used to study transient flow effects. This difference is acknowledged to a certain extent in Table XXV by combining the entries for the tracking bandpass filter and frequency counter (both with real-time outputs), and the entries for the swept oscillator wave analyzer and the Fabry-Perot interferometer (both with an output that is related to the spectrum of the photodetector signal). The schematic diagram shown in Fig. 50 should help to clarify the difference between the various types of signal processors. All the processors, with the possible exception of the period counter (see Section 1.1.4.4.7.6), are subject to errors due to ambiguity noise in the photodetector output.* In the case of those processors that have an output which is the photodetector spectrum, it appears as spectral broadening (see Section 1.1.4.4.7.1), and for the processors with a real-time output, it is in the form of small amplitude, broadband noise in the power spectrum of the processor output. This difference is due to the character of the processor outputs. For the spectral analyzer output, the fluctuations in the photodetector output caused by ambiguity noise have probability densities that make a contribution to the probability density function comparable to the probability density of the turbulent velocity fluctuations. However, the energy content of the ambiguity noise is small compared to the energy content of the velocity fluctuations, and, in
* This is ignored in the definitions of the flow parameters given in Table XXV. However, the ambiguity noise does affect the signal analysis, as will be discussed.
TABLEXXV. Techniques for Analyzing the Output from Laser Velocimeter Signal Processors
Required quantity
Practical definition
f / 'oD(r)dr 0
Autocorrelation
nolds' shear stress)
f/'
oD(t)OD(t
+ 7 ) dr
Swept oscillator wave analyzer and Fabry-Perot interferometer
Tracking bandpass filter and frequency counter
Period counter
Exhibit the processor output as a function of time
Output of a sampleand-hold circuit applied to the processor output
Assumd O D = o m , where m y is estimated from the exhibited PDF (p)
Digital and analog methodsdre
OD
Assume 3 = d , where u*is estimated from the exhibited PDF(p)
Digital and analog methodsd*
Digital computerf*'
Digital and analog methodsdSe
Digital computerfa
Digital and analog methodsdSe
Digital computerLo
= 2~N/Tpnu, wherek TNmis estimated from the exhibited PDF
Photo counting correlator
Estimate from Eq. (1.1.122) (ormore accurate forms) fitted to processor outputh,' Estimate from Eq. (1.1.122) (or more accurate forms) fitted to processor output".'
-
Digital and analog methodsd.=
Digital computerr*O
-
Digital and analog methodsd,‘
Digital computerr.#
distributionC Requires two photodetectors, one for each velocity component (ud,uPj),[P. J. Bourke, C. G. Brown, and L. E. Drain, Disu Inf. No. 12, p. 21 (1971)], or a single photodetector placed successively at two (or more) stations [F. Durst and J. Whitelaw, Disa Inf. No. 12, p. 1 1 (1971); W. J. Yanta, Purdue Univ. (Indiana)E n g . Experimenf Sr. Bull. 144, 115 (1974)l. However C. Greated [ J . Phys. E3, 753 (1970)l has demonstrated a technique in which measurements are made by varying the angle of the fringes in the measuring volume and thereby avoiding the use of two photodetectors. Requires inputs from each of the two stations (1 and 2) between which the correlation is to be measured: see P. J. Bourke, L. E. Drain, and B. C. Moss, Disa Inf. No. 12, p. 17 (1971). The power spectral density is also defined by the cosine of the Fourier transform of the autocorrelation function. T h i s approach would be used when the processor output is being subjected to analysis on a digital computer. It may also be the required procedure if the data rate is very low, as shown by M. C. Whiffen and D. M. Meadows [Purdue Univ. (Indianu)Eng.Experiment Sf. Bull. 144 (1974)l. In the definition of the power spectral density distribution w,,Cf,r) is that portion of w,(r) passed by a filter of bandwidth B (hertz) and center frequency f(hertz). P. Bradshaw, “An Introduction to Turbulence and its Measurement.” Pergamon, Oxford, 1971. The hot wire anemometer signal analysis equipment described in this reference may be used to carry out the analog analysis of the signal processor output. J. S. Bendat and A. G. Piersol, “Measurement and Analysis of Random Data.” Wiley, New York, 1966. These statistical parameters may be obtained from the sample-and-hold output at very high data rates (corresponding to particle flow rates of the order of 105 particles/s). For lower flow rates it may be possible to obtain the power spectral density of the velocity fluctuations from the autocorrelation of the Doppler burst data (see J . M. Roberts and M. Gaster, in “Stochastic Problems in Mechanics” (S. T. Ariaratnam and H. H. E. Leipholt, eds.), p. 301. Univ. of Waterloo Press, Waterloo, Ontario, 1974), provided a record of the occurrence time of the bursts is maintained and a sufficiently large data set is obtained so as to minimize bias effects. J. A. Asher, Prog. Astronauf, Aeronaur. 34, 141 (1974). * A. D. Birch, D. R. Brown, J. R. Thomas, and E. R. Pike, J . Phys. D 6, L71 (1973). C. Y. She and J. A. Lucero, Opt. Commun. 9, 30 (1973). wM is the value of wD at the maximum value of p ( o ) . TNMis the value of TNat the maximum value of do). f
1.
232
MEASUREMENT OF VELOCITY
=
FABRY-PEROT INTERFEROMETER
OPTICAL I ELECTRONIC DOMAIN -DOMAIN
I
' DP (WldW
I
I
I I I I
I
PHOTON COUNTING CORRELATOR I
k (T) I '
I I SIGNAL PROCESSING SIGNAL --II ANALYSIS II FIG. 50. Schematic diagram showing the relation among the various signal processors. Asterisk indicates that the technique is applicable provided the flow tracing particle number density is sufficiently high.
consequence, the contribution of the former to the power spectral density of the processor output is small. The ambiguity noise in the signal processors with a real-time output is caused by fluctuations in the quantity yl [Eq. (1.1.109)], and since these are uncorrelated with the velocity fluctuations, the power spectral density of the velocity fluctuations may be obtained from the power spectral density However, at high turof the signal processor by subtraction.220~268~288~280 bulence intensities, which Durrani and GreatedZz0estimate to be in excess of 15%, the power spectral density of the ambiguity noise is significantly affected by the presence of the turbulence, so the estimation of the power spectral density of the velocity fluctuations by subtraction could be in error. 1.1.4.8. Summary of Velocimeter Characteristics. The preceding sections contain a detailed exposition of the characteristics of the laser Doppler velocimeter. In view of the extent of the material, it seems appropriate to summarize the various factors that can affect the operation of the velocimeter. These factors, in turn, influence the choice of optical
1.1. TRACER METHODS
233
configuration and the latitude the experimenter has in adjusting the instrument to obtain the best possible performance. A preliminary choice of a laser Doppler velocimeter optical configuration will depend on the following factors: (a) the photomultiplier output signal-to-noise ratio, since this plays a role in the choice of the signal conditioning and signal processing techniques, (b) the ambiguity noise (or aperture spectral broadening) that can be tolerated (this has a direct affect on the precision of the fluid velocity measurements), (c) the spatial resolution of the measurements, (d) the ease and certainty with which the position of the measuring volume can be determined, and (e) the ease with which the incident light beams can be aligned on the receiving aperture. Detailed discussions of alignment procedures will be found in Durst and white la^,^^' and Bossel and O r l ~ f f . ~ ~ ~ The various optical configurations are classified in terms of these factors" in Table XXVI. A careful examination of this table suggests that the balance of these various factors favors the dual incident beam arrangements, for the following reasons: (a) The measuring volume can be located by observation. (b) In the homodyne arrangement, the signal-to-noise ratio can be improved by increasing the receiving aperture without affecting the ambiguity noise. In general, because of the low level of the scattered light, the signal-to-noise ratio is much more dependent on the receiving aperture than it is on the transmitting aperture because the light incident in the measuring volume can always be increased, in order to improve the signal-to-noise ratio, by increasing the laser power. An additional advantage of the dual incident beam configuration is the better optical efficiency with regard to the scattered light. Single incident beam arrangements have to pass the scattered light through beam combiners, with attendant losses. Such losses have a far more serious effect o n the scattered light than on the incident laser beams, which pass through the corresponding beam dividers in the dual systems. %' 588
F. Durst and J . H . Whitelaw, J . Phys. E 4, 804 (1971). H. H. Bossel and K . L. Orloff, J . Hydronaur. 6 , 101 (1972).
* Spatial resolution is not included in this table because it is probably more dependent on the choice of the light beam forming technique (see Section 1.1.4.4.4).
TABLEXXVI. Comparison of Single Incident Beam and Dual Incident Beam Velocimeters” ~
Single incident beam Factors Photodetector output SNR; effect of: i. increasing the flow tracing particle number density. ii. increasing the receiving aperture (decreasing 7D). iii. vibration (decreases vD. deviates light from receiving aperture). Ambiguity noise (aperture spectral broadening) dependence on: i. receiving aperture. ii. transmitting aperture. Ease and certainty with which the position of the measuring volume can be determined. Alignment of incident light beams on receiving aperture.
Heterodyne
Homodyne
Dual incident beam Heterodyne
Homodyne
Yeh and Cummins
Increases
Decreases
Increases
Decreases
Increases
Cannot be improved
cannot be improved
cannot be improved
Improves (qD not aBected)
Cannot be improved
Decreases
Decreases
Decreases
Usually not important
Decreases
Yes
Yes Yes By observation
No
Yes Yes By inferrence from dimensions of LDV
Can be difficult
Straightforward wfforward or backward scattering
Yes Yes By inferrence from dimensions of LDV Can be difficult
No By inferrence from dimensions of LDV straightforward w/forward or backward scattering
Assumes division or combination of amplitude is used to form the light beams.
Yes By observation
Micult
1.1. TRACER METHODS
235
The above considerations have resulted in the single incident beam systems being superseded by the dual incident beam configurations, so that the former are now really only of historical interest. Table XXVI also shows that the heterodyne arrangement is to be preferred over the homodyne configuration when the flow tracing particle number density is high. However, the exact definition of a large number density is uncertain, and in some situations this may require an actual comparison of the photodetector output signals in the heterodyne and homodyne configurations in order to decide the better arrangement. Experience shows that to obtain the best performance from a given velocimeter optical configuration will require careful adjustment of the various components. Furthermore, this adjustment is difficult to carry out effectively unless the operational characteristics of the velocimeter are understood. In adjusting the velocimeter, the experimenter has at his disposal: (a) the size of the flow tracing particles, (b) the size of the receiving aperture, (c) the size of the transmitting aperture, (d) the laser power, (e) the light scattering angle, and (f) the beam angle a (see Fig. 13). From the preceding sections it is known that varying these parameters of the instrument affects the photodetector output signal-to-noise ratio, the ambiguity noise, the dimensions of the measuring volume, and the signal frequency. This is illustrated in Table XXVII. From the table it can be seen that improving the velocimeter performance in one respect, may result in a decline in another aspect of its performance. Clearly the adjustment of the velocimeter involves certain compromises in order to locate the optimum instrument performance. Optimization of velocimeter performance has been considered by Davis,270B r a y t ~ n and , ~ ~Durst and white la^.^^ The reader is also referred to Section 1.1.4.9, where the relation of the velocimeter performance to its optical parameters is demonstrated by a numerical example. 1.1.4.9. Example of System Design. As with the design calculations for the chronophotography system described in Section 1.1.3.10, this is a preliminary estimate which will indicate the probable feasibility of making the proposed measurements. It must be emphasized that even if the calculations indicate that the planned measurements are possible, there will still be a substantial trial and error effort required to select the components and adjust the system (see Section 1.1.4.8). The system design will be demonstrated using the example employed in sm D. B. Brayton, in "Proceedings of the Technical Program-Electro-Optical Systems Design Conference (K.A. Kopetzky, ed.), p. 168. Ind. Sci. Conf. Manage., Chicago, Illinois, 1970. F. Durst and J. H. Whitelaw, Opro-electronics 5, 137 (1973).
TABLEXXVII. Adjustment of Dual Incident Beam Velocimeters" ~
Adjustment
Wotodetector output SNR
Increase particle size
Increases
Increase receiving aperture
See Table XXII
Increase transmitting aperture Increase laser power
Increases
Increase scattering angle 4
Decreases (light scattering properties of particles also important) Depends on light scattering properties of particles
Increase beam angle a
Ambiguity noise
Increases in heterodyne conliguration Increases
Measuring volume dimensions
Signal frequency
-
-
Affects dynamic characteristics of flow tracing particles.
Decreases
-
-
Increases
-
-
-
May increase laser noise, and decrease heterodyning efficiency (see Section
-
Increases (but see Note)
Notes
1.1.4.4.7.4).
Decreases
Decreases
-
See Fig. 20 for angle 4.
Decreases
Decreases
Increases
See Fig. 20 for angle a.
Assumes division of amplitude is used to form the incident light beams.
1.1.
TRACER METHODS
237
Section 1.1.3.10, which refers to a velocity measurement originally described by E i ~ h o r n (see ' ~ ~ also Section 1.1.2.9). Since the measurements are being made in air with artificial flow tracing particles added, it is probable that the number density of the particles in the measuring volume will be low enough to warrant, from signal-to-noise ratio considerations (see Section 1.1.4.4.6),the use of a dual incident beam homodyne configuration. However, unless the particle number density is very low, it will be assumed that the velocity measurements can be made using a tracking bandpass filter. To maximize the signal, the photodetector will be arranged to receive light scattered from the flow tracing particle in a direction which is as close to the forward direction as possible. This will require the laser and the photodetector to be located so that they lie in a plane parallel to the plane of the heated plate (see Sections 1.1.2.9 and 1.1.3.10). The measuring volume would then be located at the center line of this plane, so that the fluid flow in its vicinity is unaffected by conditions at the edge of the plate. A traversing mechanism will be required to move the measuring volume up and down the plate in the plane parallel to the plate, and away from and toward the plate in a plane perpendicular to the plate. The photodetector will be an RCA 4526 photomultiplier tube, and light scattered by the flow tracing particles will be collected by a lens of focal length 250 mm and aperture 67 mm. This collected light will then be projected onto the photocathode of the photomultiplier tube by a lens of focal length 200 mm and aperture 67 mm. The light source will be an unapertured 5 mW helium-neon laser (A = 0.6328 pm) with an 8 mm diameter [at point where I/Zo = exp(- l)] beam. The light will be projected into the measuring volume by a lens of focal length 250 mm and aperture 67 mm. A beam splitter will be arranged between the laser and the lens, so the single lens will be used to project both beams, thereby providing the simplest means for adjusting the beams in the flow field. The dimensions of the measuring volume will be selected to be as small as possible.* This will maximize the spatial resolution of the measurements, so that the effects of the anticipated velocity gradients in the fluid will be a minimum.
* Where the particle velocities are very high (say, 250 m/s), it may be necessary to design the measurement system so as to ensure that the data processing system is able to handle the signal frequency and to provide the desired time resolution. This could result in a rather larger measuring volume being required than was desirable from the point of view of spatial resolution.
238
1.
MEASUREMENT OF VELOCITY
The steps in the calculation are as follows: (a) match the optical system to the particle size (see Section 1.1.4.4.3);(b) determine the measuring volume dimensions; (c) estimate the signal frequency, so as to ensure that the signal processor can handle the data; and (d) estimate the signal power and compare it to the system noise power. The mean particle diameter dDis 1 pm (see Section 1.1.2.9). According to the earlier discussion (see Section 1.1.4.4.3) of matching the optical system to the particle size, the particle diameter should be smaller than the fringe spacing df in the measuring volume. The fringe spacing is given by (see Section 1.1.4.4.2) X = ?r/ko sin(a/2) = ho/2 sin(a/2).
( 1 .1 .156)
According to this expression, the angle 4 2 should be about 5" to match the flow tracing particle and the optical system. Selecting a/2 = 5", the fringe spacing is found to be 3.64 pm. The measuring volume dimensions are given in Section 1.1.4.4.4,Table XXI. These involve the diameter of the laser beam at its focus, and this is given by Table XXI. For a lens of 250 mm focal length and for a beam diameter of 0.8 mm, the effective f-number of the lens is 312. Then the beam diameter at the focus is 0.251 mm, and the -maximum diameter Ax of the measuring volume is 0.252 mm and its ,length Az is 2.89 mm. The very small beam angle ( 4 2 = 5") causes the measuring volume to be long and slender. However, the length is across the flow, as if measurements are restricted to points far from the edges of the plate, the variation of velocity in the long direction of the measuring volume should be negligibly small. Correspondingly, the dimensions of the measuring volume are small in the vertical direction of the plate, in which direction maximum spatial variations of the velocity are anticipated. The size of the measuring volume is, from Table XXI, 1.0 mm3. The maximum signal frequency is given by fmax = up,max/X.
(1.1.157)
For an anticipated maximum particle velocity of 21 cm/sec (see Section 1.1.3.10) and with the chosen fringe spacing of 3.64 pm, the maximum signal frequency will be 59 kHz. This is within the range commercially available period counters. To calculate the light scattered by a single flow tracing particle, we use Eq. (1.1.26). Since the scattering particle is located in the measuring volume, which is at the focus of the receiving lens, l / W is very small, and hence P = 1rTF~1/4Nz. We take the total lens transmission T as 0.8 (see Section 1.1.2.8) and N = 2.73. For a 5 mW laser and with the beam
1.1.
TRACER METHODS
239
diameter at the focus, as calculated above, of 0.251 mm, Fi = 1 x W/pm*. The angular scattering cross section I is obtained from Eq. ( I . I .23). The intensity functions il and iz needed in this formula will be obtained from L ~ w a n . ~For ~ ' the material of the particle (DOP), we will assume, as in Section 1.1.3.10, that m = 1.44. We will carry out the calculations for the mean particle diameter of l b m (a = 5 ) , and for particles with a diameter of 0.5 p m (a = 2.5). The latter are sensitive to Brownian motion (see Section 1.1.2.9). It is hoped that the signal originating with these particles will be weak enough so that its contribution to the total signal can be ignored (as was possible in the application of chronophotography to this situation in Section 1.1.3.10). Using the data in Lowan et ~ 1 . ~and ~ 'the above formula, we get* P = 2.03 X W/particle (a = 5 ) . W/particle (a = 2.5), and P = 4.75 x In view of the proposed flow tracing particle concentration of 1 x los particles/cm3 (see Section 1.1.2.9), it is anticipated that the measuring volume will contain more than one particle at any instant. All the particles in the measuring volume contribute to the luminous energy incident on the photodetector; so P = PsCV,,,, where C is the particle concentration, and V , , the volume of the measuring volume. We then have for W. d = 0.5 pm, P = 1.96 x W, and for d = 1 bm, P = 4.58 x The signal-to-noise ratio associated with the photomultiplier will be calculated. This will then be compared with the ratio of the signal power to the ambiguity noise power in order to assess their relative importance, and the feasibility of making the desired measurements in the presence of the electronic and ambiguity noise. To estimate the SNR associated with the photomultiplier tube, we will use Eq. (1.1.97a). We will assume the bandwidth Af of the signal processor is 50 kHz, k = 4 (assumed), and a = 80 mA/W (from published 391 A. M. Lowan, "Tables of Scattering Functions for Spherical Particles," U.S. Natl. Bur. Stand. Appl. Math., Ser. 4. US Govt. Printing Office, Washington, D.C.. 1948.
* It is interesting to compare the luminous flux density at the photocathode in the laser velocimeter measurements with that on the film in the chronophotographic measurements that were considered in Section 1.1.3.10. For a single 1pm diameter particle we have, for the laser velocimeter case (assuming that the scattered light gathered by the receiving lens is focused to a diffraction limited spot),'F, = 1.18 x 10' erg/(cm' . s), and for the chronophotographic case, F, = 1.05 x 10' erg/(cm2 . s). The difference between these two values is a consequence of the larger incident light intensity in the laser velocimeter and the fact that forward scattering is not used in the chronophotography case. This suggests that it would be highly desirable, where possible, to use both lasers and forward scattering in chronophotography .
240
1. MEASUREMENT
O F VELOCITY
data on the RCA 4526 photomultiplier tube). The calculated* SNR is then, from Eq. (1.1.97a) (with the power of both beams equal to the previously calculated scattered power), 54 dB (a = 2.5) and 68 dB (a = 5 ) . Signal-to-noise ratios of this magnitude are sufficiently large to allow signal processing by conventional techniques, e.g., a tracking bandpass filter. ACKNOWLEDGMENTS The authors would like to thank the following: Dr. R. J. Emrich for the opportunity to prepare Sections 1.1. I - 1.1.4 and for his guidance and encouragement during the writing: Dr. G. M. Friedman, Dr. A. L. Fymat, and Professor W. T. Welford, who reviewed Sections 1.1.2.2.3, 1.1.2.8, and 1.1.3, respectively; Dr. A. T. Hjelmfelt, who provided computer programs that assisted in the calculation of the numerical results given in Figs. 3,5, and 7; Drs. H. C. van Ness and F. F. Ling, who provided the facilities that made the writing possible; and Mrs. Susan Hams, who typed and retyped several versions of the manuscript, and very efficiently prepared the final manuscript for the publisher.
1.2. Probe Methods for Velocity Measurement 1.2.1. lntroductiont A simple device placed on the end of a stick to measure how fast the wind is blowing, or how fast a boat is moving through the water, would be very useful. The Pitot tube almost satisfies the need, but its use by one who does not understand its principle of operation leads to disappointment. Thus it is not an instrument readily employed like a Bourdon pressure gage or a thermometer. One difficulty is that a velocity meter, often called an anemometer, necessarily disturbs the velocity field it is trying to measure; the reaction of the disturbance on the meter is the driving mechanism. Because of its small size, the hot-wire anemometer reduces this difficulty immensely. A second difficulty relates to velocity being a vector quantity having direction as well as magnitude. The instrument must be pointed properly to read the magnitude of the velocity-called the speed. The hot-wire is useful in that it is mainly sensitive only to the component of velocity normal to the wire. The Pitot tube has a maximum reading when it is
* In these calculations the number N , of particles in the measuring volume was estimated from CV,, which is equal to about one hundred. It was assumed that with this value of N, in Eq. (1.1.97a): (i) 1 / N , is small enough to ignore; (ii)&,* = 1 (negligible loss of signal strength due to imperfect mixing of light scattered from different flow tracing particles); and (iii) Ps = N P D . t Sections 1.2.1-1.2.3 are by R. J. Emrich.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
24 1
aligned parallel to the velocity, but it requires some time to find the correct reading and the flow must be steady. The third difficulty is the most serious: fluid velocities are seldom constant; the direction and magnitude are changing from time to time and from place to place. The Pitot tube in its usual form takes seconds to respond, and its average reading in an unsteady flow is not the average velocity magnitude, even when it is aligned with the mean flow velocity. Hot wires and hot films have been adapted to instruments which have a much faster time response, but they are fragile and complicated. Simple probes to show the direction of the velocity are useful. The weather vane gives the direction of the wind by supporting a plate or other airfoil on a vertical axle so that there will be a net torque to rotate it parallel to the velocity when it is not aligned. Hairs or tufts of yam held in a stream align themselves with the velocity. Viscous substances coating a surface develop streaks showing the direction; the introduction of dye, smoke, or hydrogen bubbles has been developed and used extensively.’ Instruments combining up to five Pitot tubes or three or more hot wires are capable of measuring flow direction quite accurately in steady flows without strong pressure or velocity gradients. A probe to measure vorticity would be extremely useful in research in fluid dynamics. A crude float with radial vanes is useful for demonstration purposes, but its ability to respond to the (vertical component of) vorticity is poor. Assemblies of hot wires to measure vorticity have been built, but are also complicated and not readily available. They are described in Section 1.2.4.6.4. Measurement of fluid velocity is so important in atmosphere and ocean studies, in ship hydromechanics, in aeronautics, in chemical and mechanical engineering, and in biology that many special instruments have been devised for various situations. When the principles described in this section are understood, the usefulness and reliability of the special instruments can be evaluated. All have been evaluated or calibrated by primary methods using tracers or towing, and some have the capability of making precise measurements. Flowmeters are, in a sense, integral velocity meters in that they measure the rate of the total flow and hence the average velocity if the flow cross sectional area is known. On the other hand, velocity magnitude can be defined as the volume crossing a unit area (the area oriented normal to the velocity direction) in unit time as the area is reduced to zero. Since flowmeters are inserted in a canal or duct to measure the
W. Merzkirch, “How Visualization.” Academic Press, New York, 1974.
242
1.
MEASUREMENT OF VELOCITY
total flow rate, it is appropriate to treat them as probes. In addition, flowmeters and velocity probes share important practical and commercial similarities. Flowmeters are discussed in Section 1.2.6.
1.2.2. Velocity Measurement by Pitot Probe
Figure 1 shows a simple Pitot probe measuring the speed of air impelled by a fan. A straight piece of metal or glass tubing with its open end pointed into the air stream is connected by rubber tubing to a glass U-tube manometer containing water. The pressure ps inside the Pitot tube is higher than the atmospheric'pressure p, and the difference is registered in the difference in height h of the manometer columns. As indicated in Fig. 1, the air speed v before the tube was inserted at the point where the tip of the tube is now is given by a simple formula. By moving the tube about, the velocity of the air stream can be surveyed by reading the manometer. Whenever the direction of the tube is within 5 5" of true alignment, the indicated value of v is accurate to within a few percent, if h can be read with sufficient precision. The instrument is amazingly simple and is used for indicating aircraft flight speed for wind speed in hurricanes and in wind tunnel tests. Because of its simplicity it is often used together with accurate pressure measuring devices for calibrating other velocity gages in wind tunnels. When the Pitot and rubber tubing are completely filled with water, the Pitot tube is used for surveying the water current near a dock or bridge pier, e.g., by measuring the rise of water h in the tube above the level of the river surface; the density pairis then replaced by pwater in the formulas given below Fig. 1. This application was actually its first known use, as described by Pitot in 1732 for measurements he made in the Seine River in Paris. On the other hand, the Pitot tube is not recommended as a device for measuring ordinary wind speeds, blood flow, or chimney draft, or for metering gasoline, fuel gas, or water, although it is occasionally used for some of these. In understanding the basis on which it works, we will find why it works so well for some applications and why it fails for others. Before proceeding to the principle of the Pitot tube, we will survey qualitatively the limiting factors. The Pitot tube obviously disturbs the flow at the point of measurement. It is important that it not disturb the flow much at other points. It can be used when it is a small tube in a large flow. While it is not very sensitive to direction when aligned, the direction must be known and not changing continuously. The speed should not be changing faster than the changing pressure can be read. This limitation is
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
243
-
- - - - - - -- - - - -- - - + ------_ -- _ __ __ __ _ _ -+ 4
---D
- - - - - - - -* duct dischorge
U-tube water monometer
FIG. 1. Pitot tube and manometer used to measure air speed of ducted fan discharge.
where u is the speed of air just ahead of Pitot tube; pair and pwater are the mass density of air and water, respectively; psis the pressure of air in Pitot; p , the atmospheric pressure; g , the weight per unit mass, and h , the difference in heights of water in U-tube. The stippled region shown ahead of the tube where the air has not yet begun to be deflected is placed schematically, and is actually about a third of the tube diameter in front of the opening.
not severe if the pressure is determined with a diaphragm gage or a piezoelectric gage. The accuracy rapidly decreases as the turbulence level increases. The density of the fluid must be known. Stack and engine exhaust gases vary in composition and temperature. Particulates carried by the gas can invalidate the Pitot tube reading unless they are so small that they follow the gas path accurately. Their contribution to the density must then be included. See Section 1.1.2.3. A difference in pressure is read. Use of the Pitot tube is greatly complicated if the pressure in the undisturbed fluid is changing, e.g., very near a fan blade or in a blood artery. A great deal of effort has gone into combining the Pitot probe with a static probe, namely, one which can simultaneously sense the undisturbed fluid pressure. This can be done successfully in many cases. The Pitot tube operation is based on, among others, two assumptions which are often met, but in other cases are obviously not met. Viscous (shear) forces are assumed to play no role; in honey, bread dough, and extremely small tubes viscous forces are important. The other assumption sometimes obviously not met is that velocity and pressure gradients are absent; separated flow, such as that leaving sharp comers, contains gradients. 1.2.2.1. Principle of the Pitot Tube. The basis of operation ofthe Pitot tube is the Bernoulli equation, which is a line integral of the Euler equation at an instant in time taken along a streamline. Since the Euler equa-
244
1.
MEASUREMENT OF VELOCITY
tion is the fundamental equation of motion when pressure and body forces are the only forces acting, this means that there can be no shear stresses, i.e., no viscous forces. The Bernoulli equation is, in the case of no electric or magnetic force, g dz = const.
(1.2.1)
In this equation, g dz is the result of integrating - g * dl, where g is the weight per unit mass and 2, the vertical coordinate, i.e., the coordinate in the direction of -g. dl is parallel to the streamline. Surface tension forces may be present so long as the integral path does not go through a curved meniscus. The explicit limits are written to call attention to the integral being instantaneously taken from the place where the speed v = 0 to the place where u = v. At these places p , p, z, etc. have certain Values. Now we shall proceed to deal with each of the terms in Eq. (1.2.1) in turn. The equation cannot be used at all when viscous forces are large, but we know when viscosity is a problem; if the Reynolds number pDul(2p) is below about 30, we are warned that the effects of viscous forces must be looked into. The outer diameter of the Pitot tube is D , and v is the speed we are trying to measure; p and p are the density and viscosity coefficients of the fluid. The Pitot tube with viscosity is considered in Section 1.2.2.5. We would like to have the dv/dt term small enough so that the first term in Eq. (1.2.1) can be omitted. The warning to watch for is the presence of sound waves; in a sound wave dv/dt is large and changes sign within every wavelength. While Eq. (1.2.1) is correct, it is not useful in acoustics because we cannot evaluate this integral easily. For the Pitot tube to be useful, the flow must be sufficiently steady so that the first term can be omitted from the Bernoulli equation. Sometimes Jg dz enters, but we put it aside as the hydrostatic correction or buoyancy. It is usually easy to recognize whether pg(z - zo) is large enough to worry about in a pressure measurement. The third term in Eq. (1.2.1) is the important term and the one most likely to cause misunderstanding. When p is everywhere constant, as in a homogeneous and incompressible fluid, e.g., water, we get the simple form of the Bernoulli equation P + fpv2 = P s ,
(1.2.2)
wherep, is the value o f p at the place where ZI = 0, as used in Fig. 1; p s is called the stagnation pressure in this case. When the fluid is homogeneous and compressible, we often have
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
245
which derives from the assumption that all the fluid on the streamline has changed its p and p adiabatically and reversibly in the thermodynamic sense (i.e., isentropic processes). The enthalpy per unit mass is H (J kg-’). If there is heat flow, e.g., at a missile nose cone or flame, Jdp/p is not AH. However, if we can assume constant entropy, then often further simplification is possible for an ideal gas. Air is an ideal gas to a good approximation if it is not liquid because T is too low, or if it is not dissociated because T is too high. Then H = C,T, where C , is a constant property of the gas, the specific heat at constant pressure, T is the Kelvin temperature, and the Bernoulli equation becomes
C,T
+ tu2 = C,Tu,o.
(1.2.3)
Now remember that in Eq. (1.2.1), the integral is along a streamline. Figure 1 shows orderly streamlines, but if the flow is eddying or turbulent we do not know, at a given instant when we measure pmat the entrance to the tube (where u = 0), to what point on the stagnation streamline the p and u in Eq. (1.2.1) are referring, and dv/at is not zero. If the turbulence is such that at successive times the streamline comes from the same general region, shown stippled in Fig. 1, where the p and u values are about the same at successive times, we might hope the integral of av/at would average to zero and that we might acquire data on an average u in the region ahead of the probe tip. This hope is not quantitatively borne out, but most workers believe in the validity of corrections to pm - p amounting to some fraction of p ~ ’ where ~ , u’ is the rms fluctuation of u from its mean zi, and p is a mean of p in the region far enough ahead of the probe tip so that the deflection of the gas by the probe has not yet started. See Sections 5.3.2 and 1.2.2.6. In spite of this ambiguity in the use of Eqs. (1.2.2) and (1.2.3) in real flows, which are practically always turbulent, the Bernoulli equation is referred to-and the Pitot tube used for velocity measurement-a great deal in some approximate sense. In the remaining parts of Section 1.2.2 we will use the Bernoulli equation, calling the flow steady, meaning the turbulence level U ’ / O is small: perhaps less than lo%, preferably less than 1%. Finally, in summary, the principle o n which the Pitot tube operates is the Bernoulli equation in the forms of Eq. (1.2.2) if the fluid is incompressible, and of Eq. (1.2.3) if the fluid is an ideal gas. A Pitot tube employs a measurement of a pressure difference, and additional thermodynamic ideal gas relations are utilized to obtain the gas velocity from Eq. (1.2.3) and pressure measurements, as discussed next.
246
1.
MEASUREMENT OF VELOCITY
1.2.2.2. Stagnation Pressure and Other Terminology. Even though no sound waves need be considered in treating the flow of air at high velocities, a derived variable of an ideal gas called sound speed c and defined as cz = y p / p is introduced. y = C,/Cv is the ratio of the specific heats and is assumed constant. The temperature of an ideal gas, T = p / R p , where R is the gas constant per unit mass, is also available for use in the expressions describing the flow, but T is eliminated from the equations in favor of c . Using cz = yRT, the Kelvin temperature of the gas is readily known ifc is known. A real gas, with more complicated behavior than an ideal gas with constant heat capacities, will have an actual sound speed different from that defined above, and if sound waves are considered, one must remember to distinguish the actual speed of sound from the quantity defined above, which is only approximately the actual speed. In discussing and using Pitot tubes, effects of errors due to lack of steadiness in the flow as required for the validity of the Bernoulli equation are more prominent than those due to departure from the isentropic ideal gas equations. Equation ( I .2.3) is therefore further modified and takes the formZ
(1.2.4)
For notational simplicity the Mach number M = u / c is introduced, which seems to suppress the variable u which we are trying to measure. This result of this procedure is that a new flow variable ps called the stagnation pressure* is defined by Eq. (1.2.4)and is known if the u , p , and p of the air are known. The Pitot tube tries to measure p s , and if p and p are also known, u can be obtained from Eq. (1.2.4). Actually, since this is algebraically tedious (an expression for p in terms of c, p , and M has to be used), and since p s - p is the quantity we try experimentally to determine, an alternate expression of Eq. (1.2.4) is obtained:
and a power series expansion used which is valid even up to M = 1: u = , + - + * M M24 + . +pv2 2.2! 2 * 3 !
* L. D. Landau and E. M. Lifshitz, “Fluid Mechanics,” 0 80.
...
(1.2.6)
Pergamon, Oxford, 1951.
* This terminology is not uniform in the literature; at least two other definitions of the term stagnation pressure are common.
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PROBE METHODS FOR VELOCITY MEASUREMENT
247
For Pitot tube purposes, there is no need to have an expression valid for M > 1, because there will be a shock standing in front of a tube and hence M at the Pitot tube is always less than unity. The undisturbed stagnation pressure in the stream is obtained from the Rankine -Hugoniot relation between the stagnation pressures defined for the gas ahead of the shock p s and behind the shock pi3:
In this formula, M is the Mach number of the air ahead of the stationary shock. Referring back to Eq. (1.2.6), we see that for M << 1, the stagnation pressure defined is exactly the same as that appearing in the incompressible Bernoulli equation Eq. (1.2.2). On this account, air is often said to be incompressible at Mach numbers below 0.2; the second term of the right hand side then remains less than 1%. Stagnation pressure is also called impact pressure and total pressure in the literature. Also, frequently authors use some or all of these terms to mean the actual pressure at a stagnation point in a particular flow, regardless of whether or not the pressure there is related by Eq. (1.2.4) to the flow variables upstream. In literature dealing with velocity measurement by Pitot tubes, one of these terms is reserved for the definition given by Eq. (1.2.4), but there is no uniformity in the choice. The symbol q and the term dynamic pressure are often used for the expression bpu2 so that Bernoulli’s equation in the form of Eq. (1.2.2) can be thought of as a conservation relation. The stagnation pressure or total pressure is conserved on a streamline in the sense that stagnation pressure
=
pressure
ps
=
p
+ dynamic pressure
+ bpu2.
When Eq. (1.2.2), is not valid, some authors prefer to maintain the conservation relation and define the dynamic pressure in such a way that it will be maintained, i.e., they define dynamic pressure as p s - p, or even sometimes p m - p , where p m may be subject to unknown viscosity, yaw, gradient, and turbulent corrections. With introduction of so many terms called pressures, it is natural that confusion arises not only between them but also with the pressure. One attempt to avoid confusion is to call pressure static pressure; this is quite common, and most unfortunate and confusing. The reasoning behind this terminology is that if a pressure gage were connected to a tube and the end of the tube moved along with the stream (static with respect to the L. D. Landau and E. M. Lifshitz, “Fluid Mechanics,” 0 85. Pergamon, Oxford, 1951.
248
1. MEASUREMENT OF VELOCITY
stream), it would read the pressure and not the stagnation pressure of the stream. Measurement of the pressure (“static pressure”) in a moving stream requires care. Strictly it is only possible to measure pressure alongside a solid surface. Pressure can be measured only in an approximate sense when the pressure is sought at points away from solid surfaces. The techniques are discussed in Chapter 5.3 of this volume of “Methods of Experimental Physics” in Part 5 on pressure measurement. Some discussion is also contained in the sections (1.2.2.3and 1.2.2.6)relating to the Pitot tube, because velocity measurement requires simultaneous knowledge of both the stagnation pressure and of the pressure. 1.2.2.3. Tube Construction; Angular Sensitivity. Since the tube is inserted in the flow and is to interfere with the flow as little as possible, usually a long uniform metal tube bent in the form of an “L” and supported by a larger structure is used. Both arms of the “L” would need to be as long as possible to achieve least interference by the supporting structure. There is a limit on the length of fine tubing that will assure the mouth of the tube maintaining a fixed position. For a given outer diameter D, there is a gain in rigidity as the tube wall is chosen thicker, but little is gained when the inner diameter d is smaller than D/2. Practical values of D range from 1 to 8 mm, and length to bend, 1OD to 15D. The stem beyond the bend is usually less than 10D and is often mounted via a larger diameter tube on a traverse mechanism driven by micrometer screws. Almost any shape, wall thickness, and size of tube will accurately read the stagnation pressure in a steady uniform (no velocity gradients) flow from low to supersonic speeds if the tube is aligned with the flow. The tube mouth must not be slanted with respect to the tube axis. Viscous effects must also be absent. The main effect of tube shape and wall thickness is in the sensitivity to yaw, i.e., the angle of inclination of the tube to the flow direction. When the flow is not uniform, e.g., in a bend or in the neighborhood of a solid surface, the size of tube is naturally chosen small to permit spatial variations to be studied. Since sizes down to hypodermic tubing are available, viscous effects can begin to appear, and when the distance from a wall is comparable to the tube size, flow diversion effects appear. These effects are discussed in the following sections and are assumed absent in the chart of Fig. 2, which shows the deviation of measured pressure from the stagnation pressure for various designs of probes. For the purpose of comparing the yaw insensitive range of the various designs of stagnation probes, Chue4defines a critical angle as the angle of yaw at which the error in the measured pressure from the stagnaS . H.Chue, Prog. Aerosp. Sci. 16, 147-223 (1975).
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249
0 0.1
0.2
0.3 0.4 0.6 0.6
0.7 0.8
40
30
20
10
0
10
20
30
40
Angle of yaw (deg)
t
110
0 5 23'
a0 = 0.96 at
mouth
Q: 0.3
FIG.2. Yaw characteristics for various stagnation pressure probes. D is the outer diameter; d , the bore diameter. The difference in measured pressure at yaw angle a from the measured pressure at zero angle for probe type 5 (Kiel type) is plotted against a. Type 6 is not a Pitot probe, but a cylinder with a small hole in its side facing the flow. [From Chue.']
tion pressure reaches a value of 1% of the dynamic pressure q = bpv2 measured at zero yaw. These critical angles are given beside the probes shown in Fig. 2. The Kiel probe designated 5 in that drawing is a design worked out over the years to have an especially small sensitivity to yaw so that it can be used in situations where the direction of the flow is not well known; it has a shroud, rigidly mounted to the stem and internal tube, which redirects the stream so that the inner tube is always nearly aligned with the flow it is measuring. The Kiel probe has been made commercially with an outer shroud as small as 2-mm diameter. The yaw dependence shown in Fig. 2 is that of tubes large enough and speeds high enough so that viscous effects are avoided. The yaw dependence is different for different tube sizes and different wall thicknesses. Yaw corrections cannot be made on the basis of the data in this figure alone, but the actual dimension will have to be taken into account. The error is roughly inversely proportional to the diameter of the tube.4 Often the Pitot tube is used in a flow where the pressure p is not known and must be measured. Since p is desired at the same spatial point as p s , a combination of a Pitot tube and a static tube called aPirot -static rube is often used. The dimensions of a type recommended by workers at the National Physical Laboratory, Teddington, England, are shown in Fig. 3.
250
1. MEASUREMENT OF VELOCITY Detollr ef ellipsoidal nose NOW
I D*7.87mm
FIG.3. Pitot-static tube. Three variations, showing coaxial tubes: the inner Pitot tube to measure stagnation pressure p s , and the outer static tube with a ring of holes on the circumference well behind the nose to measure the pressure p. [From Chue.']
(Note the confusing notation: p s is stagnation pressure and not static pressure.) The conditions for construction and use of the static part of the device are more stringent than those for the Pitot part. Burrs on the static orifices are especially difficult to avoid, and the yaw sensitivity is greater. The measurement of pressure in a moving fluid is discussed in Section 5.3. Recommendations for construction and use of Pitot -static tubes may be found in a publication by Bryer and P a n k h ~ r s t . ~ 1.2.2.4. Pitot Tube Use in Velocity Gradients and Near Wall. In boundary layer studies, one wants to make velocity measurements very close to the wall. The combination of tiny tubes, changes in velocity over the mouth of the tube, and asymmetrical deflection of the flow because of the wall presence on just one side has led to a great deal of work trying to assess these effects separately. One technique has been to use Pitot tubes of different shapes, sizes, and bores to survey the strong shear in the flow behind an airfoil trailing edge, and compare results with surveys made in boundary layers. Combining these results with calculations of inviscid model flows of shear layers deflected by a sphere, a displacement correction is used to display the effects believed to alter the Pitot tube velocity readings. The displacement correction 6 is defined as the amount of displacement needed to find the place where the indicated velocity is the actual velocity without the measuring tube. The displaces D. W. Bryer and R. C. Pankhurst, "Pressure-probe Methods for Determining Wind Speed and Flow Direction." HM Stationery Office, London, 1971.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
25 1
ment is from the center of the tube mouth toward the side where v is higher. Chue4 reviews these measurements and summarizes: (i) The correction is essentially governed by the outside diameter D and not by the bore size d or the shape of the tube nose. (ii) 6 increases with the velocity gradient, but less rapidly than linearly: 6 / D = 0.10 and 0.16 for Av/v = 0.20 and 0.52, respectively, where Av is the change in v over the distance D . (iii) In spite of a great deal of effort, the corrections to be applied to Pitot tube measurements in boundary layers are subject to a greal deal of uncertainty. 1.2.2.5. Corrections for Viscous Effects. There is no theory for a Pitot tube used in viscous flow other than dimensional analysis, which tells us that
brn- p)/4ppvz = C,(Re, Sh, d / D ) , Sh, Vg, a , . . .I, where the various dimensionless combinations of variables influencing the pressure are Re = pDv/(2p), M = v / c , Sh (a dimensionless shape factor), Vg (a dimensionless velocity gradient factor), etc. (See Chapter 10 regarding the method of dimensional analysis.) Equation (1.2.5) gives the Mach number dependence, e.g., if there is no yaw a,no velocity gradient, and no viscous force. Then the measured pressure p m is the stagnation pressure p s . At small Mach number, no yaw, and no velocity gradient, the functional dependence on Re, nose shape, and d / D has been investigated in liquids and low density gases. Since at large Re, i.e., when viscous forces are negligible, pm = p s , then defining (Pm -
P ) / ~ P v '= C,(Re, Sh, d / D )
(1.2.8)
and plotting C, as a function of Re should yield curves all asymptotic to C , = 1 at high values of Re, and the parametric dependence on Sh and d / D can be displayed. Chue4 has examined the published results and found that if Re is based on the tube inrernal diameter d , data for various nose shapes and various d / D ratios fall fairly close to a universal curve for 3 < ReD < 30, while for ReD > 30, C , is within +2% of unity. Using Chue's correlation of data as a basis for corrections at ReD < 30, it seems to the present author that a feasible procedure is to compute Red = pdv/(2p) using the probe internal diameter and find C, from Cp = 4.1/Red for Red < 0.7;
Cp = 1 + 2.8/Rea6 for
Red > 0.7.
252
1. MEASUREMENT
O F VELOCITY
Low Re associated with very small density p in rarefied gases produces additional effects. These are discussed by C h ~ e . ~ 1.2.2.6. Corrections for Turbulence. As mentioned in Section 1.2.2.1, the basis for use of the Pitot tube for measurement of velocity
rests on the flow being steady. Evidence of ubiquitous turbulence in nearly all flows we want to measure presents a serious challenge to the validity of the method. In an eddying, unsteady flow, both the pressure and the velocity of the fluid are changing spatially and temporally. The only approach to description of such flows that can be said to be successful is to suppose that at each point in space, a time average of the pressure and of each velocity component exists, so that at any one instant the pressure and velocity components (as well as other flow variables) can be represented as a superposition of a mean and a fluctuating part. The mean part is then, by definition, steady, but in some flows the question of what time span to use for the time average remains unanswered. For example, is the mean wind velocity at the airport to be based on a time span of 1 s, 1 h, 1 d or 1 yr? When we insert a Pitot tube in a flow and measure a pressure and a stagnation pressure in order to use the formula in Fig. 1, we have in mind time spans of tens of seconds during which we find the difference in manometer column heights. Experiments in which a fast response pressure gauge is used and pressure changes recorded with time resolutions as small as tenths of milliseconds could use time averages over spans of tenths of milliseconds. Pressure and velocity changes occurring in times shorter than this are considered fluctuations. What is the effect of the fluctuations on the pressure read by the Pitot tube? Since the Pitot tube formula requires knowledge of both p s and p, are we justified in using a time average of p? Definitive experiments to answer these questions have not been performed to the author’s knowledge, and users of Pitot tubes and Pitotstatic tubes must be satisfied with suggestions based on assumptions of the isotropic statistical character of turbulence. Consider the flow in the region of the Pitot tube mouth to be represented by component D + u1 parallel to the tube axis, where D is the mean flow velocity, and components u2 and us perpendicular to the axis. Further assume that the incompressible Bernoulli equation, Eq. (1.2.2), is applicable at each instant, so that ps = p
+ t p [ a + u1)2 + uf + 4 3
=p
+ tp&.
Instantaneously the velocity vector will not be aligned with the tube axis,
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
253
so that it effectively has yaw angle a and, as depicted in Fig. 2, the measured pressure will differ from p s by an amount B(cos a - 1)3pu;,, where B depends on the tube nose shape and B(cos a - 1) is an empirical fit to the curves of Fig. 2. Therefore the pressure in the Pitot tube, a function of time, is pm =p
+ 3p&f + B ( c o ~LI!
- l)bp&f.
Over the time span we have chosen for averaging, we now seek the mean of this pressure. Using cos a = (D + u l ) / u e Bmaking , series expansions according to powers of ul/D, u 2 / 0 , and uJO, and neglecting terms of fourth and higher order, we obtain,' calling u ' ~= u: = uf = u% in isotropic turbulence,
-
(pJm =
+ 3pD2[1 + (uf2/02)(3- B)].
(1.2.9)
B may be larger or smaller than three, depending on nose shape, and therefore may be larger or smaller than Similar considerations applied to a static tubes allow an expression for the effect of turbulence on a Pitot -static tube to be obtained:
- bs)m -
(P)m = ipDZ[l + (~"/@)(3- B
+ B')],
(1.2.10)
where B' is an empirical factor representing cross-stream effects on the static probe. Results of an interesting unpublished study of the turbulence effects on the reading of a static probe are summarized in Section 5.3.2. In this study, u 2 , u s , and pinside were measured with time resolution of 30 ps, and the signals used to compute by analog circuits a poutside based on considerations similar to those leading to Eq. (1.2.9). The study showed that the calculated corrections to the unsteady inside pressure needed to yield the outside pressure were as much as 100% of the measured fluctuating pressure, but it also showed that the rms value of the measured fluctuations only needed corrections of up to 20%. Since the constants B and B' in Eq. (1.2.10) are presumably separately controllable by altering the nose shape and static hole position, it is conceivable that a probe might be designed so that, in certain kinds of turbulent flow, the Pitot-static tube reading would need no correction. Reference to the actual shapes of the curves in Fig. 2 suggests that the curve fitting assumption of (cos a - 1) might be inapplicable at angles a > 20°, at which value u2/D = 0.06. In summary, it seems that interpretation of Pitot tube readings in turbulent flows of turbulent intensity u'/D = 10% or higher is not possible at present. J. 0.Hinze, "Turbulence," p. 137. McGraw-Hill, New York, 1959.
254
1. MEASUREMENT OF VELOCITY
1.2.2.7. Multitube Probes for Direction Measurement. Reference to Fig. 2 shows that while a Pitot tube is not very sensitive to angle change when it is nearly aligned with the flow, it is quite sensitive at 45". When two square-ended tubes, each bent to make an angle of 45" with the flow but in opposite senses, are mounted rigidly together, the pressure difference between them is very sensitive to alignment. Such arrangements have been used to find flow directions in a steady flow by adjusting the structure until a sensitive differential pressure gage connected to the two tubes reads zero. This claw probe is easily fabricated, but awkward to use and easily bent so that it loses its calibration. More common are the Conrad probes, which consist of two or more parallel tubes in a bundle; the mouths of the tubes are not normal to their axes but slant back, making angles of opposing sense with the flow. The pressure difference between opposite pairs is even more sensitive to the alignment of the bundle than is the claw probe. When the total apex angle is 70"for two tubes mounted side by side and the pressures are measured separately, the sum is used for the stagnation pressure and the yaw angle is found from the pressure difference by calibration. The procedure of merely recording the pressures rather than adjusting to a null value saves considerable time. The probe is made as small as 3-mm overall diameter. Sensitivity to direction can be a small fraction of a degree. When five separate pressures are recorded from a five-tube Conrad type probe, it is possible to determine both magnitude and direction of the velocity as well as the pressure (see Chue, Ref. 4,p. 210). Calibration of such a probe must be carried out in a wind tunnel where the velocity is known from previous measurements. Other complex probes, e.g., a cylinder with a hemispherical nose with five taps placed in the nose and led with separate piping to pressure gages, are described in the l i t e r a t ~ r e . ~ 1.2.3. Propeller and Vane Anemometers
These devices are simple, rugged, mechanical, portable, and not very accurate. Two general types can be discerned. One, typified by a tree, bends more in a wind the harder the wind blows, employs a spring to resist the aerodynamic force on a stationary body in the flow. The other, typified by the propeller, spins faster the harder the wind blows, allows vanes to whirl unobstructed as fast as required for the net torque produced by the aerodynamic force to become zero. The first type responds to the square of the velocity, while the second type has an output varying linearly with the velocity.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
255
1.2.3.1. Principle of the Vane Anemometer. The aerodynamic (or hydrodynamic) force on a rod, a plate, or an airfoil, variously called lift, drug, or thrust depending on the situation, is to a first approximation proportional to the surface area of the body and to the square of the velocity of the fluid stream. The direction as well as the magnitude of the force on a plate or other flat body depends also very strongly on its orientation with respect to the fluid stream direction. The aerodynamic force is the integral over the surface of the normal and shear stresses acting in the fluid where it contacts the surface, and the stresses in turn are dependent on the flow pattern assumed by the fluid stream as it passes the plate or other flat body. Measurement and calculation of the flow patterns constitute the sciences of hydrodynamics and aerodynamics to a large extent. There are a number of simplifying assumptions which apply to many special cases; one of the most fruitful is the assumption of inviscid, irrotational flow combined with Prandtl’s boundary layer approximation, which says that the shear stresses in the fluid play a role only in a thin layer of the fluid at the boundary of the plate or other flat body. With the assumption of inviscid, irrotational, steady flow, the Bernoulli formula stated in Section 1.2.2.1 by Eq. (1.2.2) is usable both in water and in air at Mach numbers below 0.2: p + bpu2 = p s .
(1.2.11)
The mass density of the fluid is p , and p and u are the pressure and the fluid speed at any point in the,flow. The stagnation pressure p s (sometimes called the impact pressure or the total pressure), is the same constant number everywhere in the flow field because of the assumptions of inviscid, irrotational, steady flow. With these assumptions, the aerodynamic force on a plate or other flat body can be calculated once the fluid speed u at various places over the surface is known; for where the speed is high, the pressure is low, and vice versa. Finding the flow pattern in the stream about a plate or other flat body is the result of both experiment and theory. Generally speaking, the flow divides on the front of the body somewhere near the leading edge and comes together again after completing its path about the body. Except behind a streamlined body, the flow path leaves the body (this is called separation) before rejoining the fluid passing around the other side, and a region of stagnant fluid more or less at rest relative to the body is trapped and keeps the stream from smoothly rejoining the streams passing around the other sides. The stagnant wake is actually a violently churning and intermixing mass of trapped fluid which was kept from passing downstream earlier, and fluid captured from the passing stream.
256
1.
MEASUREMENT OF VELOCITY
In the wake the pressure, while rapidly varying, is on the average about the same as the pressure in the undisturbed stream moving at speed urn, which is lower than ps and lower than the pressure in the parts slowed in negotiating the front of the plate or other flat body. The net effect of the generally higher pressures in the slowed parts of the flow and the lower pressures in the speeded up parts and in the turbulent wake is the aerodynamic force. This is the force that is used to steer a boat or airplane with rudder, elevator, and aileron. It is also the force that is used to measure the speed vco in the stream passing the plate or other flat body used as a vane anemometer. Measurement of rapidly fluctuating velocity components was undertaken by Siddon and R i b n e P by sensing aerodynamic forces on a disk or on an ogival nosepiece mounted on an elastic beam whose bending was sensed by piezoelectric strain gauges. The use of this anemometer in turbulent flow measurement is described in Section 5.3.3. WHIRLING VANEANEMOMETER.In the discussion of lift and drag, urnis the free stream speed relative ro the plate or flat body to which the aerodynamic force is applied. In the whirling vane anemometer, the body is one of several arms free to rotate and to change its speed relative to the stream. The anemometer operates on the principle that the speed of rotation of the vanes adjusts itself until the aerodynamic force falls to zero, or actually until the total torque on the rotating system due to the aerodynamic forces balances the total torque produced by the bearings and any sensing element used to detect the speed of rotation. As with propellers used to provide thrust for an air vehicle or for a fan or compressor, it is advantageous to choose an airfoil shape to reduce the size of the wake as much as possible. This is, of course, because the blade is constrained by the rotor to which it is attached to move in a direction making an angle with the stream, and it can practically never move in the direction of the free stream. It moves so as to adjust its angle of attack a to the value where the net torque about the rotor axis is small or zero. Other whirling arms carry cups or cones which have different aerodynamic forces on them depending on whether the convex or concave side of the cup or cone faces the stream. At the steady state, the greater speed of the convex face relative to the stream gives the same torque magnitude as the smaller relative speed of the concave side, so that the speed T. E. Siddon and H. S. Ribner, AIAA J . 3,147-749 (1965).
8T.E. Siddon, Rev. Sci. Instrum. 42, 653-656 (1971). T. E. Siddon, A new type turbulence gauge for use in liquids. I n “Flow-Its Measurement and Control in Science and Industry” (R. B. Dowdell, ed.), Vol. 1, p. 435. Instrument Society of America, Pittsburgh, Pennsylvania. 1974.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
257
of rotation can be calibrated to indicate the stream speed in which it is immersed. If the stream is unsteady (gusty), the cup anemometer's rotation is not the same as that in a steady stream having the same average speed, but higher, as discussed in the next section. If the axis of the whirling cup anemometer makes an angle other than 90" with the stream velocity, the calibration is affected. Since this device is used mostly in meterological field studies and monitoring where steady winds are only horizontal, a vertical axis assures a reasonable estimate of the wind speed regardless of the wind orientation in the horizontal plane. Whirling vane anemometers may be mounted on swivels so that the axis about which the vanes whirl aligns with the wind. Whirling vane anemometers are also calibrated at various angles of yaw so that three mounted at mutually right angles may be used to monitor the direction and magnitude of the wind by appropriate combination of the readings of the three propeller speeds. Calibration curves of vane and cup anemometers are quite linear and they are sensitive down to fractions of a meter per second in air, and down to centimeters per second in water. In comparison to the Pitot tube, the other simple and unsophisticated anemometer, they are more reliable in the low speed ranges, mainly because of their linear response. A variety of vane and cup anemometers is available commercially. 1.2.3.2. Dynamic Response. Since the wind and other fluid streams in which measurements are made are often unsteady, the response time with changing speed and direction is of importance. Often, in meteorology and many other applications, one is not interested in detailed recording of fluctuating velocities and prefers an instrument giving speed and/or direction averaged over some time. As seen, there are many factors which control the size of the aerodynamic force other than the free stream velocity and the surface area, but for a given size and shape of body at a given orientation in the stream, the Bernoulli formula implies that the net force will vary in proportion to 0%. Dimensional analysis and modeling principles (see Chapter 10) suggest the aerodynamic force F is given by
F
=A
t p u f f ( a , Re, Shl, Shz,
. .
.),
(1.2.12)
where a is the angle the plate or other flat body makes with the stream, Re is a Reynolds number playing a role in the boundary layer which has been overlooked in the simple discussion above, Sh (of which there may be several) are shape factors characterizing the flat body, A is the area of the plate or other flat body, and p is the mass density of the fluid. Equation (1.2.12), used often in hydrodynamics and aerodynamics, while con-
258
1. MEASUREMENT OF VELOCITY
structed by dimensional analysis, is strongly suggested as well by the Bernoulli equation, Eq. (1.2.11). The dependence on a,the orientation angle relative to the stream, is often quite complicated since the place on the body where separation occurs is determined by several factors in addition to a. RESISTINGVANE.A plate suspended and free to tilt about its top edge deflects from the vertical by increasing amounts as the restoring torque due to the weight balances the increasing aerodynamic force associated with larger wind speed or water current. A scale marked on an adjoining bar can be calibrated in meters per second, either by calibrating it in a stream of variable and known speeds, or by towing it at known speeds. Instead of suspending the plate, it may be supported by a spiral spring connected to a rigid member, A more sophisticated version supports a fiber or hair and observes the deflection with a microscope.1oThe inertia of a suspended plate, and the moment of inertia of a vane or cup anemometer, are important factors in the response time; since in general the net force on a vane varies with &, the transient force driving the inertia when exceeds the average is larger than the transient force when v is less than the average. The net effect is to produce a reading larger than the average, i.e., the resisting vane deflects more and the whirling vane whirls faster than they would in a steady flow having the average speed. In addition to needing to know what correction is needed for average speed readings, experimenters also need to interpret records of anemometer readings as a function of time for studying turbulence. McMichael and Klebanoff l 1 have proposed a method of calibrating an instrument in a wind tunnel to determine three characteristic parameters: K, the meter factor, i.e., the slope of speed of revolution versus steady fluid speed; U o , the fluid speed intercept for zero revolution; and the fengrh constant L = 721,where T is the time constant for the instrument and is roughly the time after the rotor, when released from a locked state in the steady stream of speed v , reaches l / e of its final value. See Ref. 1 1 for the precise definitions of K, U o , and L. Given these three characteristics, one can calculate the over-registration of mean speed, and the fundamental and second harmonic of the response, for a periodically fluctuating stream speed of given relative amplitude E = u'/ti, frequency 27r/o, and harmonic content Ck. The fractional error in mean speed (overspeeding) is k= 1
(1
1 + 2p0 + PO)*+ ( k W 7 ) 2
1
'
( 1.2.13)
K. W. Bonfig, "Technische Durchflussmessung," p. 79. Vulkan-Verlag, Essen, 1977. J . M. McMichael and P. S . Klebanoff, "The Dynamic Response of Helicoid Anemometers," Rep. No. NBSIR 75-772. National Bureau of Standards, Washington, D.C. (available from National Technical Information Service, Springfield, Virginia). lo
I'
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
259
In Eq. (1.2.131, Po = Uo/Orepresents the friction in the bearings of the instrument. For negligible friction, and for very high frequencies, this expression reduces to 5 = c2/2, i.e., a high frequency turbulence level of E = 0.5 will cause the anemometer to whirl 12.5% too fast. Another example is a square wave variation in which the velocity of the stream is zero every half cycle and one the other half. For a frequency such that w7 = 1, the overspeeding is about 20%. Examples of the response of the fundamental and the second harmonic, both according to the theory and of an instrument studied in a controlled unsteady wind tunnel, raise serious doubts concerning the use of steady calibration data for extraction of quantitative data in turbulent flows, and doubts concerning linearized methods of correcting for the nonlinear response of whirling vane instruments. The level of agreement of the nonlinear theory of McMichael and Klebanoff” with the experiments on an instrument in the unsteady wind tunnel is good, however, and indicates that some meaningful corrections to a time-dependent output from a whirling vane instrument can be made with confidence. The theory in its current form predicts the performance of an instrument when the unsteady flow is characterized (by hot wire and hot film methods; see Section 1.2.4), but the inverse problem of learning the unsteady flow characteristics from the readings of the whirling vane instrument is not considered. 1.2.4. Hot-wire and Hot-Film Anemometers* A
B C F Gr K Kn L M Ma Nu P
Pr
Q R Re
List of Symbolst S area, amplifier gain, constant constant capacitance, constant T heat flow U Grashof number Z gain parameter a Knudsen number OR inductance aT time constant b Mach number C Nusselt number d power e Prandtl number g heat h resistance Reynolds number k
* Section 1.2.4 is by Ron F. Blackwelder, t For Section 1.2.4.
surface area, sensitivity coefficient temperature velocity impedence constant resistive overheat ratio thermal overheat ratio constant specific heat diameter voltage attenuation factor convective heat transfer coefficient thermal conductivity, constant
260 I m n P
4 r S
t
u,u,w X.YJ
a
P Y 6 c
c
e K
A CL
z
P
1.
MEASUREMENT OF VELOCITY
length exponent, mass exponent linearizer constant heat loss, linearizer constant radial direction distance time, thickness velocity components in ( x , y , z ) directions Cartesian coordinates temperature coefficient'of resistivity nonlinear coefficient of resistivity ratio of specific heats unbalance parameter empirical constant damping coefficient angle thermal diffusivity wavelength time constant length fluid density
U
7
4
x
JI 0
(
)c
(
)e
( )I ( )L ( I,, ( )r
( s,.,) ( )s ( )t
( )" (-)*
0 A( ) ( )-
material density, StefanBoltzmann constant time constant angle resistivity angle radial frequency SUBSCRIPTS-SUPERSCRIPTS conduction, characteristic end fluid linearizer output, reference condition, cutoff radiation root-mean-square substrate test velocity dimensional calibration constant temporal average fluctuation infinity
1.2.4.1. Introduction. Hot-wire and hot-film anemometers are among the most versatile and widely used instruments for measuring fluctuating velocities in the field of fluid mechanics. Their popularity stems from their small size (order of lmm) which results in high spatial resolution, a fast time response up to 1 MHz, high sensitivity, low cost, and ease of fabrication. In principle, both hot-wires and hot-films operate by electrically heating a resistive element and electronically monitoring the amount of heat transfer which occurs as the fluid passes over the element. The heat loss depends upon the fluid and the geometrical and physical properties of the sensing element and is a strongly nonlinear function of the velocity. The sensors can easily be calibrated and, when operated at a constant temperature, their output signals can be linearized to give an electrical signal directly proportional to the instantaneous velocity over large ranges. Under different operating conditions, these devices can be used to measure other fluctuating flow parameters, e.g., temperature and density.
1.2.4.2. Physical Characteristics 1 . 2 . 4 . 2 . 1 . HOT-WIRES.The sensing element of a hot-wire anemometer is a short length of small diameter wire that is heated electrically. Each sensor is supported by two large prongs, as seen by the examples in Fig.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
26 1
4. The most common wire materials are tungsten, platinum, and platinum alloys, e.g., platinum-rhodium. Other conductors, e.g., nickel, silver, and copper, can be used for hot-wire material: however, they are less desirable because they cannot be drawn into such small diameters, their resistivity is nonlinear, their melting point is too low, etc. Some pertinent properties of the most popular materials used in making hotwires are found in Table I. The diameter of hot-wires has decreased considerably since their early use in the 1920s. Today, wires of platinum and its alloys are commercially available with diameters as small as 0.25 pm. These wires are made by the Wollaston process, which consists of coating the original large diameter platinum wire with a sheath of a different metal that has similar ductile properties, e.g., silver. This combination is then drawn through a small die and stretched to yield a smaller diameter. The outer silver sheath is removed by etching with an acid which does not effect the noble platinum wire. One-tenth normal solutions of hydrochloric or nitric acid can be used. The etching process may be accelerated by passing a small current from the wire through the acid bath, with the wire connected to the positive electrode. Tungsten cannot be processed in this manner, because a suitable sheath material has not been discovered with the appropriate ductile properties that can be easily removed after stretching. Thus, small tungsten wires are made by first drawing them to a minimum diameter and then decreasing the diameter further by electrically etching them in an acid bath under controlled conditions. Tungsten wires with diameters as small as 1.9 pm have been produced by this manner. This process often leaves a
FIG.4. Five hot-wire sensors with different support geometries. The distances between the prongs is 3 mm, and the sensing elements are 1.25 mm long. (Courtesy of DISA Electronics, Franklin Lakes, New Jersey.)
TABLEI. Physical Properties of Hot-wire and Hot-Film Materiala
Resistivity xo cm)
Temperature Tensile coefficient strength (dyne of resistivity, a (I-]) cm-2 x 109
Mass
Specific heat, c
density (Gal (g ~ m - ~ ) g-1 ‘C-1)
cu xoa
(cal cm-* 106)
Melting point (“C)
Comments
3410
Oxidizes above 350°C; cannot be soldered Soft and weak Nonductile Nonductile; good far high temperatures Stronger than platinum Stronger than platinum Resistance is nonlinear at high temperatures
Tungsten
7.0 (5.5)
0.0036 (0.0048)
22 (11)
19.3
0.034
Platinum Rhodium Iridium
9.8 (9.6) 5.5 (4.3) 6.0 (4.7)
0.0039 (0.0039) 0.0029 (0.ooSS) 0.0031 (0.0042)
4.1 (1.7) 19 (8.3) 23 (12)
21.5 12.4 22.6
0.032 0.058 0.037
18 (18) 45 (36) 45 (42)
1769 1966 2443
PI-1WoRh Pt-ZO%Ir Nickel
18.9 (18.4) 32.0 (30.9) 6.5 (6.1)
0.0016 (0.0017) 0.0007 (0.0008) 0.0064 (0.0067)
6.6 (3.2) 13 (6.9) 6.9 (3.3)
19.9 21.6 8.9
0.035 0.032 0.105
23 (22) 31 (28) 22 (23)
1830 1840 1452
26 (25)
Data are for the hard metal or alloy with the annealed values in parentheses. References: V. A. Sandborn, “Resistance Temperature Measurements,” Chapters 5 and 12. Metrology Press, Fort Collins, Colorado, 1972, and the brochure “Properties of Metals and Alloys” from Sigmund Cohn Corporation, Mount Vernon, New York.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
263
FIG.5. Portions of the hot-wire sensing elements as seen by an electron microscope with 10,000 magnification. (a) A 2.5-pm-diameter platinum wire. (b) A 3.8-pm-diameter tungsten wire.
rough surface on the wire which can be seen by an electron microscope, as shown in Fig. 5 . At such small diameters, the wires are quite fragile and the tensile strength of the material is an important parameter. Tungsten has the highest strength, as shown in Table 1, and its popularity stems from that property. The platinum alloys have about twice the strength of platinum alone, and thus are often more popular. The inadvertent destruction of
264
1.
MEASUREMENT OF VELOCITY
these elements during use is due to improper handling, to dust particles striking the element, or to excessive surges of current through the wire. In air, the wires rarely break because of simple aerodynamic loading, as discussed by S a n d b ~ r n , ' although ~~ vibration can destroy the sensors if the supports do not have sufficient rigidity. Another important property of the hot-wire material is the resistivity x, which is the fundamental measure of the electrical resistance independent of the geometrical shape. As seen in Table I, the resistivity at a fixed temperature can vary by a factor of three among the common hot-wire materials. The resistivity is adequately related to the temperature by
x
=
xo[l
+ a(T - To) + P(T - Toy. . .I,
( 1.2.14)
where xo is the resistivity at a reference temperature To, and a and p are constants. In the following analysis, the quadratic and higher order terms will be neglected, which is a valid assumption except whenever large temperature differences are used or when p is sufficiently large, e.g., with nickel. The temperature coefficient of resistivity a varies by a factor of two among the common hot-wire materials, as seen in TaMe I. The electrical resistance of a wire or film of length 1 is
R
=
I i g d z ,
(1.2.15)
where A is the cross sectional area and z, the coordinate along the wire axis in the direction of the current flow. Since the temperature will vary along the length of the wire, (see Section 1.2.4.6.1),the measured resistance is an averaged quantity, as discussed in Section 1.2.4.3.1. The behavior of these metals at high temperature is important because in application they will often be heated to several hundred degrees Celsius. Above 350°C tungsten begins to oxidize. This condition should be avoided since it can change the properties of the sensor and hence the calibration of the anemometer. Platinum and its alloys are relatively inert metals; however, they have lower melting temperatures than tungsten and burn out more easily. It is usually desirable to have a sensing element that is several hundred diameters long. This insures that most of the heat loss is due to the velocity dependent convection and not due to the conductive losses to the end supports. On the other hand the desire to measure the velocity at a point in space necessitates that the length be small. This consideration is especially important in turbulent flow fields, where the velocity is a random ISa V. A. Sandborn, Resistance Temperature Measurements, Chapters 5 and 12. Metrology Press, Fort Collins, Colorado, 1972.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
265
variable and is associated with a multitude of spatial scales. As the fluid is convected past the sensing element, the heat loss changes in time, and hence the output signal from the hot-wire or hot-film anemometer varies accordingly. To insure that the anemometer responds correctly to the instantaneous velocity fluctuations, Wyngaardl4 has shown that the length of the sensing element should be no longer than the Kolmogorov length scale.* This criterion can not easily be satisfied in most turbulent flow fields. Fortunately, this does not introduce any significant measurement error into the larger length scales, i.e., the lower frequencies. Statistical corrections such as that proposed by Uberoi and K o v a ~ z n a y can ' ~ be used when studying the small scale (high frequency) data. For standard wind tunnel and atmospheric tests, platinum and its alloys and tungsten have been the most popular hot-wire materials. This stems from their relatively low cost, ductile properties, ease of fabrication, and availability in small diameters. In extreme environments, other materials have relative advantages; for example, the high melting point of iridium has prompted its use in combustion studies. The choice of the diameter of the sensor will involve a compromise between the desire for a small sensor with good spatial resolution and frequency response, and the practical considerations of cost, ease of fabrication, and strength. The hot-wire sensors are supported by attaching their ends to prongs which can be fabricated from sewing needles or jeweler's broaches. These supports should be rigid to prevent vibration of the sensing element, yet they should be small to minimize probe interference with the flow. Needles can be obtained with diameters as small as 200 pm, and jeweler's broaches of 75-pm diameters are commercially available. Both of these work satisfactorily at low dynamic pressures, )pW. In transonic and supersonic flows, larger forces on the prongs necessitates that sturdier supports be used. Often hot-wires are made so that the sensing element does not span the entire distance between the prongs. As seen in Fig. 4, larger diameter sections consisting of the wire and a conducting sheath extend from the sensing element to the supports. For Wollaston processed wires, this configuration is easily constructed by first attaching the wire and its sheath to the supports, and then removing the sheath from the sensor by a small jet or a capillary bubble of the etching fluid. If there is excessive
*' la
J. C. Wyngaard, J . Phys. E . 1, 1105-1108 (1968). M. S.Uberoi and L. S. G . Kovasznay, Q. Appl. Math. 10, 375-393 (1953).
* The Kolmogorov scale is a measure of the distance over which the viscosity is effective in dissipating the velocity fluctuations. It is usually much smaller than 1 mm.
266
1. MEASUREMENT OF VELOCITY
TAINLESS STEEL
ATING DEFINES L Q U A R T Z COATED PLATINUM FILM SENSOR ON GLASS ROD (0.001'' TO 0.006" DIA.) la)
QUARTZ COATED PLATINUM FILM
GOLD FILM ELECTRICAL LEADS (C)
QUARTZ COATED HOT FILM ON SURFACE
STAINLESS STEEL TUBE SHIELDING QUARTZ RO Id)
FIG. 6. Hot-film probes with various geometries: (a) cylindrical hot-film, (b) wedge hot-film, (c) conical hot-film, and (d) flush mounted hot-film. (Courtesy of Thermal Systems Corporation. St. Paul, Minnesota.)
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
267
tension in the wire before etching, the wire may break as the tension is supported by a smaller and smaller diameter during the etching process. For tungsten wires, a sheath of a good conductor must be deposited onto the wire, usually before attaching to the prongs. By immersing the portions to be plated into a solution of copper sulfate and establishing a small current through the wire, a uniform copper jacket is deposited around the immersed portion. The thickness can be accurately controlled by the magnitude of the current and the amount of time allowed for the process. For low speed flows, the hot-wires can be attached to the broaches by either soft solder or by welding. Platinum and all of its alloys can readily be soft soldered by using standard solders. Tungsten cannot be soldered directly; thus its ends must first be coated with a solderable material. On the other hand, all of the materials can be easily welded by a small spot arc welder. When hot-wires are used in flows with high stagnation temperatures, welding is usually mandatory as the solders may melt at the high temperatures. 1.2.4.2.2. HOT-FILMS.There are two primary distinctions between hot-wires and hot-films; they are constructed differently, and hot-films have an additional means of heat loss. Both of these differences are manifested in the substrate, which is a thermal insulator onto which the submicron thick film is deposited. The substrates are usually quartz, but mica and other nonconductors can be used. Three principal substrate shapes are in use today; the wedge, cone, and cylinder, as shown in Fig. 6. Parabolic and flush mounted hot-films are also commercially available. The films are deposited onto the quartz by vacuum sputtering, which ensures a uniform thickness of the sensing element. Only in the case of the cylinder is the substrate almost completely covered with the thin film. By carefully controlling the sputtering process, film thicknesses of approximately 0.1 pm are obtained. The films are usually made of platinum or nickel. After the thin sensing portion has been deposited, a thicker layer of a conducting material is sputtered or otherwise attached to the ends of film to supply the electrical current. This provides a well-defined sensing area in the middle of the cylinder or near the tip of the wedge or cone. It is standard practice to coat the film with a 1.0 pm thick deposited layer of quartz or similar insulating material. This not only protects the film from abrasion, but provides greater electrical stability as well. If the probe is to be used in a conducting fluid, such as salt water or mercury, a thicker 2-pm layer of quartz is sputtered onto the film to guarantee that the electrical conduction is only through the film and not through the fluid. This external coating also protects the film from corrosion and electrolysis. Since the sputtering process is more complicated than sol-
268
1.
MEASUREMENT OF VELOCITY
dering or welding, hot-films are more difficult to construct but are less fragile than hot-wires. The cylindrical hot-film probes are typically 25-50 p m in diameter and are 1-2 mm long. The sensing elements of other film probes have nominal dimensions of 1 mm in the cross-stream direction and 0.1-0.2 mm in the streamwise direction. The thickness of the films are usually controlled to give a cold resistance of 5-20 a. Although the substrate material is more brittle than most hot-wire metals, its larger cross sectional area gives it considerably more strength under ordinary operating conditions. This is especially true of the wedge and cone type probes. These configurations also are less sensitive to particulate contamination since dust and other impurities will not adhere as readily to the slanted surfaces. Because of the geometry, hot-films have much smaller length-to-diameter ratios than hot-wires. Consequently the temperature distribution along the sensor is less uniform. A more fundamental distinction between the two probe types is that the substrate of the hot-film provides an addition path for heat loss, i.e., heat can be conducted through the substrate and lost by convection to the ambient fluid. This effect is greatest in the noncylindrical probes because of the more massive and exposed substrate. Combined with the small length-todiameter ratio and the difficult geometry, the hot-film’s temporal response is more complicated than the hot-wire’s, as discussed in Section 1.2.4.4.3. 1.2.4.3. Principles of Operation 1.2.4.3.1. BASICRELATIONS. Hot-wire and hot-film anemometry utilize the convective heat transfer from the heated sensor to the surrounding fluid to measure the velocity. The amount of heat transferred per unit time is inferred from an electric signal that is related to the temperature of the sensor. This signal results from the unique relationship between the temperature and resistance of the sensor. From Eqs. (1.2.14) and (1.2.13, the resistance of the sensing element R is related to its temperature T by R = RJl
+ a(T - To)],
(1.2.16)
where R, and To are the average resistance and temperature at a reference condition, and a is the temperature coefficient of resistivity. When the thermodynamic properties of the flow are constant, R, and To will refer to the ambient conditions. Variable ambient conditions are discussed in Section 1.2.4.6.5. It is assumed in Eq. (1.2.16) and throughout that R and T denote the average resistance and temperature of the sensing element. The variation of these parameters along the axial direction of the sensor is discussed in Section 1.2.4.6. I. An important parameter governing the operation of hot-wire and hot-
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
269
film anemometers is the overheat ratio defined as UT
= (T - To)/To,
(1.2.17)
where T is the temperature of the heated element and To is a reference temperature in Kelvins. Usually, the ambient temperature is used for the reference conditions. In operation, it is more practical to use a resistive overheat ratio defined by UR =
( R - Ro)/Ro,
( 1.2.18)
where R is the resistance of the heated sensor and R, is the resistance in the reference state, often called the cold resistance. Note that uR = aToa,. 1.2.4.3.2. BASIC HEATTRANSFER MECHANISMS. Heat is introduced into the sensing element by Joule heating and is lost by convection, conduction, and radiation. Usually the sensing element radiates only a small amount of energy to its surroundings. If the sensor radiated as a blackbody, the radiated heat would be q r = aS(P -
GI,
( 1.2.19)
where u is the Stefan-Boltzmann constant (56.7 nW m-2 K-4); S, the radiating area; T, the temperature of the sensor; and TI, the temperature of the fluid environment. Typical hot-wires radiate only about 10% as much heat as a blackbody. Under normal operating conditions, the radiation losses are much less than 0.1% of the convection losses and hence will not be considered further. However, if the sensor is operated in low density fluids, the convection losses become quite small, and the radiation can become a significant percentage of the total heat loss. 1.2.4.3.2.1. Convection. Most hot-wire and hot-film anemometers operate in a forced convection regime in order to maximize their sensitivity to the velocity. However, at low velocities, i.e., small Reynolds* numbers, free convection is important. Collis and Williamslghave shown that the free convection is small if the Reynolds number of the sensor is related to the Grashof number by Re > 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.
MEASUREMENT OF VELOCITY
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
0.24 0 0 0 0 0.32
0.56 0.48 0.89 0.82 0.62 0.43
0.45 0.51 0.33 0.38 0.47 0.52
0.17 0.17 0.08 0.09 0.12 0
From D. C. CoJlis and M. J. Williams, J . Fluid Mech. 6, 357 (1959).
* From R. Hilpert, Waermeabgabe von geheizten Draehten und Rohren, L u f , Forsch, Arb, Ing. Wes. 4, 215 (1933).
‘ From W. H. McAdams, “Heat Transmission,” 3rd ed., Chapter 10. McGraw-Hill, New York, 1954. cated here. The role of the aspect ratio l / d is minimized by assuming that it is large enough so that the heat loss is essentially two-dimensional. The effect of the angle of inclination 8 will be discussed in Section 1.2.4.6.2. By assuming that 8 and y are constant for the present analysis, the last equation reduces to Nu
=
Nu(Re, Pr,u T ) .
(1.2.21)
This is the functional form of the convective heat transfer that is valid for incompressible fluids under the above restrictions. Even after the restrictions imposed in the previous paragraph have been made, there is still a lack of agreement in the literature concerning the exact form of the relationship expressed in Eq. (1.2.21). Many authors have sought an explicit form for the convective heat loss, and there is considerable evidence that NU = [ A ( h ,
UT)
+ B ( R , U,)Re”](l + U T / ~ ) ”
(1.2.22)
correlates most of the available data, where A , B, and n are functions of the Prandtl number. In addition, the exponents vary slightly with the Reynolds number. The values of the constants appearing in Eq. (1.2.22) obtained from the several authors are listed in Table 11. It can be seen that the heat transferred is not a strong function of the temperature loading factor ( 1 + aT/2). Since King20was the first to recognize that the convection losses could be written in the form of Eq. (1.2.22), this relationship is often referred to as King’s law. It is distressing (especially to the uninitiated) that there is such a lack of agreement concerning the basic equation of hot-wire anemometry. Indeed, if there were a more exact and reliable form of Eq. (1.2.22), the use 2o
L. V. King, Philos. Trans. R . SOC.London, Ser. A 214, 373 (1914)
272
1. MEASUREMENT OF VELOCITY
of hot-wires and hot-films would be easier. The values of the coefficients and the form of the heat transfer equation become even more diverse when the other variables, e.g., Ma and lid, are included. These variations and deviation from an exact universal equation necessitate that each hot-wire and hot-film be calibrated individually, as explained in Sections 1.2.4.4.5and 1.2.4.5.3,before it can be used to record and interpret data. 1.2.4.3.2.2. Heat Conduction within Hot- Wires The wire can change its temperature distribution by conduction in either the radial or the axial direction. Since the wire is an excellent heat conductor, and its diameter is small compared to its length Bensen and Brundrett2*have shown that radial conduction within the wire can be neglected in comparison to the axial conduction. The axial conduction along the length of the wire can be an important mechanism by which heat is lost from the sensor. The importance and magnitude of the heat lost by conduction to the prongs of a hot-wire or cylindrical hot-film can be estimated. The rate of heat transfer to an end support is given by
(1.2.23) where k is the thermal conductivity of the probe material and A is the cross-sectional area. With a uniform flow field over the sensor, the temperature distribution given by Eq. (1.2.72) yields a loss through both ends of qe = [ k d ( T - To)m/21c]tanh(1/21,),
where d and 1 are the diameter and the length of the sensor. The length 1, is a characteristic distance along the wire over which the effect of the supports is felt, and (T - To)= is the asymptotic temperature difference for a wire of infinite length. Comparing this to the heat loss to the fluid by convection from Section 1.2.4.3.2.1 results in
where the subscript f denotes the properties of the fluid environment. For a platinum wire in air, the thermal conductivity ratio is roughly 2500, which necessitates a large l / d ratio in order to minimize the end losses. For cylindrical film probes with diameters greater than 25 pm in water, the conduction to the end supports is primarily through the substrate because of its larger cross-sectional area. The ratio of the thermal conductivR. S. Bensen and G . W. Brundrett, “Temperature, Its Measurement and Control in Science and Industry” (C. M. Herzfeld, ed.), Vol. 3, p. 631. Van Nostrand-Reinhold, Princeton, New Jersey, 1962.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
273
ity of the quartz to that of water is approximately ten. To maintain the same loss ratio in Eq. (1.2.24) the probe does not need to have f / d as large as does the hot-wire in air. Typical length-to-diameter ratios for hotwires are 200-500, whereas values of approximately 20-50 are used for hot-films. For film probes of other geometries, the conduction to the massive substrate far exceeds that to the end supports of the cylindrical probe. The heat conducted through the substrate will ultimately be lost to the fluid environment. Due to the complicated geometry, these losses are impossible to accurately describe analytically. Thus Eq. (1.2.24) can only indicate how the various parameters influence these losses. The heat lost through the substrate and ends can be decreased by minimizing the conducting area, i.e., making the substrate small. Residual losses must be accounted for experimentally by calibrating each sensor. 1.2.4.3.2.3. Heat Conduction within Hot-Films. The heat conducted from a hot-film probe is of greater consequence than that from a hot-wire because the backing material provides an additional path by which heat can be transferred. The heat conducted into the substrate per unit time is given by 4 s = kss(aT/ar)lr=,
9
where ks is the thermal conductivity of the substrate material; S, the surface area across which heat is transferred; and (aT/ar)),=,, the temperature gradient normal to the surface between the film and the substrate. Although this mechanism is the same as that governing the end losses of a wire, the conduction is into a heat insulator, usually quartz, whereas the conductive end losses are usually into a heat conductor. In spite of this difference, the geometry of hot-film probes provides larger areas through which conduction to the supporting material can occur, and thus more heat is lost into the substrate than to the ends of hot-wires. A more important difference between these two conduction mechanisms is that some of the heat transferred into the substrate may not be lost but only momentarily stored. This affects the frequency response of the hot-film as explained in Section 1.2.4.4.3. The total heat flow from the sensor is given by the sum of the convection, end conduction, substrate conduction, and radiation as: F = 4 c + 4 e + 4s + q r ,
where it has been implicitly assumed that each type of heat loss is independent of the others and the conduction losses are only to the prongs and substrate. Under the conditions stated earlier, i.e., high dynamic pressure and large I/d, the radiation and end conduction are small and can be neglected. For a hot-wire, the substrate conduction does not exist and
274
1. MEASUREMENT
OF VELOCITY
hence F = qc; for a hot-film, F = qc + q s . Since the heat conducted through the substrate is ultimately lost by convection to the ambient fluid, the functional form of both of these relationships can be written as F = F ( U , T - To),i.e., the heat loss depends only upon the convection velocity which enters through the Reynolds number in Eq. (1.2.22) and the temperature difference between the sensor and the fluid. Note that all of these parameters may be implicit functions of time. If the reference temperature To is constant, then F = F ( U , T). 1.2.4.3.3. HEATING.By definition, hot-wires and hot-films require a means of adding heat to the sensing element. This is accomplished by passing a current through the sensor from an external power supply. The power input to the sensor, i.e., the Joule heating, is P = ie = PR,
(1.2.25)
where e is the voltage drop across the sensor; i , the current through it; and R, its resistance. The simplicity of Eq. (1.2.25) is misleading because all of the parameters may be functions of time. Hence to obtain an electric signal which can be interpreted in terms of the fluctuating flow quantities, the hot-wire and hot-film are usually operated so that either the current or the resistance is held constant, as discussed in Sections 1.2.4.4 and 1.2.4.5. The power input can be implicitly represented by P = P(i, R ) , or by using Eq. (1.2.16) as P = P(i, T). 1.2.4.3.4. THE ENERGY EQUATION. The time rate of change of energy within the sensing element must equal the difference between the power input and the heat leaving the device. As discussed in the last section, the energy input is through Joule heating given by P . Likewise, we can represent the energy loss by the sum of the various types of heat flows described in Section 1.2.4.3.2 as F ( U , T). Following Kovasznay,22the energy equation can be written as
d Q / d t = P - F,
(1.2.26)
where Q is the total heat stored in the element. Letting c , m , and T represent the average values of the specific heat, mass, and temperature of the probe, Q can be written as Q = cmT. For a hot-film, the value of the specific heat and mass should be some appropriate average over the sensing element and the substrate. Normally the mass of the substrate is so much greater than that of the film that the specific heat and mass of the substrate alone are used. For sufficiently small temperature fluctuations, the specific heat is a constant and Eq. (1.2.26) can be written as
cm(dT/dt) = P ( i , T ) - F ( U , Z). L. S. G. Kovasznay, Acra Tech. Hung. 50, 131 (1965).
(1.2.27)
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
275
This relation provides the basic differential equation for the operation of the hot-wire and/or hot-film anemometer. Of course, the appropriate functional form of F which describes the heat losses under the operating conditions must still be determined by calibration. From an operational viewpoint, the desired output from the hot-wire or hot-film should be an electrical signal proportional to the velocity U . This is thwarted by two major obstacles: first, the relationship in Eq. (1.2.27) is a nonlinear function of the velocity; and second, some additional constraints must be used to eliminate one of the three variables i, T, and U . The nonlinearity is discussed below, and the latter problem is discussed in Sections 1.2.4.4 and 1.2.4.5. ENERGYEQUATION.By examining the 1.2.4.3.5. THE LINEARIZED general forms shown earlier for the power and heat losses, it is easily seen that none of the three variables, i.e., velocity, temperature, and current, enter Eq. (1.2.27) linearly. Even though there is no agreement on the best analytical form for the heat flux F, there is general agreement that it is highly nonlinear. Although many different specific forms for the heat flux and energy equation have been proposed, no closed form solutions have been obtained, primarily because of the nonlinearity. Such obstacles are familiar in the field of fluid dynamics, and the first attempt at circumvention is to linearize the equation about some appropriate mean values. As long as the deviations about the chosen mean values are small, a linear version of the equation can be obtained which treats the mean values as constants. To linearize Eq. (1.2.27), let the average of the resistance R be denoted by R . The fluctuating time dependent part is given by AR(t) = R(r) - R. Similar expression can be written for the velocity, current, and temperature. If a temporal average is used to generate the mean values, the averages are independent of time but can vary as the sensing element is moved in space. Since the variables U , i, and T have mean values plus fluctuating components, the heat loss F and the power input P can be treated similarly in spite of their nonlinearity, i.e., F
=
F+
AF,
P =
P
-k
AP.
(1.2.28)
By using a Taylor expansion, A F and A P can be shown to depend linearly upon the mean and fluctuating temperature, current, and velocity if higher order terms are neglected. Although this linear dependence is only an approximation, the results are accurate as long as (1) the functional forms of F and P are relatively smooth, and (2) the fluctuations are small compared to the mean values. The mean values will be governed by a time independent nonlinear equation derived from (1.2.27), and a linear equation will describe the temporal fluctuations. This latter equation will govern the dynamic response of the instrument.
276
1.
MEASUREMENT OF VELOCITY
The equation for the mean values is found by substituting the relations in Eq. (1.2.3.13) into the differential equation (1.2.27) giving
em d AT/dt
=
P
+ AP - F -
AF,
(1.2.29)
since dT/dt = 0. The time average of this equation is
o=P-F
( 1.2.30)
since the averages of the fluctuating quantities are zero by definition. Thus over a long period of time, the mean heat input P must equal the average heat loss F. The equation for the fluctuations is obtained by subtracting Eq. (1.2.30) from (1.2.29), yielding cm d AT/dt = AP - AF.
(1.2.31)
This equates the instantaneous changes in the heat stored within the sensor to the difference between the changes AP in the heat input and AF in the heat loss. The fluctuating heat loss and heat input are in this approximation linear functions of the changes in temperature, current, and velocity. An explicit representation of the heat loss can be obtained from the functional form F = F ( U , T ) as If the flux changes are sufficiently small, AF = dF. Doing the same for the other variables and substituting into Eq. (1.2.31) yields
em d AT/dt
+ (aF/aTI, - aP/aTJi)AT
aP/ailT Ai - aF/aUIT AU. (1.2.32) This expression states that the temperature fluctuations of the sensing element respond as a first order system with forcing functions due to the fluctuating current and/or velocity. Note that for a given operating condition, the temperature response will be identical for either current or velocity fluctuation. Thus, either an additional constraint must be imposed on the system or some information must be added before a complete solution can be obtained. The usual procedure is to operate the hot-wire or hot-film with either a constant current through the sensing element or by maintaining a constant temperature within the element, as discussed in the next sections. =
1.2.4.4. The Constant Current Anemometer. The earliest scientific use of hot-wire anemometry utilized a constant current mode of operation. SchubauerZ3gives an account of the early developments of the con-
= G . B. Schubauer, in “Advances in Hot-wire Anemometry” (W. L. Melnik and J. R. Weske, eds.), p. 13. University of Maryland, College Park, Md., 1968.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
277
stant current aneomometer in the United States, which he dates back to the first use of a compensation circuit by Dryden and K ~ e t h e .In ~ ~the same volume, describes the development of the hot-wire in Europe, which had been used much earlier for mean flow measurements. Both authors provide an insight into the difficulties and development of hot-wire anemometers before electronic amplifiers were readily available. 1.2.4.4.1. BASICCIRCUITRY. One of the simplest circuits for constant current operation, shown in Fig. 7(a), consists of a voltage supply, variable resistor R 1 , limiting resistor R 2 , and sensor R . The output of the circuit is the voltage drop el across the sensing element. If the supply voltage is constant, a change in the resistance R will change the current by Ai = i A R / ( R , + R2 + R ) . To reduce the current fluctuations to a relatively small value requires AR/(Rl
+ R2 + R ) << 1 ,
so that R1 + R2 must be large. The fixed resistor R2 should be sufficiently large to eliminate the possibility of accidentally burning out the wire when either the variable resistor R1 or the velocity is decreased. If the value of R2 is accurately known, the current through the sensor can be determined by measuring the voltage drop across Rz and using Ohm’s law. With the current thus determined and the output voltage known, one can compute the resistance of the sensor and thus the overheat ratio in situ. A more functional constant current circuit is shown in Fig. 7(b). This circuit places the sensing element in a Wheatstone bridge with a series resistor R, to provide protection against burnout and to adjust the current through the bridge. The resistors R2 and R3 are usually chosen so that the bridge can be balanced, i.e., el = e2, at some reference condition. The current through the sensor can be obtained by an ammeter or from the voltage drop across R, and Ohm’s law. Together with the voltage e l , the operating resistance can be obtained. The overheat uR can be controlled by the resistors R3 and R 4 . To reduce the power consumption, the resistors in the bridge should be larger than R . The voltage output from both circuits is quite low. The current of a few milliamperes and the wire resistance of approximately 10 R gives a mean output of typically 50 mV. The voltage fluctuations due to turbulence, for example, will usually be at least one order of magnitude smaller; thus some means of amplifying the signal is necessary. It is desirable to use a low noise dc amplifier having a wide bandwidth and H . L. Dryden and A. M. Kuethe, NACA Tech. Rep. 320 (1929). J . M. Burgers in “Advances in Hot-wire Anemometry” (W. L. Melnik and J. R. Weske, eds.), p. 25. University of Maryland, College Park, Md., 1968. I4
27 8
1.
MEASUREMENT OF VELOCITY
r
operating the ser
(bl
sufficient gain to increase the signal to levels compatible with the other circuits following the anemometer. If a compensation circuit is used as discussed in Section 1.2.4.4.4, a second amplifier is usually mandatory. 1.2.4.4.2. GOVERNING EQUATIONS. The linearized dynamic response of the hot-wire or hot-film can be written from Eq. (1.2.32) as
M d AT/dt
+ AT = f ( r ) ,
(1.2.33)
where M is the time constant (1.2.34) and f ( t ) is the forcing function (1.2.35) In a constant current mode of operation, the current fluctuations are zero and the system is forced by the velocity fluctuations alone. To determine the time constant M , the velocity is usually held constant and the current is allowed to fluctuate, as discussed in Section 1.2.4.4.4. Equation 1.2.33 can be solved to yield (1.2.36) The fluctuating temperature of the sensor is forced by the current and/or the velocity fluctuations through the function f ( t ) . For example, the
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
279
response AT(t) to the step function shown in Fig. 8 is
so the temperature change follows behind the forcing fluctuation. The time constant M is a measure of this lag, and hence the reason for its name and importance. The linearized response of the hot-wire in the frequency domain is the same as that of a low pass filter, as seen by lettingf(r) = A cos(wt) and substituting into Eq. (1.2.36). For t >> M, the result is AT
=
1
+
A w2M2
(cos wt
+ ~w
sin w t ) .
(1.2.37)
The magnitude of the temperature fluctuation of the wire is less than the magnitude of the forcing function by the factor 1/(1 + w 2 W ) . In addition, the temperature response of the sensor is shifted in phase with respect to the forcing function by an amount depending upon the time constant and the frequency. For radial frequencies much less than 1/M, the magnitude approaches unity and the phase shift approaches zero, so that the temperature fluctuations are approximately the same as those of the forcing function. For w = l/M, the root-mean-square magnitude of the temperature fluctuations is smaller than that of the forcing function by 4, and the phase shift is -45". For higher frequencies (a >> l / M , the magnitude of the temperature fluctuations goes to zero and the phase shift approaches 90". 1.2.4.4.3. THETIMECONSTANT. The time constant determines how well the temperature fluctuations in the hot-wire correspond to the fluctuations in the forcing function. To have the temperature fluctuations correspond to the velocity fluctuations uniformly for all frequencies, the time constant must be zero. This could only occur if the sensing element had zero heat capacity, as shown by Eq. (1.2.34). However, it is possible to make the time constant quite small. To explore this possibility,
FIG.8. When forced by an instantaneous change in current or velocity (a), the temperature of the sensor rises exponentially with a time constant M (b).
AT(t)
2Lf7-7 M
280
1.
MEASUREMENT OF VELOCITY
explicit expressions must be obtained to evaluate the heat flux F and the Joule heating P in Eq. (1.2.34). For a hot-wire, the heat is lost primarily by convection, and Eq. (1.2.20) shows that F can be written as
F
=
hS(T - To).
(1.2.38)
Furthermore, with the exponent rn = 0, the coefficient of heat convection according to Eq. (1.2.22) is h
=
(k,/d)(A + B Re"),
so that F
=
rrkfI(T - To)(A+ B Re").
The heating P can be obtained from Eqs. (1.2.16) and (1.2.25) as
P
= ?RJl
+ a ( T - To)].
Evaluating the derivatives in Eq. (1.2.34) and using the steady state average condition F = P yields the time constant of a hot-wire
M
=
(Ca/X,a)[(rrd2/4)21(uR/T2),
(1.2.39)
where a is the density of the wire material. As pointed out by Corrsin," the three groupings in this equation represent the effects of wire material, geometry, and operating conditions upon the time constant. The data in table I indicates that different wire material may change the time constant by a factor of two, with platinum having the smallest value. The time constant is independent of length but depends strongly upon the wire diameter. However, for a fixed overheat ratio, the diameter cannot be decreased without decreasing the current. At high velocities, Corrsin" has estimated that this effect causes the time constant to vary as dJ/2. Hence one method of reducing the time constant is to decrease the wire diameter. The effect of the operating conditions can best be seen by solving the steady state equation (1.2.30) with the above conditions, resulting in uR/P
= aRo(l
+ u,)/rrlkf(A + B Re").
At a constant velocity with low overheat, uR << 1 , the operating conditions have no effect upon the time constant, as pointed out by Weidman and Browand.26 As the velocity, i.e., Re, increases, however, the time constant decreases for all overheat values, since the faster velocity will convect heat away more quickly. For a fixed velocity and for high overheats, i.e., uR = 1 , the time constant increases with the overheat. In pa
P. D. Weidman and F. K. Browand, J . Phys. E 8, 553 (1975).
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
28 1
practice, the time constant can not be varied by more than a factor of two by changing the operating conditions. Experience has shown that Eq. ( I .2.39) does describe the variation of the hot-wire time constant with the various parameters under the conditions of convection cooling. Because of the approximations used and the difficulty in measuring the parameters accurately, it should be emphasized that Eq. 1.2.39 yields only an estimate of the time constant and not its absolute value. Weidman and BrowandZs found that Eq. (1.2.39) overestimated the time constant of their wires by a factor of three. They attributed part of this discrepancy to the diameter of the wire being different than stated by the manufacturer. To determine the correct value of the time constant, direct measurement by a square wave test as described in Section 1.2.4.4.5 is advisable. A 2.5-pm diameter platinum hot-wire has a nominal time constant of 0.4 ms. Estimation of the time constant for the hot-film is more complicated than for the hot-wire because the heat flux from the sensor must include the conduction losses to the substrate. Thus there are two mechanisms by which the hot-film can lose heat: (1) by convective loss directly to the ambient fluid, and (2) by conduction through the substrate and indirectly to the fluid. Two time constants are associated with these different mechanisms of heat loss. For slow temperature variations, adequate time is available for the heat to be conducted to an exposed surface and escape, thus both mechanisms are operative. For rapid variations, the heat has insufficient time to penetrate the substrate, and only the convection losses directly from the sensor are of importance. The losses due to convection at low frequency are governed by the same principle as for the hot-wire. However, since the heat capacity of the substrate is much greater than that of the film, it dominates the thermal inertia, and hence the specific heat and mass of the substrate must be used to evaluate the time constant. For a cylindrical hot-film of thickness t mounted on a substrate of diameter d, an analysis similar to that above reveals that the time constant due to convective losses to the ambient fluid is
where mS and c, are the density and specific thermal capacity of the substrate. (For noncylindrical probes, d is a characteristic dimension.) The time constant for hot-films will be greater than for hot-wires primarily because the dimensions of the probes are much greater. Consequently the frequency at which rolloff occurs will be correspondingly lower than for a hot-wire. Typical values of the time constant for a platinum film on a
282
1. MEASUREMENT
OF VELOCITY
0.05-mm-diameter quartz substrate is 0.015 s, corresponding to a frequency of 10 Hz. The effect of the substrate conduction on the frequency response of hot-films is determined by the distance a temperature disturbance penetrates into the backing material. The simplest model illustrating this effect is that of a heater, sinusoidally varying in time, mounted on a semi-infinite solid. Carslaw and JaegeF show that the temperature disturbance decays with depth, as e-2rrC‘A,where 4 is the depth, A is the “wavelength” of the disturbance given by A = 27r(2~,/o)’’~; w/27r is the frequency of the heat source; and K , is the thermal diffusivity of the substrate (0.0084 cmz s-’ for quartz.) To be transferred to the fluid, the heat is conducted through the substrate over some path of length &. If A >> 27&, only a small decay is experienced, and most of the heat will be conducted through the substrate and convected away from the exposed surface. At these low frequencies, i.e., o = 8 7 r ~ , / As~? 2 ~ , / & the heat is essentially conducted directly through the substrate and into the fluid; hence the cylindrical probe behaves as a hot-wire sensor. For 27rtO>> A, the temperature fluctuations decay considerably before reaching the exposed surface, and consequently the heat is not transferred to the environment. For these frequencies, o >> 2~,/5:, the heat conducted into the substrate is momentarily stored during part of the cycle and then returned to the sensor when its temperature is lower than that of the substrate. This response is different from that of hot-wires and accounts for proportionally less heat loss in hot-films at high frequencies. The actual path length 6, followed by the heat will depend strongly upon the geometry of the substrate. Small diameter cylindrical probes have a short path length and behave like hot-wires over a broader frequency domain than cones and wedges. As mentioned in Section 1.2.4.2.2, a 1-2-pm coating of the substrate material is often deposited externally over the film and probe in order to electrically insulate and protect it from the environment. This offers a negligible amount of resistance to the heat transfer except at very high frequencies. For example, the attenuation due to a 2-pm layer of quartz becomes significant only for frequencies greater than 60 kHz. An example of the effect of the time constants on the response of hot-wires and hot-films is seen in Fig. 9. For the hot-wire, the single time constant corresponds to approximately 400 Hz and is responsible for the 6 dB/octave first order rolloff. For the cylindrical hot-film, the time constant due to convection corresponds to a frequency of approximately 30 Hz. Notice that between 30 and 700 Hz the slope of the hot-film transfer
*’
H. S. Carslaw and J . C. Jaeger, “Conduction of Heat in Solids,” p. 64. Oxford Univ. Press (Clarendon), London and New York, 1962.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
283
Frequency ( H z )
FIG. 9. Frequency response of a hot-wire and cylindrical hot-film sensor; 0,. data from a Pt-10% Rd wire o f 2.5-pm diameter with a R = 0.2; 0 ,data from a Pt film of 25-pm diameter with u R = 0.4.
function is identical to that of the hot-wire. The change of slope to 3 dB/octave at approximately 700 Hz indicates that the heat penetrates less deeply into the substrate and not all of it is lost by convection to the ambient fluid. 1.2.4.4.4. COMPENSATION. The problems associated with the thermal lag at high frequencies have been recognized since hot-wire anemometers were first used in the 1920s. Their earliest use was in measuring only mean values because the wires at that time were greater than 25 p m in diameter with corresponding time constants greater than 1 s. It was not until electronic compensation of the thermal lag was introduced by Dryden and KuetheZ4that full advantage could be taken of the hot-wire anemometer. The need for some form of “compensation” is seen in Fig. 9; namely, for a fixed amplitude velocity fluctuation, the output of the constant current anemometer decreases as the frequency of the fluctuation increases. For small velocity fluctuations, the output from the hot-wire anemometer can be compensated by passing the signal through a simple attenuator with high pass characteristics. The output voltage e, from the compensation network shown in Fig. 10 satisfies RsC de,/dt
+ ( 1 + Rs/R4)e, = (R,/R,)(R,C de,/dt + el).
FIG. 10. A passive Compensation circuit used to extend the uniform frequency range of a constant current anemometer.
(1.2.41)
284
1.
MEASUREMENT OF VELOCITY
FIG.1 1 . Schematic of a constant current anemometer with an amplifier and compensation network.
For constant current operation, the fluctuating voltage across the sensor in Fig. 7b is related direthy to the temperature by Ael = i l a R o A T . It follows from Eq. (1.2.33) that M d P e l / d t + el = ilRoaf(t). The effect of the compensation becomes apparent by allowing the capacitor in Fig. 10 to assume the value C = M / R , . Rewriting Eq. (1.2.41) for the fluctuating voltages and combining with the last equation yields a first order equation with a new time constant of R,C. By allowing R4 >> Rs,
he, = ( R 5 / R r ) a ~ l R d f ( f )
(1.2.42)
for frequencies such that the derivative is not important, i.e., for w < l / R 5 C . Thus the frequency range of operation is increased by R4/R5and the cutoff frequency is controlled by the choice of circuit elements and not by the time constant of the sensor. If a variable capacitor is used as shown, the gain of the circuit, expressed as R5/R4in Eq. (1.2.42), remains constant for all values of compensation. However, the compensation capacitor must be reset whenever changes in the overheat, velocity, etc., cause a change in the time constant. Since the gain of the circuit is always much less than unity, a low noise amplifier is needed to increase the signal amplitude to a usable level. A complete circuit for a constant current anemometer with compensation is shown is Fig. 11. Although the use of electronic compensation marked the beginning of the modern use of hot-wires, it is interesting that there is a present trend to omit the compensation network. This is feasible because wires of much smaller diameter with correspondingly smaller time constants are now available. LaRue and Libbyzs used a 0.25-pm-diameter platinum wire 0.25 mm long that had a time constant of 16 ps. The frequency response was flat to 10 kHz, which was more than sufficient for their application. J. C. LaRue and P. A. Libby, Phys. Nuids 17, 873 (1974).
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
285
1.2.4.4.5. CALIBRATION A N D THE SQUARE WAVETEST. Calibration is the general procedure whereby a user determines how to translate recorded voltages and/or data into physical quantities, such as velocity. For constant current anemometers, this process usually proceeds in three steps; determining the sensitivity to the steady mean flow, d Z , / d 8 , obtaining the local linear sensitivity for the fluctuations, d AeJd Au, and establishing the frequency response. The static calibration for the mean flow is obtained by neglecting the temporal dependence of the heat transfer. If forced convection is the primary means of heat loss, the energy equation becomes ( k f d / a R d : ) ( A+ B Re”) = R / ( R - I?,).
( 1.2.43)
The lumped parameters on the left-hand side are extremely difficult to determine individually. For example, the length of the wire, 1, cannot be accurately measured, and its effective length is dependent upon the temperature distribution as explained in Section 1.2.4.6.1. The temperature coefficient of resistivity, a,is a property of the wire material. However, it has been determined that for extremely small diameter wires, a has a large variation from its nominal value. As noted earlier, there is no general agreement on the values for the coefficients A and B , and in addition, these parameters change as the temperature of the sensor varies. Consequently, the coefficients in Eq. (1.2.43) must be determined empirically by calibration. This can proceed by two different methods. First, the current can be held absolutely constant throughout the calibration and data gathering. This technique has the advantage that the sensitivities for the mean and fluctuating velocities are obtained simultaneously. To study this case, the last equation is written as A*
+ B*W
= e,/(e, -
C*),
(1.2.44)
where e, is proportional to the voltage across the sensor, e.g., the output voltage in Fig. 1 1 , and A * , B*, and C*are unknown dimensional constants which must be determined by the calibration. As the velocity is varied in an isothermal flow field, the calibration curve at constant current resembled the dashed lines in Fig. 12. Then in a flow field with an unknown velocity, the mean velocity at the probe is readily obtained by recording the mean output voltage and using the calibration curve. The sensitivity for the velocity fluctuations around the mean velocity is obtained by assuming that the fluctuations are sufficiently small that the slopes of the calibration curves in Fig. 12 are approximately linear in the range of interest. For example, a Taylor’s expansion around a mean value 0 yields the following equation for the root mean square fluctuations
286
1.
MEASUREMENT OF VELOCITY
I
U
FIG. 12. Two types of calibration curves obtained by varying the velocity and recording the output voltage. The dashed lines represent three different constant values of the sensor current; io, i l , and io. The sensor resistance was maintained at the constant values of R,, , R 1 , and R, along the solid lines.
where erm is the measured root mean square voltage fluctuation and (de,/dLT(o,,, is the magnitude of the slope of the constant current calibration curve at 0. The major disadvantage of this technique is that as the velocity increases, the wire’s overheat ratio decreases, and the time constant varies according to Eq. (1.2.39). If there are energetic fluctuations in the flow field at frequencies higher than (27rM)-’, a compensation network must be readjusted at each value of the mean velocity. The second calibration technique avoids this difficulty by manually readjusting the overheat at each value of the mean velocity so that the time constant does not vary. With constant resistance, Eq. (1.2.43) can be written as [A*
+ B*o]1’2= go,
where A* and B* are different dimensional constants than before. When data are taken in this manner, calibration curves represented by the solid lines in Fig. 12 result. When placed in an unknown flow field, the mean voltage can be used with the calibration curve to determine the mean velocity. The disadvantage of this method is that the fluctuations about any mean
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
287
velocity follow the constant current curve; thus, the sensitivity to the mean and fluctuating velocities are different. To fully calibrate the instrument in this mixed mode requires the determination of the local slope at constant current about the mean velocity. During operation of constant current anemometers at moderate and high overheats, care must always be taken to decrease the current whenever the cooling velocity decreases. Otherwise the temperature of the sensor continually increases as the velocity decreases, until ultimately it reaches the melting point and burns out. As part of every calibration procedure, the frequency response of the system must be determined. This is a necessary procedure before the compensation can be set. Ideally one would like to impose a known fluctuating flow field upon the sensor and observe the electrical response of the anemometer. The Karman vortex street behind a circular cylinder is one such possible flow. However, it and other natural and artificially generated fluctuating flow fields, have found only limited use because their amplitudes and frequencies cannot be accurately determined and/or varied. A mechanical vibrator that will induce an artificial simulation of the velocity by oscillating the probe can be constructed but is often cumbersome and generally has a limited frequency range. The most widely used technique for measuring the time constant has been to inject a current perturbation into the sensing element. Equations (1.2.33) and (1.2.35) show that the dynamic response of the sensor is identical for either a current fluctuation or a velocity fluctuation, thus suggesting that a current perturbation be used to measure the time constant. One method is to add a sinusoidal perturbation to the mean current, determine the corresponding voltage fluctuation, change the frequency, and repeat. It is easier operationally to use a single perturbation that has many harmonic frequencies simultaneously. A square wave was originally proposed by ZieglerzSaand was first used by K o v a s ~ n a y . ~ ~ This method allows one to quickly determine the time constant from an oscilloscope trace and immediately set the compensation parameters. The procedure is to first set the overheat, velocity, and other operating conditions. This prescribes a mean current through the wire to which is added the square wave perturbation. By observing the voltage difference across a balanced bridge, e.g., el - e2 in Fig. 11, a repetitive exponential decay as in Fig. 8 can be observed on an oscilloscope. The current fluctuation must be small compared to the mean current in order to avoid nonlinear effects. M. Ziegler, Proc. R . Neth. Acad. Sci. 34, 663 (1931). L. S. G . Kovasznay, NACA Tech. Memo. 1130 (translated from the original Hungarian
28a
report) (1947).
1. MEASUREMENT
288
OF VELOCITY
Analytically, this is explained by analyzing the constant current anemometer in Fig. 11. The voltage e l across the sensor is
el
=
iR(R2 + R3)/(R1 + Rz
+ R3 + R ) .
(1.2.45)
Let the current and the resistance have a mean and fluctuating component, so that i = i + Ai and R = R + AR. Substituting these expressions into (1.2.45) and assuming that the fluctuations are small compared to the mean values so that the products of Ai and AR can be neglected, results in
el + bel
=
iR(R2 + R 3 ) / 2 R + (R, + R3/XR)(iAR
+ R Ai),
(1.2.46)
where ZR = R1 + R2 + R3 + R and AR << XR. The first term on the right is the mean voltage e l , and the remaining terms are the fluctuations. The voltage perturbations Ael has two parts, the mean current multiplied by the change in the sensor resistance and the mean resistance times the fluctuating current. We are only interested in the fluctuating resistance term since it alone is related to the temperature changes in the sensor. A convenient method of subtracting the R Ai term is to use a differential amplifier with el and e2 as input voltages. The circuit in Fig. 11 yields
el - ez
=
i(RR2 - RlR3)/XR.
m2
If the resistance R3 is set so that = RlR3, the bridge is balanced and the mean voltage difference is zero. Applying a current perturbation yields Ael - Aez = (SR2/XR)AR. Using Eq. (1.2.16), the output of the amplifier in Fig. 11 with voltage gain A is
Aei(t) = (AaR,R,i/CR) AT(t). Since the temperature fluctuation AT satisfies the differential equation [Eq. (1.2.33)], Ael does also. For a square wave with a period greater than approximately five times the time constant, the upward rising portion of the voltage as seen on an oscilloscope resembles the AT response in Fig. 8(b). The time constant can easily be found since it equals the time at which the initial slope intercepts the final value as shown in Fig. 8(b), or the time when the voltage attains 63% of its final value. Hence the square wave is ideal for this measurement because with sufficiently large periods, the asymptotic value is easily seen on the oscilloscope and the time constant is obtained immediately. The square wave test also provides a convenient technique for determining the circuit parameters in the compensation circuit. By observing
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
289
the output of the compensation on an oscilloscope, the circuit parameters can be varied until a square wave is obtained, as illustrated by KOvasznay .30 After the Compensation parameters have been suitably chosen, the compensation network can extend the frequency response by several decades, and hence its output should closely resemble the velocity fluctuations with good fidelity. A few precautions are necessary whenever the square wave technique is employed. First, the amplitude of the current fluctuations must be small compared to the mean current through the sensing element to insure that a linear response will be obtained. Second, care must be taken to insure that other fluctuations in the forcing function are small, i.e., during the square wave test, the wire should be in a flow field that has small velocity and temperature fluctuations. Third, the fundamental frequency of the square wave should be considerrably lower than the characteristic frequency of the wire; otherwise the compensation may appear to be working properly whenever its corner frequency is below the frequency of the square wave. 1.2.4.5. The Constant Temperature Anemometer. An alternative method of operating hot-wire and hot-film sensors that has many advantages maintains the sensor at a constant temperature. To accomplish this, the current must be rapidly varied to instantaneously balance the heat loss from the sensor, as seen by Eq. (1.2.27). To interpret the output signal of constant current anemometers, a linearized version of the heat flow equations was required. This restricted its use to velocity fluctuations small in comparison to the mean velocity. In addition, manual adjustments of the anemometer and/or compensation parameters may be required when the sensor is moved to a location where the mean velocity is different. The constant temperature method of operation makes these adjustments automatically. Although it requires a more complex circuit, its output voltage can be linearized to be proportioned to the velocity. The advantages of constant temperature operation were recognized early in the history of hot-wire anemometers, but the disadvantage was that it required a good current amplifier as explained later. Despite this limitation, ZiegleP and later W e ~ k and e~~ O s ~ o f s k ywere ~ ~ able to construct and test constant temperature anemometers using vacuum tubes. Semiconductor devices became available in the early 1950s and provided L. S. G. Kovasznay, "Princeton Series on High Speed Aerodynamics," Vol. 9. Princeton Univ. Press, Princeton, New Jersey, 1954. 31 M . Ziegler, Verh. K . Neth. Akad. Wer. Amsterdam 15, 3 (1934). sz J. R. Weske, NACA Tech. Note 881 (1943). 35 E. Ossofsky, Rev. Sci. fnstrum. 19, 881 (1948).
290
1. MEASUREMENT OF VELOCITY
the current amplifier that was necessary for advancing the technique to its current state. 1.2.4.5.1. BASICCIRCUITRY. The basic elements of a constant temperature anemometer circuit are a differential dc amplifier, the sensing element, and a reference voltage. A typical circuit is shown in Fig. 13 where the sensor is placed in one leg of a Wheatstone bridge. The current through the element gives a voltage e l . A reference voltage is used by the feedback amplifier to determine how much the resistance, and hence temperature, of the sensor has changed. This can conveniently be obtained as the bridge voltage e2. These two voltages form the inputs to the differential amplifier. The amplifier's fluctuating output current is inversely proportional to the resistance change in the sensor. This current is fed back into the top of the bridge as shown and restores the sensor's resistance and temperature to their original values. Since the time response of the amplifier is quite fast, its fluctuating current maintain the sensor at a constant temperature except at very high frequencies. Explicit circuits for constant temperature operation have been reported by various authors, e.g., Kovasznay et F r e y m ~ t h Kaplan ,~~ (see Weidman and BrowandZ6),and Wyngaard and L ~ m l e y .Useful ~ ~ features for various applications include overload protection against burning out the sensor, filtering to decrease the noise, and subcircuits for determining the overheat ratio. All utilize the basic circuit shown in Fig. 13; hence the following analysis will use this model bridge and feedback amplifier. It will be assumed that the amplifier has a single time constant and the bridge consists only of resistive elements. Deviations from this ideal will be discussed in Section 1.2.4.5.4. Hot-wire and hot-film sensors typically have a low resistance of 10-100 fl and require 5 -50 mA from the amplifier to generate moderate overheat values. To efficiently use the available current from the amplifier, the resistances in the left leg of the bridge in Fig. 13 are usually larger than that in the right leg. R2/Rl is called the bridge ratio and typically has a value of 10, although values between unity and 50 have been used. The hot-wire bridge circuit is balanced whenever RR2 = RIR3.
(1.2.47)
To accomodate differing wire resistances and/or to set the overheat ratio, at least one leg has a variable resistor as shown. To measure the cold resistance of the sensor, R o , the circuit can be run in an open loop configuL. S. G . Kovasznay, L. T. Miller, and B. R. Vasudeva, "Project SQUID," Tech. Rep. JHU-22-P. Dept. of Aerospace Engineering, University of Virginia, Charlottesville. 1963. P. Freymuth, Rev. Sci. Instrum. 38, 677, (1%7). J. C. Wyngaard and J. L. Lumley, J . Sci. Instrum. 44, 363 (1967).
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
29 1
FIG.13. Schematic of a constant temperature anemometer circuit with provision for a test signal e , .
ration and a small current fed extraneously to the bridge. With zero offset voltage, the amplifier output is proportional to the bridge unbalance. The variable resistor R3 can be adjusted to bring this output to zero, and the value of the sensor’s resistance can be determined by using Eq. (1.2.47). In closed loop operation, as shown in Fig. 13, the amplifier strives to maintain el = e 2 . Consequently any overheat ratio can be obtained by changing the variable resistance R 3 . The output impedance of the anemometer is essentially that of the bridge and hence is quite low. This eliminates loading problems when additional instruments at the circuit output are used, as long as they have high input impedances. The gain of the amplifier needs to be high to provide high frequency response, as explained in the next section. The frequency response of the anemometer circuit varies inversely with the 3-dB rolloff point of the open loop amplifier, as will be seen later. Hence an amplifier with a small time constant is needed, consistent with the other imposed requirements. If initially the two sides of the bridge have equal voltage, the differential input to the amplifier is zero, giving a zero output. Hence there would be no current provided to the bridge, and no operating point would be established. Thus an offset voltage must be provided somewhere in the circuit as shown in Fig. 13. This voltage establishes a mean current through the sensor, and determines its temperature and overheat. The offset voltage is important dynamically because it determines the frequency response of the system. The offset voltage can be introduced into the circuit in several ways: it can be provided as a bias voltage at one input of the amplifier; it can be inserted between two stages of amplification; it can be an additional current added into the bridge, etc. Irrespective of how the offset voltage is introduced, its effect can be considered as an unbalance in the bridge, which makes the bridge seem to have a slightly higher resistance
292
1. MEASUREMENT OF VELOCITY
in one leg. Thus one can consider the mean resistance in the sensor leg to be R(1 S), where S is the unbalance parameter. Provisions are also included in Fig. 13 for a test signal e , to be injected into the sensing element. This is essential for electronically determining the frequency response of the system, as discussed in Section 1.2.4.5.3. To inject this signal, a resistor R , which is large compared to the bridge elements is used to isolate the bridge from the test signal generator. 1.2.4.5.2. THE CONSTANT TEMPERATURE ANEMOMETER EQUATIONS. The constant temperature mode of operation of a hot-wire or hot-film means that the temperature or resistance fluctuations of the sensing element are negligible compared to the current fluctuations. Mathematically this is expressed by
+
ARIR << Ail/il,
(1.2.48)
where il is the current through the sensing element. The resistance is not held to an absolute constant but varies slightly at the highest frequencies, as will be shown later. However, there exists a large frequency range over which the resistance and temperature are constant for all practical purposes. This range extends from dc up to some characteristic cutoff frequency f, = 0,/27r. The relationship between f, and the system parameters will be defined more precisely later in this section. Since the resistance is approximately constant for all frequencies less thanf,, the time derivative of the resistance, and hence the temperature, must be approximately zero. It can thus be neglected with respect to the other terms in the energy equation (1.2.27) so that
PGl,
n = F ( U , n.
(1.2.49)
Here both the current and the velocity are functions of time, and the temperature is constant over the frequency range of immediate interest. The advantage of the constant temperature mode of operation is that the current through the sensor is related directly to the velocity through the nonlinear relationship above. Using Eq. (1.2.25) to write the power input to the sensor and assuming that the heat loss is due only to convection given by Eq. (1.2.22), with m = 0, one can obtain (1.2.50) In an isothermal flow, the overheat is held constant, and hence the current is related directly to the Reynolds number and velocity by Eq. (1.2.50). The bridge voltage is related to the voltage across the sensor in Fig. 13 by (1.2.51)
1.2.
293
PROBE METHODS FOR VELOCITY MEASUREMENT
Combining the last two equations and using Ohm's law results in
e,
=
( R + R,) [rr'kf -~ uR (A ffR, U R f 1
+ B Ren)]'''.
(1.2.52)
After determining the constants in this equation as described in Section 1.2.4.5.3, the velocity at the sensor can be obtained from the recorded voltage. The last equation is valid for both small and large amplitude velocity fluctuations as long as the resistance is constant. Since the equation is nonlinear, errors are introduced if the fluctuating voltage is assumed to be proportional to the oscillating velocity, as discussed by F r e y m ~ t h .If~ ~ ~ the voltage is first linearized as discussed in Section 1.2.4.5.5,these errors are usually negligible. Freymuth3' investigated the linearized response of a sinusoidal velocity perturbation and concluded that for wires with small time constants and optimally adjusted broadband amplifiers, the nonlinear errors are only of significance when analyzing higher order statistics. This is in contrast to the constant current anemometer and makes the constant temperature anemometer much easier to use in shear flows and more desirabie in flows where the velocity varies over a wide range, e.g., in the outer regions of jets, in the atmosphere, and near solid boundaries. The limiting frequency of the constant temperature anemometer will always be higher than that of the uncompensated constant current anemometer. Because of the complexity of the constant temperature anemometer, it is more difficult to determine the limiting cutoff frequencyf, and to ascertain its governing parameters. However, it is known that for any constant temperature anemometer operating system, the upper frequency limit must be determined by an analysis of at least a second order system, in contrast to the first order response of the constant current anemometer. This higher order response arises because there are several different time constants that can be of importance and hence govern the system. In addition to the time constant of the sensing element discussed earlier, the feedback amplifier has a time constant associated with its rolloff frequency. More complicated amplifier responses and the effects of stray capacitance and inductance in the circuit are discussed in Section 1.2.4.5.4. Since the bandwidth of the constant temperature anemometer is quite large, one is interested only in learning the upper frequency response of the system and not in correcting the response for higher frequencies, as in constant current anemometers. The dynamic response of the constant temperature anemometer and its upper frequency limitation are determined by three basic equations that
J7
P. Freymuth, Rev. Sci. Instrum. 40,258 (1969). P. Freymuth, J . Phys. E 10, 710 (1977).
294
1. MEASUREMENT OF VELOCITY
describe (1) the bridge, (2) the sensing element, and (3) the amplifier. First, consider the bridge with only resistive elements, as shown in Fig. 13. The voltage at the top of the bridge, eo(t),is related to the current through the sensing element il(t) by eo(t) = [ N t ) + Rllil(t).
Writing all of the variables as the sum of a mean and fluctuating parts, substituting, subtracting the mean value, and neglecting nonlinear terms of order AZ yields Aeo(t) =
(R + R,) Ail(t) + r, AR(t).
(1.2.53)
The voltage difference across the two legs of the bridge in Fig. 13 is
Under balanced conditions, this voltage difference is zero, necessitating that an offset voltage be added to the amplifier, as indicated in Fig. 13 As discussed earlier, this voltage can be considered as an additional input to the amplifier so that eoffset= SR2RPo/(R+ R,)(R2 + Rs), where S << 1. Although the bridge is assumed to be operated in a balanced condition, the effect of the offset voltage is similar to maintaining the bridge with an unbalance given by 6
=
(R2R - RlRS)/RzR.
(1.2.55)
For completeness, a test signal is introduced through a resistor Rt in Fig. 13. If Rt is large compared to the other resistors in the bridge, the voltage into the amplifier is given by gtet(t), where et(r) is the test signal generator voltage, and g, is the attenuation due to Rt and the bridge. Under these conditions, the voltage difference at the amplifier input is
Writing the voltages and resistance as a mean and fluctuating component, averaging, and subtracting the mean value results in
(1.2.56) since R z / ( R 2 + R S ) = R J ( R + R , ) under balanced conditions. The dynamic response of the sensor is given in Eq. (1.2.32), which can be written as
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
M-
dt
+ AR
= a Ail
+ b AU.
295 (1.2.57)
By using Eq. (1.2.25) and the definition of the time constant, the constants a and b are a = 2RaR/r1
( 1.2.58)
b = -(a R/q)(aF/aU)lT.
(1.2.59)
The dynamic response of the amplifier can be modeled as p d Ae,/dt
+ Ae,
=
(1.2.60)
-A, Ae,,
where A, is the dc gain of the amplifier and p is its time constant. The amplifier is assumed to have a constant gain A, out to frequency f = (27rp)-l, at which point the amplitude response rolls off at 6 dB/octave, i.e. 20 dB/decade. The basic response of amplifier is given by theirgainbandwidth product defined by A0/277p, which is typically lo6 to lo' Hz. For anemometers, large gains of A = 1000-10,000 are required to insure sufficient response, and hence p = lo+' s. Although some amplifiers have more complicated response functions, they can usually be simplified to first order systems by the use of standard compensation techniques. These equations can be combined to yield the relationship between any two of the variables. Operationally, it is desirable to have the relationship between the output variable e, and the input variables u(t) and et(t). Combining Eqs. (1.2.53) and (1.2.57) to eliminate Ai,(t), using Eq. (1.2.56) to eliminate AR(t), and applying Eq. (1.2.60) yields the following relationship: 0 dL Ae
dt2
+ 250,- d dtAe, + w', Ae,
= W:
[S, A W )
+ St [M%
+
(1 + -
2Ra +
)
R 1 Pet]
)
(1.2.61)
The natural frequency u,,the damping coefficient 5, and the sensitivity coefficients for fluctuations of the velocity and test signal, S, and St, are given below. The equation in the above canonical form shows that the constant temperature anemometer behaves like a second order linear system as long as the resistance fluctuations are small compared to the current fluctuations and there are no reactive elements in the circuit. The characteristic frequency and damping of the system are governed by the bridge unbalance parameter S and the system gain parameter, defined as KO = A , i?Rl/(R + R,)',
(1.2.62)
296
1.
MEASUREMENT OF VELOCITY
For constant temperature anemometers at typical operating conditions,
m,/(R+ R1)2is of order one, and thus KO is much larger than unity, since A, is quite large. The unbalance parameter S is small compared to unity, as discussed earlier. For a balanced bridge under these conditions, the characteristic frequency is 0,
= (2U~K,/phf)"~.
The frequency range defined by o,can be maximized with a unity bridge, Le., R = R1,as seen by examining Eq. (1.2.62). Since K o / p is proportional to the gain-bandwidth product of the amplifier, it is a constant for each amplifier and cannot be changed by the operating conditions. Hence for a given circuit and sensor, the only means of altering the natural frequency o, is by varying the time constant of the sensor M . Since the time constant of the amplifier is typically less than that of the sensor, p << M, the damping parameter is 25 = (1 4- K , ~ ) ( M / ~ K , ~ U ~ ) ~ ' ~ . (1.2.63)
For a given sensor and amplifier, the response of the system depends critically upon the value of the unbalance parameter 6; thus it is one of the most important variables to be set during calibration. A flat frequency response with maximum bandwidth dictates that S be set to a critically damped position. The sensitivities to the velocity and test signal are (1.2.64)
and (1.2.65)
The amplitude response of a constant temperature anemometer to velocity fluctuations and a test signal are sketched in Fig. 14. It is assumed that the wire has an overheat of 0.5 and a time constant of 0.4 ms, the amplifier has a time constant of 25 ps, and K O = 1250, yieldingf, = 55 kHz. The response to three different values of the unbalance parameter S are shown. As S -+ - l / K o , the amplitude becomes quite large at f, = 0,/27r and the anemometer oscillates at that frequency. The amplitude is, however, limited because as the current increases in the element, its resistance increases. This decreases the output current. The heating and cooling of the sensor are 180 degrees out of phase with the amplifier output, and consequently the system oscillates. Although this ocillation is self-limiting, it will continue until either the sensing element bums out or the operator changes the damping of the system by increasing S toward
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
297
r - 4 7 Amplifier Alone
4 Frequency
FIG.14. Frequency response of a constant temperature anemometer to a test signal with critical damping and to velocity fluctuations with under-, critical, and overdamped conditions.
the positive side. Fortunately, the sensing element usually does not burn out during oscillation because of this self-limiting feature. However, very small diameter hot wires may not be able to withstand the increased current arising from the oscillations and may be destroyed by this phenomenon. 1.2.4.5.3. CALIBRATION A N D THE SQUARE WAVETEST. With a variable speed flow facility, the static calibration of a constant temperature hot-wire or hot-film is similar to that of the constant current mode. After the sensor is installed and the instrument set to operate at the desired overheat ratio, the output voltage at several known velocities is recorded. If King's convective cooling adequately describes the heat transfer, Eq. (1.2.52) can be written as
e: = A*
+ B*U".
(1.2.66)
Since the temperature of the sensor is a constant, A* and B* are dimensional constants and must be determined by the calibration. They are most readily obtained by plotting the square of the recorded voltages
298
1.
MEASUREMENT OF VELOCITY
FIG. 15. Three calibration curves for a constant temperature anemometer. The sensor's resistance is constant along each trace, but the overheat ratio is different on each curve.
versus U",as shown in Fig. 15, so the constants correspond to the slope and intercept of the straight line through the data. If the value of the exponent n is not known a priori, it must be determined also. This can be accomplished by first assuming a value of n and plotting the data to determine A*. Then, if ln(e2, - A*) is plotted versus ln(U), the slope determines a new value of n according to Eq. (1.2.66). If n differs significantly from the assumed value, an iteration is required until a convergence is obtained. For hot-wires, the nominal values of n are given in Table 11. The exponent varies considerably for hot-films, with values between 0.25 and 0.50 having been reported. Note that the data near U = 0 in Fig. 15 deviate from a straight line. Even though there is no velocity imposed by the flow facility, the heating of the fluid by the sensor induces a small velocity due to buoyancy. Thus A* cannot be obtained by recording the output voltage as U = 0. The discrepancy between the intercept A* and the output voltage at U = 0 increases at higher overheat ratios. The dynamic calibration is obtained by using an electronic test signal. However, unlike the constant current system, the response of the constant temperature anemometer to a test signal is different from that due to a fluctuating velocity. Even though the system is still second order, its response to a test signal includes a contribution from the derivative of the test signal, as seen in Eq. (1.2.61). For a sinusoidal test signal, this implies that the frequency response is constant out tof = 1/(27rM). As the frequency increases further, the amplitude likewise increases at 20 dB/decade, as seen in Fig. 14. The amplitude response peaks at the natural frequency fo, and then decreases to an ultimate asymptote of - 20 dB/decade as shown by Weidman and Browand.26 In spite of the fact that the constant temperature anemometer responds differently to velocity fluctuations than it does to a test signal, an induced electrical fluctuation can be used to set the unbalance parameter and
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
299
hence the frequency response. Traditionally, a square wave has been used because it is rich in higher harmonics. For the underdamped case, the system’s response given by the solution of Eq. (1.2.61) is
where f(t)=
S, AV(r) + St{M d Ae/dt
+ [l + 2RaR/(R + R , ) ] Act}.
(1.2.68)
With no velocity fluctuations, the injection of a square wave test signal simulates a jump in the velocity (i.e., the square wave) and a delta function pulse of velocity (i.e., the derivative of the square wave). For a square wave with a fundamental frequency less than 1/(27M), the predominate feature of the output signal is the pulse component associated with the term M d Aet/dt because its amplitude is much larger. Three examples of the output voltage due to an injected square wave appear in Fig. 16. These traces were obtained from the same sensor and anemometer by varying the unbalance parameter to give an underdamped 5 < 1 , a critically damped 5 = 1, and an overdamped condition 5 > 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.
MEASUREMENT OF VELOCITY
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.
PROBE METHODS FOR VELOCITY MEASUREMENT
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.
MEASUREMENT OF VELOCITY
time constants. If p << 7 , a second order analysis using the time constants M and T will describe the system. In either case, the bridge unbalance S determines the damping of the system. However, if p << T , the response of the system to the square wave test will be different than discussed in Section 1.2.4.5.2because the operating system will respond to the test signal and its first two derivatives. 1.2.4.5.5. LINEARIZING THE ANEMOMETER S I G N A L . Since hot-wires were first used, efforts have been made to obtain a signal that was linearly proportional to the magnitude of the velocity over a wide velocity range. For the constant current anemometer, proportionality over a large range is impossible because as the velocity changes, so does the overheat and thus the time constant; hence either the compensation or the current must be continually reset, and the overheat adjusted in order to obtain a flat frequency response. Thus, the use of constant current anemometers in shear flows will always have some definite disadvantages. In uniform flow fields, the constant current anemometer gives a response proportional to the velocity fluctuations only as long as the fluctuating velocities are small compared to the mean velocity. The constant temperature anemometer does not have this inherent disadvantage, and techniques exist to produce a signal proportional to the velocity over a very wide velocity range. Processing the output voltage in order to obtain a signal proportional to velocity is called linearization, and a circuit that performs this task is called a linearizer. In principle the task of the linearizer is to invert Eq. (1.2.52) so that the output voltage is directly proportional to the velocity. Therefore, the specific type of heating and cooling of the sensor must be known and specified before linearization can be accomplished. Linearizers provide no temporal corrections and thus are only useful below the cutoff frequency of the anemorneter or the linearizer, whichever is lower. If King’s law describes the heat lost from the sensor, the anemometer output e, is given by Eq. (1.2.66). The linearizer accepts this voltage as an input and performs the following algebraic function: eL = (4eE - p)””,
(1.2.69)
where eL is the linearizer output, and p and 4 are adjustable constants. Combining with Eq. (1.2.66) and setting p = A,/B, and 4 = l/Bo, the linearizer output eLis found to be directly proportional to the velocity for the range over which Eqs. (1.2.66) and (1.2.69) are valid. Linearization can be assigned to an electronic circuit which has a nonlinear transfer function as given by Eq. (1.2.69) or it can be done digitally on a computer. In either case, at least two constants, e.g., p and 4 , have to be determined and set before linearization can occur. If the heat
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
303
transfer is a more complicated function of velocity than that given in Eq. (1.2.66),other constants may be necessary as well. In addition, the value of the exponent n may have to be determined as part of the calibration process before linearization can be achieved. Analog linearizer have been used from the early days of constant temperature anemometers. The earliest linearizers assumed that the exponent was 2 and thus used two squaring circuits together with linear operational amplifiers to perform the inversion described by Eq. (1.2.69). A more sophisticated linearizer utilizes a diode chain as used by CompteBell~t,~O or a transistor chain as described by Chevray and K o ~ a s z n a y . ~ ~ These linearizers make use of the threshold voltage characteristics of diodes and transistors. By proper biasing, any nonlinear function can be obtained by this method; however, it is awkward to change the functional characteristics or the operating range of these linearizers after they have been constructed. A third type of linearizer uses nonlinear logarithmic diodes. These linearizers have the advantage that the exponent can easily be changed, which is useful in circumstances where it is significantly different from 0.5. Another linearizer utilizes a generalized polynomial to invert the anemometer output. This linearizer has at least as many constants as the order of the polynomial and can be used to invert more complicated heat transfer relations than that expressed by Eq. (1.2.69). This method is particularly adaptable at very low velocities, where free convection becomes important. Linearization can be performed on digital computers if the functional dependence of the heat transfer on the velocity is known. Even if the explicit form of the velocity heat loss dependence is not known u priori, the computer can be used to linearize the anemometer signals by fitting a polynomial through the calibration data or by using a table look-up and interpolation technique. Although digital interpolation can yield highly accurate results, extrapolation outside the range of validity can lead to large errors, because the computer generated function may not be valid there. 1.2.4.6. Additional Topics 1.2.4.6.1. TEMPERATURE DISTRIBUTION ALONG A SENSOR OF FINITE LENGTH.In the previous sections, it has been assumed that the temperature of the sensing element was represented by a single value T, which corresponded to a unique resistance R. This idealized state corresponds to a sensor of an infinite length with uniform cross sectional area and therG . Compte-Bellot, Contribution a l'etude de la turbulence de conduite. Ph. D. Thesis, L'Universite de Grenoble (1963). " R . Chevray and L. S. G . Kovasznay, Rev. Sci. Insfrum. 40, 91 (1969).
304
1.
MEASUREMENT OF VELOCITY
modynamic properties. The finite length and cooler end supports of real sensors introduce a nonuniform temperature distribution T ( z ) into the sensing element. [T(z) will be used to denote the explicit dependence upon the axial distance, z; T alone will represent the mean value of the temperature averaged over the length of the sensor.] The temperature and resistance variation are important to the understanding and operation of these sensors, especially as the trend toward smaller and smaller sensing elements continues. Consider the hot-wire shown schematically in Fig. 18. Given a steady laminar flow field around the sensor, the energy equation expresses a balance between the heat input and the losses due to convection and conduction. The supports are much more massive than the sensing element and
FIG. 18. (a) A hot-wire and supporting prongs in a uniform velocity U. (b) The resulting axial temperature distributions for several operating conditions.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
305
act as a heat sink. Although the measurements of Champagne ef ~ 1 in- . dicate that for high overheats, the prongs may be as much as 50°C above the ambient temperature To, it will be assumed that the supports are at the fluid temperature, and the temperature difference T - To is small. Assuming no radiation losses, infinite radial conduction, and no variation of the cross-sectional area and convective heat transfer coefficient, the steady state energy balance for a section dz along the hot-wire element is
(3* :3
=
(kw
dz
rzxo dz - ~ h d [ T ( z ) n(d/2)2
-
+
To] dz
=
0,
(1.2.70)
where x is the resistivity. The analogous equation for hot-films must include the conduction to the substrate as well. If the temperature difference T - To is small, the nonlinearity due to variations of the coefficients is negligible, and the equation becomes 1; d*[T(z)- T0]/dz2 - [T(z) - To] = -(T - To),,
(1.2.71)
where I, is a characteristic length over which the supports affect the sensor’s temperature distribution. The asymptotic temperature difference (T - To), is the temperature difference between the wire and the fluid as the length becomes infinite. Kingz0 obtained a solution of Eq. (1.2.71), which is (1.2.72) Following Corrsin,” this solution is plotted for several values of the parameter 1, in Fig. 18. Using the mean operating conditions, the characteristic length 1, is found to be
lcld
=
[+(kw/kr)(l + ~ R ) / N u ] ” ~ .
Since the Nusselt number is approximately unity at typical wire Reynolds numbers, the length 1, is a weak function of the operating conditions and is thus determined by the ratio of the conductivities. For platinum wire operating at 50% overheat in air, I, = 30d. In operation, the anemometer will sense the average temperature T of this distribution. Using Eq. (1.2.72), Corrsin has shown that
T - To (Tm - T o ) ,
21 1 tanh-. I 21, B e t c h o ~Davies , ~ ~ and Fisher,44and Champagne et
‘* 43
=
1-
( 1.2.73)
have sought to
F. H. Champagne, C. A . Sleicher, and 0. H . Wehrrnann,J. FluidMech. 28, 153 (1967). R . Betchov, NACA Tech. Memo. 1346 (1952). P. 0. A. L. Davies and M. J. Fisher, Proc. R . Soc. London 280,486 (1964).
~
~
3 06
1.
MEASUREMENT OF VELOCITY
extend the previous analysis to higher temperatures by including the nonlinear variation of the coefficients in Eq. (1.2.70). Champagne et have also measured the temperature distribution along several hot-wires. The effect of a nonuniform velocity on the temperature distribution has been studied by Gessner and M01ler.~~ 1.2.4.6.2. DIRECTIONAL DEPENDENCE. A well-defined effective cooling velocity has been implicitly assumed in the preceding analysis without specifying the relationship of this scalar velocity U , to the velocity vector ( u , v , w ) . Consider an infinitely long circular cylinder aligned along the z axis, as shown in Fig. 19. Prandtl,& Jones,47and Sears48have shown that for a laminar flow field, the two velocity components in the plane perpendicular to the cylinder, u and 21, are independent of the axial component z. In addition, if the cylinder is uniformly heated, the energy equation and the temperature are independent of w and z, and the heat transfer coefficient depends only upon the normal velocity component. This velocity is called the effective cooling velocity and is
u, = u, cos 8,
(1.2.74)
where U , is the magnitude of the velocity vector and 8 is the angle between U , and the normal to the axis of the sensing element. The heat transfer under these conditions is said to obey the “cosine law” of cooling. For hot-wires of finite length, several length scales enter the problem. The most fundamental length is that of the wire, 1. An associated length scale 1, characterizes the effect of the temperature distribution, as discussed in Section 1.2.4.6.1. Other length scales of importance shown in Fig. 20 are the diameter of the sheath, d,; the diameter of the prongs, d2; the distance between the prongs, sl;the distance from the sensor to the solid body of the probe, s2; and the diameter of the stem, d 3 . In addition, the effective cooling velocity also depends upon the orientation of the velocity vector with respect to the probe, as indicated by the angles 4 and 8. Thus the effective cooling has a functional form given by UelUm=f(e,
4, 1, 4,d , 4 , 4 , 4 , ~
1 SZ). ,
(1.2.75)
There is no hope of finding the general form of Eq. (1.2.75) because it depends upon the particular probe configuration. However, there are some experimental results to serve as a guide to the hot-wire designer and user. F. B. Gessner and G . L. Moller, J . Fluid Mech. 47,449 (1971). L. Prandtl, Albert-Betz Festschrift, Goettingen, 134- 141; also Report and Translation No. 64. Ministry of Aircraft Production, Voelkenrode, 1946. R. T. Jones, NACA Tech. Note 1402 (1947). Is W. R. Sears, J . Aeron. Sci. 15, 49 (1948). 46
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
307
FIG. 19. Coordinate system and velocity components with respect to the sensing element.
This evidence suggests that the function fconsists of two factors: namely,
fi,which expresses the dependence upon the length, the angle 8, and operating conditions of the wire; and f,, which depends upon the geometry and orientation of the probe support system, so that
The functionf, approaches cos 8 as l / d + ~0 and lc/d + 0 independently off,. Hence it expresses the departure from cosine law cooling. The function fi represents the probe interference effects and will be discussed in Section 1.2.4.6.3. The departure of the functionf, from the cosine law is usually ascribed either to convective cooling of the wire by the tangential velocity compo-
FIG.20. A typical hot-wire sensor and its support system.
308
1. MEASUREMENT O F VELOCITY
nent, or to the heat conduction into the end supports. The conduction can be analyzed for small overheats by assuming cosine law cooling and solving Eq. (1.2.70). Corrsin" found a solution which gave the variation of the end conduction losses with the angle of inclination. Champagne ef ~ 7 1 measured . ~ ~ the temperature distribution along the wire for several values of 8 and thereby estimated the end losses. They concluded that for the I/d values most often used, the tangential cooling is more important than changes in the end conduction losses. Many authors have experimentally studied the dependence of the cooling velocity upon the angle 6 and the length-to-diameter ratio. Several of the empirical relationships that have been obtained are listed in the accompanying tabulation, where k , E , and b are empirical parameters de-
1. Cosine law 2. Hime,'@Webster,so Champagne ct 3. Fujita and Kovasznaysl 4. Friehe and Schwarzsz
termined experimentally. These authors found that the above parameters are most strongly influenced by the I / d ratio. The residual scatter in their results possibly indicates a lesser dependence upon the operating conditions, l J d , and some probe interference effects. 1.2.4.6.3. PROBEINTERFERENCEEFFECTS. The principal advantages of hot-wire and hot-film sensors are derived from their very small physical size. Although the sensing elements themselves offer very little disturbance to the flow field, their support system consisting of the prongs and the probe body may significantly distort the flow field and introduce large errors into the measurements. By exercising care in the design and use of these probes, this interference can be minimized. Some of the anomalous effects introduced by the probe's orientation and support system are discussed by Florent and ThioletJ3 and T r i t t ~ n who , ~ ~ noticed that their data were dependent upon the orientation and configuration of their probe systems. Although Hoole and CalvertJ5 were apparently the first, many invesJ. 0. Hinze, "Turbulence," Chapter 2. McGraw-Hill, New York, 1959. C. A. G. Webster, J . Fluid M e c h . 13, 307 (1962). H. Fujita and L. S. G . Kovasznay, Rev. Sci. Instrum. 39, 1351 (1968). 5* C. A. Friehe and W. H. Schwarz, J . Appl. M e c h . 16, I (1968). s3 P. Florent and G . Thiolet, C. R . Acad. Sci., Paris, Ser. A 269, 405 (1969). s4 D. J . Tritton, J . Fluid Mech. 28, 433 (1967). ss B. F. Hoole and J. R . Calved, J . R . Aeron. Soc. 71, 51 1 (1967).
'9
M,
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
3 09
tigators have noticed that when a hot-wire is rotated around its axis, the anemometer output changes. Since the sensing element is symmetrical about this axis, the changes can only be attributed to the support system. Varying the angle 4 in Fig. 20 with 8 = 0 has become a standard means of checking the influence of the probe support system upon the measurements. Depending upon the probe geometry, 20% variations in the velocity readings have been reported. This error can be reduced to a negligible value by maintaining 4 at the same value during calibration and data gathering. When attempting to design a hot-wire probe, there are many length parameters that must be specified, as Fig. 20 indicates. Since there are so many variables, there has not been an investigation that systematically changed all of them. Champagne ef ~ 1 studied . ~ the ~ prong diameter/wire diameter ratio to determine its effect upon the deviations from the law of cosine cooling. For typical ratios from 12 to 50, no changes were observed as 8 varied with 4 = 0. They also found that the deviations from cosine cooling were unaffected by the presence or absence of the short sheaths between the sensors and the prongs. Compte-Bellot et also found that the sheaths had practically no effect on the probe interference as C#J varied from 0 to 90”. The ideal probe support system would be one which caused no velocity disturbance to the flow pattern and provided no heat conduction from the sensing element. With these ideas serving as a guide, modern probes use prongs that are as small as possible, but yet have sufficient strength to withstand aerodynamic loading. Tapered jeweler’s broaches, sewing needles, and commercial prongs are most frequently used. The tapered ends have diameters of 75-300 p m to minimize the interference effects. Usually probes are designed so the prongs are parallel to the mean flow. When they are perpendicular to the flow field, i.e., 4 = 90” in Fig. 20, they may produce a Karman vortex street, aild the accompanying vibrations can easily break a hot-wire. The aerodynamic disturbance caused by the prongs has been studied by GilmoreS7and Dahm and R a s m u ~ s e n . The ~ ~ latter authors used prongs tapered to 100 pm. They changed the spacing between the two prongs, sl, from 1 to 4 mm and altered the length of the prongs, s2, from 6 to 18 mm while keeping the diameter of the stem constant at d3 = 2 mm. At 10 m/sec and 4 = 90, s1 = 2 mm produced a 6% change in the indicated velocity for the tested values of the different lengths tested. As the distance between the prongs was increased to s1 = 4 mm, the error deG . Compte-Bellot, A. Strohl, and A. Alcaraz, J . Appl. M ~ c h 38, . 767 (1971). D. C. Gilmore, “The Probe Interference Effect of Hot-wire Anemometers,” T N 67-3, Mech. Eng. Res. Lab., McGill University, Montreal, 1967. 58 M . Dahm and C. G . Rasmussen, DISA InJ: No. 7, p. 19 (1969). 5’
310
1.
MEASUREMENT OF VELOCITY
creased to approximately 3%. Compte-Bellot et studied the effects of the length s2 and the diameter d , of the prongs as well as the stem diameter dsin a potential cone of a jet. They concluded that the changes in heat transfer are due to the aerodynamic disturbances introduced by the prongs and not due to the changes in the prong cooling. Furthermore, they found that the largest aerodynamic disturbances are due to the prongs, and the stem has a lesser effect. These investigations have shown that there will always be some probe interference effects which depend particularly upon 4. Although these effects can be minimized by design, the inescapable conclusion is that the user must always calibrate hot-wire and hot-film probes if quantitative data are to be obtained. Individual calibration essentially determines the empirical heat transfer relationship unique to each sensor and support structure, thus eliminating many of the general uncertainties. Calibration should be done with conditions and configurations that duplicate is nearly as possible those used when taking data, e.g., should remain constant between calibration and data gathering. Special caution is necessary when two probes are moved into close proximity to each other to insure that they do not interfere with the flow field measured by the other. Similar caution is required when making measurements near solid boundaries. 1.2.4.6.4. MULTIPLE PROBEARRAYS.Probes containing more than one hot-wire or hot-film have been used for many years to increase the instantaneous velocity information obtainable. The array used most often is one with two sensors in an X or V arrangement to study the streamwise and one other velocity component. Consider the sensing element lying in the x-y plane inclined at an angle with respect to the x axis, as shown in Fig. 21. Neglecting the w velocity component for the moment, the effective cooling velocity can be expressed by
+
u, = U l l U
+
u1,v,
+;
where the coeficients uU are primarily functions of the angle however, they also include effects related to the deviations from the cosine law cooling. An X-probe is obtained by placing another sensor at angle perpendicular to the z axis, as indicated in Fig. 21. A similar expression is obtained for its effective velocity. In principle, the two equations can be solved simultaneously to yield the velocity components u and v if the sensors are sufficiently close. In practice, several assumptions and approximations must be made to separate the two velocity components. Foremost among them is that the cooling velocity can indeed be represented in the linear form shown above, since Section 1.2.4.6.2 indicates that a nonlinear representation is more exact. The linearized form requires that the fluctuations of all three velocity components be small com-
+'
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
31 1
FIG.21. Sketch of an X-probe consisting of two sensors at angles $ and 9' with respect to the x axis.
pared to the mean velocity 8,and that deviations from the locally linearized versions of the formulas in Section 1.2.4.6.2 be small. In addition, the nonlinear effects associated with Eq. (1.2.32) must also be negligible. These approximations have been discussed by Champagne and S l e i ~ h e r .To ~ ~ascribe the velocity components to the spatial location of the probe, the lengths and separation of the sensors should be smaller than the Kolmogorov length scale (see footnote on page 265), as discussed by Wyngaard.so The effects of the prongs and other aspects of probe interference of X-probes are discussed by Strohl and CompteBellot .61 In spite of the above difficulties, inclined wire probes have been successfully built and used. Since so many assumptions and unknown effects are involved, the probes must always be calibrated in a flow field with known variable velocities. This can be accomplished by yawing the probe in a variable unidirectional flow field so that two different velocity components can be simulated by changing the yaw angle and flow speed. With a mean velocity gradient in the y direction (see coordinate system of Fig. 21), an X configuration has been the most popular probe for measuring the u and u velocity components. In this arrangement, the sensors are close together although they lie in different planes. In a V arrangement, the two elements lie in the same plane, but have a larger spatial separation. In a shear flow this is usually preferable for measuring the veloc-
'' F. H. Champagne and C. A. Sleicher, J . Fluid M c c h . 28, 177 (1967). Bo
J . C. Wyngaard, J . P h y s . E 2, 983 (1969). A . Strohl and G . Compte-Bellot, J . Appl. M r c h . 40,661 (1973).
312
1. MEASUREMENT OF VELOCITY
ity component perpendicular to the mean shear, because both sensors can then experience the same mean velocity. No systematic study has been made of the effects of a mean velocity gradient upon inclined wires. The wall region of bounded turbulent shear flow is very important because most of the turbulent energy is produced there by the interaction of the streamwise and normal velocity components u and u . One disadvantage of the X-probe in this region is that its dimension in the normal direction y is large compared to the characteristic length scales. The effects of the strong velocity gradient in this region are also unknown. Beguier et aleszhave used another configuration in order to reduce the dimension of the probe in the y direction. A constant temperature wire with its axis parallel to the transverse direction z was used to measure the streamwise velocity. Several wire diameters downstream and displaced slightly in the normal direction, a parallel wire with a low overheat ratio and constant current sensed the temperature of the wake from the upstream wire. The measured temperature was a function of the wake’s position which depended upon the normal velocity component. The relationship between the temperature and the normal velocity v was determined by calibration. Dual film sensors deposited on a cylindrical or wedge substrate are available and have been used in liquids to measure the simultaneous u and u velocity components. Another use of multiple sensors is to measure velocity gradients in the normal spanwise directions. For example, the gradient du/ay is given approximately by = (uz - U M Y Z - Y1)
for y z sufficiently close to y, . Hence the voltage difference of two probes sensitive to the velocity u placed at y1 and y, will be proportional to the gradient. Other spatial arrangements of hot-wire and hot-film sensors have been used. One unique configuration is the streamwise vorticity meter proposed by K o v a s ~ n a y . By ~ ~ appropriately summing and subtracting the signals from four suitably arranged sensors, a voltage proportional to au/dy - d v / d z can be obtained. WyngaardsO has studied the spatial resolution for multiple probe configurations and estimates the error incurred for finite separations of the sensors. Kastrinakis er and VukoslavCevic and Wallaceszbhave experimentally studied the signal contamination by the cross-stream velocity components. C. Beguier, C. Rey, R. Dumas, and M. Astier, C . R . Acad. Sci., Paris, Ser A 277, 475 (1973). B*a
E. G . Kastrinakis, H. Eckelmann, and W. W. Willmarth, Rev. Sci. fnsrrum. 50, 759
( 1979).
P. Vukoslavtevic and J. M. Wallace, submitted to Rev. Sci. Insrrum. (1981).
1 . 2 . PROBE METHODS FOR VELOCITY MEASUREMENT
313
Measurements of gradients in the streamwise direction defer to the use of the hypothesis introduced by TayloP3 which states that the turbulent eddy producing the velocity field does not change significantly in the time required to convect past the probe. Thus any property of the fluidfassociated with the eddy has a fixed relationship between x and t so that f = f ( x - c t ) , where c is the convection velocity. The spatial derivative is then related to the time derivative by af/ax
=
-(l/c,af/at
Rather than measure this spatial gradient directly, it is easier to form the time derivative of the velocity signals electronically or digitally and use Taylor’s hypotheses with c = 0. Since hot-wire and/or hot-film sensors respond to variations of temperature and density as well as to the velocity of the fluid, multiple sensors at different overheat ratios have been used to measure the velocity and other properties instantaneously. Chevray and Tutus4 and A P 5 used two wires at different overheats and analog techniques to measure the streamwise velocity and temperature. Keffer et ~ 1 and. Chen ~ ~and Black~ e l d e I - 6have ~ used digital techniques to accomplish the same goal. The latter authors used a three-wire sensor and measured the instantaneous normal velocity component as well. Stanford and Libby6*used a similar arrangement to measure two velocities and the concentration of a twocomponent gas. Interest in the large scale spatial structure of turbulent shear flows has necessitated the measurement of simultaneous velocity measurements at many points in space. Aided by high speed digital data processing techniques, multiple hot-wire arrays have been extensively used by Gupta et al. ,69 Paizis and Schwarz,’O Blackwelder and Ka~lan,~O” and others. 1 . 2 . 4 . 6 . 5 . TEMPERATURE SENSITIVITY. The total heat lost by the sensor described in Section 1 . 2 . 4 . 3 . 2 is a function of the fluid velocity, the temperature difference between the mean sensor temperature and the fluid, the fluid properties, and the probe properties and geometry. When used as an anemometer, one attempts to hold all of the variabIes constant except the velocity. However, if any of the other properties change, the G. I . Taylor, Proc. R . Soc. London, Ser. A 164, 476 (1938). R. Chevray and N . K . Tutu, Rev. Sci. Instrurn. 43, 1417 (1972). 85 S. F. Ah, Rev. Sci. Insirurn., 46, 185 (1975). J . F. Keffer, G . J . Olaen, and J . G . Kawall, J . Nuid Mech. 79, 595 (1977). C. H. P. Chen and R. F. Blackwelder, J . Nuid Mech. 89, 1 (1978). R . A. Stanford and P. A . Libby, Phys. Fluids 17, 1353 (1974). gB A. K. Gupta, J . Laufer, and R. E. Kaplan, J . Nuid Mech. 50, 493 (1971). O ‘ S. T. Paizis and W. H. Schwarz, J . Nuid Mech. 63, 315 (1974). Ioa R. F. Blackwelder and R. E. Kaplan, J . Nuid Mech. 76, 89 (1976).
3 14
1. MEASUREMENT OF VELOCITY
heat transfer will vary, and thus the output of the device will change. Unless the user is able to correct for this change, he cannot interpret his data. In the laboratory, the temperature in the test facility may slowly change with time. Tests in the atmosphere may have temperature inhomogeneities and/or a mean temperature drift in time. In addition, the anemometer output will depend upon the composition of the fluid because the density, specific heat, and thermal conductivity may vary. C o r r ~ i n ' ~ has discussed this problem that is important where exhaust fumes are present, where the gas composition changes because of combustion, etc. To analyze the effect of the temperature of the ambient fluid, a reference temperature different from that of the ambient fluid must be used. If the principal form of heat transfer is due to forced convection, then the heat loss varies with both the temperature of the heated element and of the ambient fluid. Equation (1.2.38) can be written as F = F ( U , T , Tf) = d k f ( A
+ B Re")(T - Tf),
where Tfis the temperature of the fluid. The coefficients A and B depend upon the fluid properties, and are thereby functions of the fluid temperature Tf. The Joule heating depends upon the absolute resistance of the sensor and is independent of the ambient conditions, so
P
=
P(i, T) = i2R.
The drift of the system is studied by assuming that the changes in the ambient temperature are much slower than the changes in the velocity. Corrsin7' discussed this situation for constant current operation, and Bearman" has analyzed it for constant temperature operation. Corrsin not only estimates the errors in velocity measurements due to temperature drift, but shows that at very low overheat ratios in air, the constant current hot-wire anemometer is relatively insensitive to the velocity. At low overheat ratios, this configuration has found wide use as a sensor of instantaneous fluid temperature variations, as discussed in Chapter 4.1.
ACKNOWLEDGMENTS
I wish to express my gratitude to the late Prof. L.S.G. Kovasznay, who originally introduced me to hot-wire anemometry and whose ideas must surely permeate this work. Thanks also to Prof. Ray Emrich for editing the manuscript and to Mr. Zhang Yuquan for checking the equations. The support of ARO-Durham under grant DA-ARO-D-3 1- 12473-Gl18 during preparation of portions of this work is gratefully acknowledged. 'I
S. Corrsin, NACA Tech. Note 1864 (1949). No. 1 1 , p. 25 (1971).
'*P. W. Bearman, DISA If.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
315
1.2.5. Velocity Measurement by Other Probes*
Even with mechanical probes, e.g., the Pitot probe (Section 1.2.2) and the Vane anemometer (Section 1.2.3), either pressures, deflections, or rotation rates may be sensed electrically when the instrument develops beyond the rudimentary form. Other physical effects which require the transmission of electrical signals and electrical recording methods are exploited for fluid velocity measurements, but experience has shown that they are limited to more special applications. Those discussed are the more practical and fully developed ones of the many that have been proposed. Descriptions of principles and devices in the literature are numerous and extensive. compendium^'^-'^ are useful for a survey and for references. A large impetus for development lies in the commercial and industrial use of instruments for metering, and probably more development has occurred for installations in pipe walls and boundaries of channels than for probes in flow fields. 1.2.5.1. Acoustic Anemometer. Sound waves can be used in several ways to measure fluid velocity. In one method, a sound pulse is timed between a sender and a receiver inserted into the fluid. A second method requires making a schlieren photograph of a beam of finite ultrasonic wave fronts propagating in a gas, and reading from the photograph the amount the beam direction changes. A third method detects the Doppler shift in the frequency of sound scattered from inhomogeneities, e.g., particulates carried with the fluid. The disposition of senders and receivers when they are mounted in the walls of a duct filled with fluid is illustrated later in Fig. 25 of Section 1.2.6 on flowmeters. Pulse timing is usually performed on a short burst of ultrasonic waves. Infinitesimal pressure waves move with a speed c relative to the fluid medium characteristic of the fluid state. If the state is known and unchanging, and the speed of a sound pulse is measured over a given path the component of the fluid velocity v parallel to AB can be inferred, if u / c << 1. If the angle p that v makes with the direction AB is known or can be otherwise determined, the measurement then determines the magnitude of v. The hypothesized situation is seldom met in practice, however. Usually the characteristic speed c , which in a gas depends strongly
a,
r3 R. B. Dowdell, ed., “Flow-Its Measurement and Control in Science and Industry.” Instrument Society of America, Pittsburgh, Pennsylvania, 1974. Peter PereI’ C. G. Clayton, ed., “Modem Developments in Flow Measurement.” grinus, Ltd., London, 1972. 7s K . W. Bonfig, “Technische Durchflussmessung.” Vulkan-Verlag, Essen, 1977.
* Sections 1.2.5 and 1.2.6 are by
R. J. Emrich.
316
1. MEASUREMENT OF VELOCITY
on the temperature, is not accurately known, and it is difficult to determine the speed of a sound pulse with sufficient accuracy. In meteorological and oceanographic studies, the direction of the stream is variable, and needs to be measured as a function of time along with the magnitude of v. Practical sonic anemometers therefore dispose a pulse sender and a pulse receiver each at points A and B, and determine the two times tl and f z for pulses to travel in opposite directions, one from A to B and the other from B to A.76 The times required for pulses to travel between the two fixed points a distance d apart are77 (1.2.77) where uII and u, are the stream velocity components parallel and perpendicular to the sound pulse path A T . The plus sign and the minus sign are for pulses traveling respectively against and with the stream component vII. The difference in the two times is At = 2 d ~ “ / -( ~u2). ~
(1.2.78)
Measurement of At will then give one component of v if d and c are known and if vI is much smaller than uII. Note that uL does affect Ar if uI/ulI is not negligible, however. Since c depends on temperature, it is fortunate that c is calculable by finding the sum of the two times, and employing the formula tl
+ r2 = (1 - v zI/ c )
( 1 - u2/c”-’(2d/c),
(1.2.79)
which is easy to do if u / c is negligible. In the instrument developed by Mit~uta,’~ three noncoplanar pairs of points A and B are employed with d = 0.20 m, and electronic signal processing techniques are used to calculate and display the three components of v and the sound speed (displayed as virtual temperature. The sound pulses are emitted from piezoelectric ceramic elements 7 mm in diameter and received by identical ceramic elements, both mounted in streamlined bodies to reduce interference with the fluid stream whose velocity and temperature are being measured. Sound pulses of 100 kHz are transmitted, and the detection of the third os-
’@ Y. Mitsuta, Sonic anemometer-thermometer for atmospheric turbulence measurements. I n “Flow-Its Measurement and Control in Science and Industry” (R.B. Dowdell, ed.), Vol. 1, Part 1, p. 341. Instrument Society of America, Pittsburgh, Pennsylvania, 1974. R . M. Schotland, J . Mereorol. 12, 386-389 (1955).
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
317
cillation in the received pulse is performed to aid in achieving the accuracy of time measurement needed to display wind velocity components to a small fraction of a meter per second. When the wind velocity is within 5 degrees of one of the sound paths, accuracy is affected by the sound pulses having to travel in the wakes of the sender and receiver. There are several sources of error in sonic anemometers, and they are rather complex instruments. Even when the system is working, it is doubtful that 5% accuracy is achieved. When two instruments developed by different groups were used side by side at a meterorological test site, the agreement was not found to be better than A pulsed Doppler ultrcisonic velocity meter for finding the velocity profile in blood arteries is described by Peronneau et A single crystal exterior to the artery operates alternately as emitter and receiver. During the interval between pulses, the crystal receives the echoes reflected from the artery walls and from the blood cells. By electronic timing, the reflection from one depth is isolated, and the frequency of the reflected signal is determined. The reflected signal is Doppler shifted by an amount (1.2.80)
wherefo is the ultrasonic frequency transmitted; c , the sound speed in the tissue (water, essentially); a, the angle that the ultrasonic beam makes with the direction of the velocity of the sound scatterer; and u , the speed of the scatterer. Detection of the Doppler shift is obtained by beating the received signal with two signals at the emission frequency with phase difference of 90”;both detectors give the absolute magnitude of the Doppler shift, and the relative phase of the two Doppler signals is used to determine the sign of the shift and thus the sign of the velocity. By simultaneously choosing sixteen separate depth intervals and processing the isolated signals in sixteen channels of Doppler shift detection, velocities at sixteen depths can be displayed on a scope face within 850 ps. The emission frequency is 8 MHz, and the duration of the pulse is 0.5 ps; the time width for selecting the depth interval in each channel is 0.5 ps. Thus the changing velocity profile in the artery is displayed in real time, and by moving the transducer-receiver the position of the profile can be moved 78 M . Miyake, R. W. Stewart, R . W. Burling, L. R . Twang, B. M. Koprov, and D. E. Kuznetzov, Boundary Layer Mrreorol. 2, 228-245 (1971). 78 P. A. Peronneau, M. M. Pellet, M. C. Xhaard, and J. R. Hinglais, Pulsed doppler ultrasonic blood tlowmeter. Real time instantaneous velocity profiles. In “Flow-Its Measurement and Control in Science and Industry” (R. B. Dowdell, ed.), Vol. I , Part 3, pp. 1367- 1376. Instrument Society of America, Pittsburgh, Pennsylvania. 1974.
318
1.
MEASUREMENT OF VELOCITY
along the artery as it curves and changes its diameter. Since the direction of the velocity of a blood cell in a pulsating flow in a curving passage is not known, Peronneau and his co-workers dispose two transducers operating separately at different angles to find the direction of the velocity in the plane defined by the two beams. In principle, three identical sixteen depth channel velocimeters disposed in three dimensions could gather data to give complete direction and magnitudes of velocities in real time. Since the time interval selected defines the depth, a narrower interval is associated with a better defined depth; on the other hand, the narrower the time interval the greater the error in frequency measurement. In such an instrument, versatility rather than accuracy is emphasized. However, when calibrated by reflecting the sound from a rotating drum in water over the range 0.2-2.0 m s-' at angles a = 30°, 45", and 60°, the indicated velocities were accurate to about 5%. 1.2.5.2. Electromagnetic Anemometers. These devices require the imposition of a magnetic field B on the flowing medium and insertion of two electrodes for measurement of a potential difference; they find wide use as flowmeters where the walls of a channel are present for mounting electrodes but are unwieldy as movable probes. Michael Faraday in 1832 showed considerable interest in one of the early attempts to demonstrate electromagnetic flow detection when electrodes were placed on the banks of the Thames River and the potential difference generated by the water current through the B field of the earth was looked for but not detected. The representation of the integral of the E field in a moving medium basically requires treating an electromagnetic phenomenon with a relativistic description. Since the electric field E,electric current density J, magnetic field B, and medium velocity v all change according to the Lorentz transformation when different frames of reference are employed, and the constitutive relations involving conductivity, permittivity, and thermodynamic properties are valid only in the frame in which the fluid is at rest, considerable care must be used in stating the equations of motion and electromagnetism. We will be concerned only with reference frames moving relative to the laboratory with speeds much smaller than the velocity of light, since we need only consider frames locally attached to the fluid. We need not consider propagation effects and may assume that fields arising from charge motions at one place are not delayed in appearing at other places. Also, for simplicity, we will assume that there is no magnetization M in the moving material. The electromagnetic equations used are highly valid, but their forms seem inconsistent with those stated for an inertial frame in special relativity. The confusion of reference frames is also compounded with the use of units other than SI. We
1.2.
319
PROBE METHODS FOR VELOCITY MEASUREMENT
recommend books and articles whose authors have been aware of this possible confusion.80-82 The form of Faraday's law that is used in a fluid medium moving with respect to the laboratory in which there is a magnetic field B isel
v
X E;nd =
-(dB/dt)
4-
v
X (V X
B),
(1.2.81)
where Elnd is the induced part of the electric field as measured in the primed frame moving with respect to the laboratory with velocity v, and it is only part of the E' felt by the moving material. There are also Coulomb fields EL,, due to charges (conservative fields). The fields are E'
=
E,',d
+ E;,,,
moving medium (primed frame)
and E = Eind+ E,,,,
laboratory
(unprimed frame).
Since velocities are all very small compared with the velocity of light, it is not necessary to worry about propagation effects, or the change in charge density due to the Lorentz contraction, and Charges in the moving medium are subject to different forces than are charges on the electrodes and other bodies at rest in the laboratory, but all charges produce forces on other charges, and their final positions depend on the conductivity and dielectric properties of the materials. The voltmeter, at rest relative to the laboratory, used to measure the potential difference between the electrodes, responds to the integral of E, which is due only to aB/dr and to charges, but the positions of the charges are, as explained above, affected by the motions of some of them. Calculation of the electrode voltage is therefore a boundary value problem of some complexity, unless assumptions of simplifying conditions can be made. One assumption is that B is controlled only by the device in the laboratory used to create the B field, and that currents induced by the flowing medium make no contribution to B. However, this assumption cannot be made with good conductors such as liquid sodium and ionized gases. Another assumption that is good in rivers, tap water, and blood is that the conductivity is large enough so that displacement currents can be W. K . H . Panofsky and M. Phillips, "Classical Electricity and Magnetism," 2nd ed. Addison- Wesley, Reading, Massachusetts, 1962. P. Lorrain and D. Corson, "Electromagnetic Fields and Waves," 2nd ed., p. 341. Freeman, San Francisco, California, 1970. la R. P. Feynman, R. B. Leighton, and M. Sands, "The Feynman Lectures on Physics," Vol 11. Addison-Wesley, Reading, Massachusetts, 1964.
3 20
1. MEASUREMENT OF VELOCITY
neglected and Ohm’s law in the form J = uE’ applies in the medium. However, Cushinge3 has described a magnetic anemometer in which the opposite assumption is made, namely, that the fluid is nonconducting, so that dielectric polarization plays the chief role in redistributing charges and affecting the electrode potentials. If there is only a steady B field so that aB/ar = 0, Eq. (1.2.81) can be written V X (E’ - v X B) = -(dB/at) = 0 . A curl-free quantity can be written as the negative gradient of a scalar V,
so E‘ - v
X
B
= - VV.
(1.2.82)
Because J = uE’,and V E’ = (I/U) V J = 0,* the equation to be solved is simply a scalar potential equatione4 obtained by forming the divergence of Eq. (1.2.82): V2V = V * (v X B). (1.2.83) Note that when we can assume B is steady so that dB/at = 0, and when we can assume that the conductivity of the moving medium is high, not only does a scalar potential V exist, but the field E, of which V is the integral, is conservative and arises only from the charges as they redistribute themselves on the boundaries of the moving medium, the electrodes, and other bodies at rest in the laboratory. Solutions to Eq. (1.2.83) with a number of configurations have been carried out. The B field must be mapped or calculated, and the velocity profile throughout the B field must be specified in order that the sources in Eq. (1.2.83), which is Poisson’s equation, be known. Contributions to the potential developed by the motions are small, and electrolytic potentials are often present and of the same order of magnitude. The use of alternating B fields helps to compensate for these unwanted potentials, but Eq. (1.2.83) must then be modified in the way that the electrostatic equations are ordinarily modified for ac circuits, whereby self-inductance and “pickup” in the leads must be taken into account. Frequencies must be kept low enough so that “skin effect” plays no role. Devices to tow behind a ship to measure its speed, to insert into a blood artery, as well as to build into a channel wall to sense flow in the neighborV. Cushing, Rev. Sci. Instrum. 36, 1142 (1965). J. A. Shercliff, “The Theory of Electromagnetic Flow Measurement.” Cambridge Univ. Press, London and New York, 1962. Iu
* One of Maxwell’sequations i s V x B = J since we are neglecting displacement currents. The divergence of a curl is zero.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
32 I
hood of the wall have been built for demonstration purposes, and several ~ n a r k e t e d . ~Most ~ . ~ ~applications have been in pipes and channels for flowmetering. Since the fluid velocity next to the electrodes is always zero, there is always a “velocity profile” in the region between the electrodes, and the solutions of Eq. (1.2.83) differ for different profiles. While it can be shown84that the potential difference between the electrodes is simply proportional to the average fluid velocity in a pipe so long as the profile is axially symmetric, this is not the case when the profile is unsymmetric due to bends or obstructions, e.g., valves. Another type of electromagnetic fluid velocity sensor, used with very highly conducting fluids such as liquid sodium or ionized gas, detects in a nearby pickup coil distortions of an externally excited B field due to curWith such very good conductors, the sparents induced in the tial extent of the B field must be large to minimize the electrical shorting effect of the conducting fluid in the region outside the B field in the configuration to which Eq. (1.2.83),or its extension to the ac circuit model, applies. While, in principle, the electromagnetic anemometer is an absolute instrument not requiring calibration if B, the separation of the electrodes, and the potential difference are measured by calibrated instruments, tests to verify the accuracy of devices show that they seldom give values correct to 2%. Lack of knowledge of the velocity profile, inapplicability of simplifying assumptions needed in order to solve the complex boundary value problem, turbulence in the fluid, and presence of spurious electrolytic and thermoelectric effects are some of the reasons for the failures.
1.2.6. Flowmeters In laboratory work, in production plants, and in commercial sales of liquids and gases, the need to meter quantities of fluids and rates of flow of quantities of fluids continually arises. In practically all chemical and most physical laboratories, one sees flowmeters as parts of experimental arrangements. Chemical engineers and instrument makers for chemical plants are most cognizant of available meters and calibration facilities. A R . C. Baker, A review of some applications of electromagnetic flow measurement. In “Flow-Its Measurement and Control in Science and Industry’’ (R. B. Dowdell, ed.), Vol. 1, Part 2, pp. 745-753. Instrument Society of America, Pittsburgh, Pennsylvania, 1974. s6 K . W . Bonfig, “Technische Durchflussmessung,” pp. 97- 116. Vulkan-Verlag, Essen, 1977. G. Thatcher, Electromagnetic flowmeters for liquid metals. In “Modern Developments in Flow Measurement” (C. G . Clayton, ed.), pp. 359-379. Peter Peregrinus, Ltd., London, 1972.
322
1.
MEASUREMENT OF VELOCITY
wide variety of special meters developed for specific uses exists, but unlike liquid-in-glass thermometers and Bourdon-type pressure gages, no standard flowmeter of general usefulness and comparable reliability exists. Probably the most generally useful are positive displacement meters, used, for example, for sales of water, gasoline, and fuel gas, and rotameters, the tapered glass tubes with a float indicating flowrate on an etched scale on the glass. All of these are subject to fouling through deposit of solids carried in the fluids, and their wide use is attributable to the mechanical ingenuity of their designers and makers for mass production. For maximum reliability and constancy of calibration, orijice-type meters, which provide a pressure drop across an insert in the pipe or duct carrying the fluid, are preferred, since they are less subject to fouling and employ the readily available pressure difference measuring instruments. Metering of liquids in open channels by means of a dam or weir, where only the height of the surface of the liquid ahead of the weir needs to be measured, falls in the same category. The intent in this section is to elucidate the principles on which these flowmetering methods work. Further explanation and presentation of detailed specifications for their use can be found in books by Hengstenberg,88B ~ n f i gand , ~ ~an ASME paneLBO Manufacturer’s literature is often the most up-to-date source of information on specific flowmetering problems. Unless one is involved in metering an entire river or a comparable huge flow, the calibration of any flowmeter is usually a simple affair of capturing the efflux in an empty tank of known volume and combining the measurements with those made by a clock. If a gas is being metered, measurement of temperature and pressure of the gas in the known volume usually is adequate to determine the volume or mass of the metered fluid; in a large tank, several hours must elapse to be certain that the gas temperature has come to the temperature of the tank and thermometer. High pressure gases supplied in steel tanks can sometimes be metered very accurately by weighing the tank on a sensitive balance; although the weight of the tank far exceeds that of the gas, very sensitive and accurate balances are available to detect the amount of gas which has left the tank. Irregular, unsteady, and unpredictable velocity profiles are present in some flowmetering situations. A river with continually changing bed and banks is one example. Blood flow, pulsating in an artery of variable size J. Hengstenberg, B. Sturm, and 0. Winkler, eds., “Messen und Regeln in der chemischen Technik.” Springer-Verlag, Berlin and New York, 1957. K . W. Bonfig, “Technische Durchflussmessung.” Vulkan-Verlag, Essen, 1977. H. S. Bean, ed., “Fluid Meters: Their Theory and Application,” 6th ed. Am. SOC. Mech. Eng., New York, 1971.
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
323
and curvature, is another. In such cases, use of velocity probes which give the velocity as a function of position over the cross section is necessary, even though one is only interested in the total flow in the channel, and all of the methods described in this chapter may be called upon. Finding the velocity profile over an entire cross section is usually quite difficult and tedious; many clever procedures to reduce the work and time are described in the literature. 1.2.6.1. Positive Displacement Flowmeters. This category of meter can be described as a pump running backwards: the fluid flow runs the pump instead of the pump pushing the fluid. The number of fillings of a scoop or chamber is counted, often in a rotary fashion. An example is the so-called wet-gas meter shown in Fig. 22. A drum is divided into sections into which the gas flows successively, causing the drum to turn. The sealing liquid keeps the gas which enters the hub from emerging into more than one section at a time, and the slightly increased pressure in the filling section causes the drum to turn in the sense indicated by the arrow. By the time the outer slot in section A emerges from the sealing liquid and the pressure in A falls, the entrance a in the hub is sealed and the next hub slot b provides an outlet for the gas into section B. The drum is connected to a revolution counter that keeps track of the volume of gas passing through. The liquid level must be maintained at the proper height and
FIG.22. Wet-gas meter. The inner drum is divided into sections A , E , C, andD and turns freely on bearings relative to the outer fixed drum. Gas entering through the hub flows out through small slots a , 6, c , and d as they rise above the level of the sealing liquid, filling the sections in turn. Liquid then displaces gas in the sections as they are again immersed, flowing through immersed slots in the hub.
3 24
1.
MEASUREMENT OF VELOCITY
the whole apparatus set on a level surface. So long as the rate of gas flow is below that causing the liquid to slosh in the hub, the accuracy of measurement can be better than 0.5%.8D The gas must have a limited solubility in the liquid, and evaporation of the liquid adds to the quantity of gas and vapor emerging; air with 100% humidity is accurately measured if the sealing liquid is water. A low vapor pressure oil should be used with dry gas. Since there are no pistons or valves, this meter is particularly useful for very low flow rates. It becomes rather large and bulky for reliable metering of large flowrates, and in general is limited to 0.1 m3 s-l. A wide variety of meters is available commercially, utilizing almost every conceivable mechanism.e1 Some of them are the double-acting piston with valves, rotary piston, interlocking gears, reciprocating bellows, and diaphragm. Varying amounts of leakage are permitted to avoid friction and loss of power due to pressure drops over the mechanism of the meter. 1.2.6.2. Turbine Flowmeters. A turbine or propeller set in the fluid line which turns faster for greater flow rates may have a geared or magnetically actuated revolution counter or a magnetically actuated rate of revolution readout. Basically it is a leaky positive displacement meter. Its chief advantage is that it requires less pressure drop to actuate it; its disadvantage is that it is not accurate at flow rates of 20% or less of its capacity. Another advantage is less likelihood ofjamming due to foreign solids in the fluid. Most have flow-straightening vanes ahead of the turning vanes. A useful table comparing different types of positive displacement and turbine types as to size, capacity, range of validity Qmax/Qmin, accuracy, pressure drop, and working pressure is given in B ~ n f i g . ~ ~ 1.2.6.3. Venturi and Orifice Flowmeters. Metering by measurement of pressure drop at, or across, a constriction in a pipe is by far the most exact and widespread of all flowmetering methods. Variously called Venturi meters, orifice meters, differential pressure meters, or metering nozzles, all make use of the reliability of pressure measuring instruments. Because of their commercial importance, orifice flowmeters are provided by many commercial suppliers, who are able to provide advice as well as the necessary devices. There is no basic foundation underlying the operation of orifice type meters, but we can understand the types of phenomena involved, and the extent to which dimensional analysis can provide confidence in extension of calibrations (see Part 10: “Dimensional Analysis and Model Testing D. M. Considine, “Encyclopedia of Instrumentation and Control.” New York, 1971.
McGraw-Hill,
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
325
Principles”) by approaching the flow as though it were an ideal frictionless flow. First let us consider an incompressible, inviscid, steady flow to which the Bernoulli formula in the form p
+ +pv2 = p s = const
(1.2.84)
can be applied. (See Section 1.2.2 for a discussion of the Bernoulli formula.) Water, and air at Mach numbers below 0.2, can be described by Eq. (1.2.84) fairly well outside boundary layers. In a pipe, boundary layers extend fairly far out from the walls in many cases, and the Bernoulli formula is not applicable to the extent that there is a noticeable pressure drop along a uniform cross section pipe but no change in speed v, which is clearly in violation of Eq. (1.2.84). Nevertheless, to see qualitatively what changes in pressure there are in a pipe constriction used for a flowmeter, Eq. (1.2.84) is helpful. One type of orifice meter used widely is diagramed in Fig. 23. First let us apply the principle of conservation of matter to the incompressible fluid supposed flowing in the pipe at a constant volume per second of
Q=
v1 dA A1
u2 dA = DIAl = &AA,.
=
(1.2.85)
/A*
Now in Fig. 23 it is supposed that fluid flow lines leave the pipe wall at a place upstream of the orifice plate, graze its edge, and contract to a minimum cross section A 2 . Shown are regions of recirculating fluid each side of the plate, and the jet of fluid breaking up into eddies as it reexpands to the original pipe cross section. In actuality the fluid in the recirculating regions is continually mixing with the passing fluid, but the situation diagramed is a sort of time average. We can measure the pressure at various stations along the pipe with wall taps (see Section 5.3.1). We know that there is a pressure gradient associated with curved streamlines, and therefore do not expect the pressure to be the same over a cross section in a section of contraction or expansion. We suppose, however, that at the region where the jet is breaking up, the time average of the pressure is about the same as at the wall tap shown as p ; . Equation (1.2.85) states that at the smaller cross section A 2 , v2 = (A1/A2)v1, and thus Eq. (1.2.85) says that the pressure p i is smaller than p i . By measuring the pressure drop, the flow rate could be calculated if the pressure and velocity were uniform over a cross section and if the Bernoulli formula were strictly valid. The result of the algebraic combination of Eqs. (1.2.84) and (1.2.85) with p constant gives (1.2.86)
326
1.
PI
PI
MEASUREMENT OF VELOCITY
t I
Workin preawPe
r:! ' ; 3\ !
- -1-
I
\\
I
A',
I 2p'p
p:
Ressure l a 8 AlP;
1
b)
I
FIG.23. Orifice flowmeter. (a) Schematic diagram of Row through a sharp-edged orifice in a pipe. Smooth Row approaches from left at speed u, and squeezes down to form a j e t at higher speed uI. Circulating pools of semistagnant fluid are ahead of and behind the orifice plate. The jet breaks up in violent diffusive motion and, farther down the pipe, the Row calms to again proceed at speed u, . (b) Schematic graph of pressure variation along the pipe near the orifice. Heavy line indicates average pressure at the wall. Dashed line indicates average pressure at the centerline. Pressures are fluctuating, particularly in the diffuser.
According to the Bernoulli formula [Eq. (1.2.84)], the pressure farther down the pipe p ; , at the place schematically designated in Fig. 23, recovers to its value p i , but in reality pi - ph is greater than zero and represents apressure loss as sketched. Althoughp; - p ; cannot be measured, p 1 - p z is easily measured with the wall taps shown and is experimentally found to be reproducibly related to Q = &Az for a given pipe and flow geometry. The relation is not as simple, however, as would appear from Eq. (1.2.86). Orifice flowmetering requires a straight smooth pipe leading up to the constriction and an unobstructed diffuser downstream, a carefully controlled shape at the narrowest part of the orifice (the measurement is upset by erosion of the orifice edge or solid deposits on it), and a low level of or freedom from pulsation in the flow. Swirling flow must be avoided by insertion upstream of aflow straightener -a bundle of (thin-walled tubes) or an eggcrate of thin plates at least two pipe diameters long. The meter requires calibration utilizing the pressure difference, usually p1 - p 2 , called the working pressure, specified for the particular unit. It is necessary to machine the parts carefully according to specifications and to refer to calibration charts such as those given in the books by
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
321
Hengstenberg,@B~nfig,~O or Bean.go With attention to details, it is possible to meter with better than 1% accuracy in a wide variety of cases. Commercial gages are supplied with their own calibration charts. A closer look at some of the details will help as a guide through the extensive changes in the references from the elementary notation employed here. Since SI units were not employed in the older references, care must be taken to convert non-SI formulas from weight density to mass density by insertion or deletion of g, the weight per unit mass. (i) First we observe that flowmetering requires the mean speed D = Q/A, whereas the Bernoulli formula [Eq. (!.2.84)] deals with local speed u and pressure p . Since in steady flow (and small scale turbulent flow, i.e., no extensive regions of secondary flow such as depicted in Fig. 23) the time-averaged pressure is essentially constant over a cross section, it is useful to form a time and space average of u over cross sections so that the speed squared term in the averaged Bernoulli formula becomes the meun kinetic energy per unit mass
2AD
v 3 d A = p -D.2 A
2
(1.2.87)
Thus p, for a steady pipe flow, takes into account the effect of different velocity profiles on the pressure variation with area changes. For nearly all turbulent pipe flows p varies only in the range 1.0 < p < 1 . 1 , but it has the value p = 2.0 for laminar Poiseuille flow. (ii) A second consideration leading to altering the application of the Bernoulli formula is that the fluid contracts to a smaller cross section Az than the area A. of the actual orifice. Thus the contraction number p = A2/Ao and the orifice ratio m = Ao/AI combine with Eq. (1.2.87) in the (still frictionless) relation of Eq. (1.2.86) to give
(iii) As sketched in Fig. 23, p , is somewhat higher thanpi and, as mentioned earlier, the pressure at the wall tap bears some uncalculable relation to the pressure in the center where Eq. (1.2.84) would apply. These uncalculable pressure differences could be represented by a coefficient 6, so the flowmetering equation becomes (1.2.89)
The virtue of writing Eq. (1.2.89) with all its coefficients, only one of which is calculable, is the support it gives to combining all of them
328
1. MEASUREMENT O F VELOCITY
together in a single discharge coejjicient a and writing (1.2.90) where a is a dimensionless coefficient. This equation might have been written on dimensional analysis grounds alone, but its form and the variables on which a depends are suggested by the approach of the Bernoulli formula [Eq. (1.2.84)]. (iv) For geometrically similar orifices, all the uncalculable effects can be comprised in a plot of empirical values of cr as a function of the pipe Reynolds number Re = D o , / , (where D is the diameter of the pipe entry section, and Y is the kinematic viscosity of the fluid being metered), and of the orifice ratio m = Ao/Al: a
= a(m,
Re).
(1.2.91)
An example of the empirical dependence of Eq. (1.2.91) is shown in Fig. 24 for a sharp-edged orifice in a pipe of circular cross section. It is noteworthy that cr only varies from 0.6 to 0.8 with no dependence on Reynolds number so long as Re > 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.
PROBE METHODS FOR VELOCITY MEASUREMENT
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.
MEASUREMENT OF VELOCITY
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.
PROBE METHODS FOR VELOCITY MEASUREMENT
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
1. MEASUREMENT OF VELOCITY
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.
PROBE METHODS FOR VELOCITY MEASUREMENT
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.
MEASUREMENT OF VELOCITY
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.
PROBE METHODS FOR VELOCITY MEASUREMENT
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.
MEASUREMENT OF VELOCITY
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.
PROBE METHODS FOR VELOCITY MEASUREMENT
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. E. E. Tompkins, Flow measurement utilizing multiple, parallel capillaries. I n “Flow-Its Measurement and Control in Science and Industry” (R. B. Dowdell, ed.), Vol. 1, Part 2, pp. 465-472. Instrument Society of America, Pittsburgh, Pennsylvania, 1974.
1.
338
FluM fla
Fluid flow
MEASUREMENT OF VELOCITY
-
-
\\
(C)
FIG.25. Acoustic flowmeter arrangements. (a) Time of flight. In steady homogeneous flows only one channel, say X I , need be used, and each piezoelement serves alternately as sender and receiver. (b) Beam displacement. The relative signal height detected by the two receivers gives information on the displacement of the directed ultrasonic beam. (c) Doppler shift of radiation from entrained scatterers.
Measurement of tl and r2 with sufficient accuracy requires some ingenuity in the electronic circuit designs. One technique employs a burst of about five ultrasonic vibration periods, and selection of one of the center oscillations of the burst received for precise timing of the crossover. Another sends ultrasonic vibrations so long as the receiver does not receive any signal and ceases as soon as a signal is received: when the broadcast train has ended its journey to the receiver, the circuit then starts a new train of vibrations, so the period of the repetitive starting of trains repre-
1.2.
PROBE METHODS FOR VELOCITY MEASUREMENT
339
sents the travel time. This latter “sing-uround” method uses two channels with and against the fluid flow and requires all four piezoelements, as pictured in Fig. 25a. A third method monitors the phase of the received continuous ultrasonic signal and reads out the fraction of a period for a precise “vernier” addendum to the time of flight in an integral number of periods determined or calculated by another method, e.g., knowledge of the range of 17 and the approximate speed of sound. Again, alternation of the sense of sending the ultrasonic signal in one channel, or simultaneously operating phase detection loops in both channels # 1 and #2 is necessary, since the speed of sound is seldom known with sufficient precision to calculate the fluid speed from one time of flight only. The beam displacement method in Fig. 25b does not require precise knowledge of the sound speed, but has a more limited range of fluid speeds which it can measure. The wavelength of a 5 MHz ultrasonic wave train in water is about 0.3 mm; a 3-mm-diameter sender broadcasts a directional beam whose intensity spreads out relatively little, so that its maximum can be located across the channel. The maximum is deflected an amount b after traversing a channel breadth D where b / D is given by1o1 b / D = ij/c.
(1.2.101)
At fluid speed 0 = 10 m s-’, since sound speed in water is 1480 m * s-l, the deflection is b / D = 1011480 or about 0.7%. Rather careful measurement of b is necessary, but speeds of 3 m s-’ or greater are measurable with 2% accuracy.’O’ The beam displacement method is also used at high gas speeds such as produced in a shock tube, where the ultrasonic beam can be photographed by spark schlieren and its deflection measured from the photograph; the sound speed c is found from the spacing of the wavefronts in the photograph and knowledge of the frequency broadcast by the sender in the channel wall. If the fluid speed or sound speed is nonuniform over the channel, the photograph reveals the variations. The ultrasonic Doppler reflection method has been described as a fluid velocity probe in Section 1.2.5, since it is uniquely able to determine a nonuniform velocity profile, and the flow rate is then calculated by integrating the perpendicular component of the velocity over the cross sectional area. This method is used, with less elaborate electronic circuit requirements, for observing the shift of frequency in the scattered acoustic radiation from entrained scatterers in the central region of a straight K . W. Bonfig, “Technische Durchflussrnessung,” pp. 141, 143. Essen. 1977.
Vulkan-Verlag,
340
1. MEASUREMENT
OF VELOCITY
channel. A separate receiver to pick up the radiation scattered from a continuously broadcasting ultrasonic sender feeds a frequency shift detector, and thus continuously monitors the flow rate so long as the fluid velocity profile remains constant.’O’ A number of configurations of the B ELECTROMAGNETIC FLOWMETER. field, of conducting and nonconducting pipe walls, and velocity profiles lead to soluble boundary value problems,102although some require assumptions of doubtful accuracy. In practice, alternating B fields are frequently employed; the spurious electrolytic and thermoelectric emf s sensed by the electrodes are thereby greatly reduced, but mutual inductances and capacitances then lead to pickup signals which must be compensated for. When the B field can be treated as homogeneous (whether time varying or not), calculation of eddy currents due to assumed velocity profiles is easier. However, it is most practical to use a B field of limited spatial extent, and then three-dimensional effects complicate the calculations so much that early resort to testing is required. Ferromagnetic material in the form of a “C” to generate a strong B field in the gap is used in one type of meter. The field may be excited by incorporating a strong permanent magnet within the “C” or by using windings carrying direct or alternating current. The fluid is moving in a metal pipe or in an electrically insulating pipe between the pole faces; electrodes may be fastened to the exterior of the conducting pipe, but larger potentials are produced by electrodes leading through an insulating pipe or pipe liner to make contact with the fluid. Conducting pipe should have lower conductivity than the fluid; otherwise, the nonmoving pipe tends to “short out” the potential developed. When the fluid is hot, the use of ferromagnetic material to produce the B field may be ruled out. In such cases, the B field may be generated by pancake coils symmetrically placed on each side of the pipe carrying the fluid. Another case in which the excited field is produced by coils without magnetic material is in cuff--type meters placed around blood arteries and vessels in animal experiments. Electromagnetic flowmeters developed by commercial suppliers of instruments are the result of many tests carried out under conditions simulating the environment in which they will be used. The electrical conductivity of the fluid metered must remain within certain bounds to meet the conditions under which the meters have been tested, but the fluid may, for example, contain large amounts of suspended materials. Flowmeters can be inserted in pipes ranging in size from 1 mm to several meters in diamel M J. A. Shercliff, “The Theory of Electromagnetic Flow Measurement.” Cambridge Univ. Press, London and New York, 1962.
1.3.
ANALYSIS: DOPPLER SHIFT OF CHARACTERISTIC RADIATION
341
ter in the chemical industry, and are used often in adverse environments such as “dirty” fluids containing solids, e.g., sewage, and with corrosive fluids. The electromagnetic flowmeter in a compact form is widely used in biological experiments for blood flow, and can be obtained from medical instrument manufacturers. Electromagnetic flowmeters have been developed for use inside nuclear reactors, where reliable measurements of coolant flows are very important.
1.3. Measurement of Velocity by Analysis of Doppler Shift of Characteristic Radiation* Tracer methods and probe methods, described in the preceding chapters, are developed and widely used in fluid dynamics experiments. Laser radiation scattered by particulate tracers is collected and analyzed for the Doppler shift in its frequency to find the tracer speed. This laser Doppler velocimeter is described in Section 1.1.4. A Doppler shift of emitted characteristic spectral radiation has been noted and used in astrophysics throughout the past century to measure the speeds of astronomical objects, but terrestrial fluid velocities are so small (in comparison to the speed of light) that the method has not been developed for fluid dynamics experiments to the degree that tracer and probe methods have. It is available in principle, and has been demonstrated to be possible; the spectroscopes used for direct spectrum analysis of laser Doppler scattered light and for laser Raman scattered light can also be used for emitted radiation. Very small frequency shifts have to be measured, and very narrow bands of emitted frequency (i.e., sharp spectral lines) have to be found in the moving fluid. The latter requirement means in general that only luminous rarefied gases are suitable objects for this type of measurement, and rarefied gases do not in general emit strong radiation. If not self-luminous because of their high temperature, gases are made luminous by intense laser beams and electron beams. The emission of radiation so induced can be analyzed for Doppler shift as well as to measure gas composition and temperature; these latter measurements are described in Chapters 3.2, 3.3, 4.2, and 6.4 of Part B of this volume. * Chapter 1.3 is by R. J. Emrich.
342
1. MEASUREMENT OF VELOCITY
1.3.1.Doppler Shift Formulas Consider a molecule moving with velocity v relative to the laboratory, in which a spectrometer is measuring the frequency of light emitted by the molecule. In the frame of reference of the molecule, light is emitted with a characteristic frequency w ’ , which may be looked up in spectroscopic tables, and goes off in a direction e6 making an angle 8’ with v in order to amve at the spectrometer inlet. In the frame of reference of the laboratory, the light has a different frequency w and is traveling in a direction e, making an angle 8 with v. The frequency measured is related to the characteristic frequency by’ (1.3.1) where c is the speed of light. If the molecule is moving directly away from the spectrometer, 8 is 27r and the measured frequency is (1.3.2) which is smaller than the characteristic frequency. Formulas (1.3.1) and (1.3.2) apply to all gas speeds u , even those approaching the speed of light c. In fluid dynamics experiments u / c << 1 and the doppler shift formula may be expressed as
+ ( u / c ) cos 8) = o’+ k,
w = o’(1
v
(1.3.3)
by retaining only terms of order u / c in a power series expansion. Here k, is the propagation vector of the emitted radiation: k, = kses = (27r/A,)e,, where e, is a unit vector with the direction and sense of k, ,and the magnitude k, = 27r/As, where A, is the wavelength of the emitted light. If laser light of frequency ooin the laboratory is incident on the moving molecule, the frequency “felt” by the molecule is 0”
= wo
- kQ’v
(1.3.4)
since, to the molecule, the laser source appears to be moving with velocity - v. The propagation vector of the incident laser radiation is ko . An isolated molecule absorbing this radiation of frequency w’’ may emit at the same frequency, or it may emit at a different frequency and be left in an energy state other than the one it was in before it absorbed the radiation. A molecule with many closely spaced vibrational states may emit J. D. Jackson, “Classical Electrodynamics,” p. 364. Wiley, New York, 1962.
1.3.
ANALYSIS: DOPPLER SH IFT OF CHARACTERISTIC RADIATION
343
radiation only slightly shifted in frequency from the incident frequency w“; these characteristic shifts do not depend on W” and are called Raman
shifts or Stokes shifts. The molecule may also emit other frequencies that are characteristic of the molecule and unrelated to the incident frequency and are called fluorescent emissions; these emissions are brighter when the incident radiation has a frequency W ” at which the molecule strongly absorbs, called a resonant frequency. The light emitted from the moving molecule, whether of frequency W ” (called Rayleigh scattered, or resonance radiation if W ” coincides with a molecular resonance frequency) or of a shifted frequency (called Raman scattered if no resonant absorption is involved), will seem to be again shifted when analyzed by the spectrometer at rest. For Rayleigh scattered light, i.e., for o’ = o r ’ ,combining Eqs. (1.3.3) and (1.3.4) W,
=
00
+ (k, - kJ
V.
(1.3.5)
This is the same formula which applies to the laser doppler velocimeter (see Section 1.1.4.2). 1.3.2. Method of Measurement of Doppler Shift
Greytak and BenedekZand Robben and Cattolica3 employed very high resolution spectrometers to detect the shift of the center of the 488-nm line of an argon ion laser which was Rayleigh scattered from a test gas. The low intensity of the scattered light requires great care to shield the spectrograph input from stray laser light scattered from windows and slits. The laser beam must be trapped after traversing the test gas, and the background seen by the spectrograph slit must also be a light trap. Robben and Cattolica studied light scattered from a subsonic and a supersonic argon jet whose gas speed at various distances from the nozzle was calculated from fluid dynamic formulas. They employed piezoelectrically controlled spacing of the plates of a Fabry -Perot interferometer to detect the center of the spectral line, achieving accuracy of 2.5 m s-’ in the flow velocity. The low intensity of the scattered radiation required photon counting for 20 s to achieve each velocity reading. The velocity of sputtered atoms has been measured by determining the
* T. J . Greytak and G . B. Benedek, Spectrum of light scattered from thermal fluctuations in gases. Phys. Rev. Leu. 17, 179-182 (1966). F. Robben and R. Cattolica, Gas velocity and temperature measurement from molecular scattering of laser light. Mechanical Engineering Dept., Univ. of California, Berkeley, Oct. 1973.
344
1.
MEASUREMENT OF VELOCITY
doppler shift of laser light which is resonantly a b ~ o r b e d .In ~ this experiment essentially only those atoms moving with the velocity giving the doppler shift to the resonant frequency, Eq. (1.3.4), absorb and reradiate. The number of atoms having various velocities is found by scanning the dye laser frequency and observing the emitted radiation of all frequencies.
A . Elbern, E. Hintz and B. Schweer, J . Nucl. Marer. 76 & 77, 143-148 (1978).
2. DENSITY SENSITIVE FLOW VISUALIZATION* 2.1. Introduction Visual inspection or observation of a fluid-mechanical process may render a comprehensive insight into the development of a fluid flow. Such observation becomes an objective research tool if one can derive quantitative flow data from the observed flow pattern. Fluids, and particularly gases, are normally transparent and therefore invisible. One needs therefore to provide special flow visualization techniques which enable one to observe the flowing fluid. Some of these techniques depend on the fact that the optical behavior of gases is related to the density of the gas. Optical flow visualization methods are therefore applied to fluid flows which exhibit a variation of the fluid density, either locally or as a function of time in the flow field. Such flows are called compressible. The most common means of visualizing a compressible flow field is to record the refractive behavior of the gas flow when illuminated by a beam of visible light. Since the fluid density is a function of the refractive index of the flowing fluid, the compressible flow field represents, in optical terms, a phase object. A light beam transmitted through this object is affected with respect to its optical phase, but the intensity or amplitude of the light remains unchanged after passage. Certain optical methods which are sensitive to changes of the index of refraction in the field under investigation can provide information on the density distribution in the flow. From the so-determined density values, one might then deduce further information on other flow parameters, e.g., temperature, pressure, and velocity, provided that an appropriate equation of state is available. This is often the case for gases. Difficulties might arise for liquids, e.g., in visualizing stratified water flows, since in this case an equation of state is either not available or very restricted in its range of applicability. Besides the refractive behavior, one can also make use of the radiative characteristics of some gases for the purpose of flow visualization. In this case, one injects energy, in the form of either an electron beam or an electric discharge, into a gas flow so that the gas molecules are excited to emit characteristic radiation. The intensity of this radiation is a function * Part 2 is by
W. Menkirch.
345 METHODS OF EXPERIMENTAL PHYSICS, VOL.
18A
Copyright @ 1981 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0- 12-475960-2
3 46
2.
DENSITY SENSITIVE FLOW VISUALIZATION
of the gas density, and one may thereby visualize regimes of different gas density in a flow field. When utilizing these methods one must provide that the energy injected into the flow (per unit time) is only a negligible fraction of the flow’s kinetic energy. Otherwise the technique could not be considered as a nondisturbing test method. Such radiation emission methods are rarely used for quantitative purposes and are applied in the study of low-density gas flows only. The major part of the present discussion will be devoted to the principles of optical flow visualization methods used for compressible flows. These principles have been known for more than a century. Toepler published his book on schlieren methods in 1864, and Ernst Mach and his co-workers were utilizing shadow, schlieren, and interferometric methods for ballistic studies in the 1870s. There has been continuous improvement of these optical methods and of insight into their principles. From time to time this development becomes a very rapid one, e.g., during the last decade, when the availability of lasers and holographic methods pushed the development forward in great strides. A major problem still requiring a solution is development of efficient evaluation procedures so that all the information contained in a flow photo can be resolved into spatial information, particularly for an asymmetric flow field. In the following sections we shall discuss the refractive behavior of a compressible fluid flow, employing viewpoints of both physical and geometric optics. The principles of the different visualization methods will then be described, and finally a survey of evaluation procedures necessary to obtain quantitative flow data will be presented. More extensive reviews are available for additional information.’-‘
2.2. Refractive Behavior of Fluids 2.2.1. Relation between Fluid Density and Refractive Index
We consider a light beam of finite width which is transmitted through a compressible flow field. The molecules of the fluid are assumed to have no net electric dipole moment. The light beam represents an electromagnetic wave with the electric field vector E,which distorts the charge con-
’ D. W. Holder and R. J. North, “Schlieren Methods,” Notes Appl. Sci. No. 31. HM Stationery Office, London, 1%3. F. J. Weinberg, “Optics of Flames.” Butterworth, London, 1963. a W. Hauf and U. Grigull, Adv. Heat Transfer 6, 131 (1970). W. Merzkirch, “Flow Visualization.” Academic Press, New York, 1974.
’
2.2.
REFRACTIVE BEHAVIOR O F FLUIDS
347
figuration of the fluid molecules. A dipole moment p thereby induced per molecule is proportional to E: p = aE,
(2.2.1)
where a is the electronic polarizability. Since E describes an oscillating field, the distortion of the electronic charge configuration is frequency dependent. The field E might be represented by E
=
Eo exp(i27rvr),
where Eo is the amplitude; Y , the (single) frequency; and t , the time coordinate. In the classical radiation interaction theory of Lorentz, p is related to E by utilizing the model of an induced harmonic electron oscillator. If one assumes furthermore that several distorted electrons per molecule with different resonant frequencies vi and oscillator strengths f, contribute to the induced dipole moment, one obtains P
=
E/(d- vz)l,
(e2E/4trZm,)
(2.2.2)
i
where e is the charge; and me, the mass of an electron. In the preceding discussion it has been presumed that the induced dipole per molecule depends on the external field vector E only. The fluid molecules in the neighborhood of the molecule under consideration also become electric dipoles and therefore generate a secondary electric field which is superimposed on the external field E. The influence of this secondary field may be neglected if the average distances between the fluid molecules are large, e.g., in a gas. If this approximation cannot be made one should use, instead of Eq. (2.2.1) a relation p = aEeff,where Eeffis an effective field vector which also takes into account the secondary field. For a dielectric medium with randomly distributed molecules, Eeff is known to be Eeff = E + $TP,
where p designates the polarization vector, i.e., the net dipole moment per unit volume of the dielectric medium. With N being the number density of molecules in the fluid, the polarization p is, according to the Lorentz theory, given by p = aN(E
+ 47rp).
(2.2.3)
By using the relation p = ( E - 1) E/47r, where E is the dielectric constant of the medium, one may eliminate p and E from (2.2.3) and express the product ( N a ) as a function of E . The number density N can be replaced by the density p of the fluid through N = 6Pp/M, 3 being Loschmidt’s number and M, the molar weight of the fluid. For the refractive index n
348
2.
DENSITY SENSITIVE FLOW VISUALIZATION
of the dielectric fluid, one has n = so that one derives from (2.2.3), with the aid of (2.2.1) and (2.2.2), the following expression:
(n2 - l)/(n2 + 2) = @2'e2/37rmJM)
[fi/(V:
-
3)]. (2.2.4)
f
This is a relation between the refractive index n of a fluid and the fluid density p expressed in terms of atomic constants and properties of the fluid, and the frequency u of the light used. Equation (2.2.4) is often called the Clausius-Mosotti relation and should be used if the fluid is not a gas. The use of the model of a harmonic electron oscillator in deriving (2.2.2) excludes the presence of damping forces or of optical absorption in the medium under study. This assumption is well justified if the resonant frequencies vf of the fluid are far from the frequency v of the light used. This is the case for most gases at normal temperatures. The assumption can be violated in some liquids and in gases at high temperatures when certain degrees of freedom are additionally excited. The use of optical visualization methods in the regime of anomalous dispersion, i.e., with Y being close to uf , will be discussed below (see hook method). The Clausius-Mosotti relation (2.2.4) can be simplified in the case of gases. The refractive index of most gases is very close to one (e.g., for air n = 1.000292 at A = 5893 A), so that in (2.2.4) one may replace (n2 - 1) by 2(n - 1) and (n2 + 2) by 3. This results in the so-called Gladstone-Dale relation, which applies to gases only: n - 1 = @5?e2/27rm&fE[fi/(V: - 3>].
(2.2.Sa)
The physical meaning of this simplification is that the spacing between the molecules in a gas is so large that one may neglect the influence of the secondary field in deriving (2.2.3). With the simplified relation (2.2.3), in the form p = aNE, one arrives at the same equation (2.2.5a). The Gladstone-Dale relation is often written in the form
n
- 1 =
Kp,
(2.2.5b)
where the Gladstone-Dale constant K has the dimension of l/p and depends on some characteristics of the gas as well as on the frequency or wavelength of the light used. Away from the immediate neighborhood of the resonant frequencies v f , K is only weakly dispersive, which can be seen if one replaces vf by the resonant wavelength At ; then, with (2.2.5a),
K
= (e2 ~/2.rrc2mJ4)8[f,AfA2/(A2- A f ) ] ,
(2.2.6)
where c is the velocity of light in vacuum. Since usually A? << A2, the sum in (2.2.6) does not depend on A2 within a first-order approximation.
2.2.
349
REFRACTIVE BEHAVIOR O F FLUIDS
From the preceding it follows that a gas is described by a linear relationship between E and P. Due to this linearity, the refractive index of a gas mixture composed of several constituents is given by
where the K f are the Gladstone-Dale constants, and p f , the partial densities of the individual components. If one defines a Gladstone-Dale constant K of the mixture by n - 1 = Kp, p being the density of the mixture, one has
(2.2.7) where ( p f / p ) designates the mass fractions of the components in the gas mixture. The Gladstone-Dale constants of several gases are listed in Table I. A dissociated gas can be considered as a mixture of neutral molecules, atoms, and-eventually-molecular residues, if the neutral gas molecule consists of more than two atoms. In the case of a diatomic gas, e.g., pure oxygen, the coefficient of dissociation aD is the mass fraction of the atoms in the gas mixture. With all the previous assumptions and neglecting the effects of excited states in the mixture, the Gladstone-Dale relation for the dissociated biatomic gas is
(2.2.8) where KM and K A are the Gladstone-Dale constants of the neutral or molecule gas and of the atom gas, respectively. The Gladstone-Dale constant of the mixture is a function of temperature, since aDis known to depend on the temperature. Values of KA for oxygen and nitrogen have TABLEI. Gladstone-Dale Constants of Different Gases Gas
K [cm”gl
Wavelength (pm)
Temperature (K)
Air Air Air Air Air Air
0.2239 0.2250 0.2259 0.2274 0.2304 0.2330
0.9125 0.7034 0.6074 0.5097 0.4079 0.3562
288 288 288 288 288 288
O*
0.190
N:
0.238 0.1% 0.229
0.589 0.589 0.589 0.589
273 273 273 273
He CO:
350
2.
DENSITY SENSITIVE FLOW VISUALIZATION
been determined by Alpher and WhiteSand by Anderson,s and are found to be independent of the values of aD. The ratio KM/KA for oxygen is nearly one at intermediate wavelengths. In performing interferometric studies of air flows, one may therefore disregard the effect of dissociation up to at least 5000 K and use the Gladstone-Dale constant of neutral air; for oxygen has a large degree of dissociation at these temperatures, but its value of KM/KA is nearly one, whereas nitrogen, although with a ratio KM/KA much different from one, has an almost negligible degree of dissociation. The same procedure can be applied to determine the Gladstone -Dale constant of an ionized gas. If we consider a monatomic gas, the ionized gas mixture consists of neutral atoms, ions, and free electrons. The degree of ionization aIis the mass fraction of ions in the mixture, and one may neglect the mass fraction of the electrons. The Gladstone-Dale relation of the ionized gas is therefore written (2.2.9) where KA and K I are the Gladstone-Dale constants of the atom gas and of the ion gas, respectively; N , is the electron number density, i.e., the number of the electrons per unit volume; and the Gladstone-Dale constant of the electron gas, KL, has dimensions different from those of KA and K I . According to Alpher and White,7 one has for argon K I = 8KA. From plasma theory it follows for the refractive index n, of the electron gas
n, - 1 = (A2N,e2)/(2rrrn,c2),
(2.2.10)
with e being the charge and M, the mass of an electron; c, the velocity of light in vacuum; and A, the wavelength of the light which is transmitted through the gas mixture. If one measures A in centimeters, one may also write n,
-1
= -4.46 x
10-14
AZN,,
and it becomes evident that the Gladstone-Dale constant of the electron gas, KL,is negative and strongly dispersive. If one compares, at a given wavelength, the Gladstone-Dale relation for the electron gas with that of the neutral atom gas, one realizes that the specific refractivity (n - 1) per electron is greater than the specific refractivity per atom by more than one order of magnitude. The optical behavR. A. Alpher and D. R. White, Phys. Fluids 2, 153 (1959). H. B. Anderson, Phys. Fluids 12, Suppl., 1-57 (1%9). ' R. A . Alpher and D. R. White, Phys. Fluids 2, 161 (1959).
* J.
2.2.
35 1
REFRACTIVE BEHAVIOR O F FLUIDS
FIG.1. Refractive index of a gas, n , with anomalous dispersion close to a resonant frequency of the gas, Y(.
1
1
V
V,
ior of an ionized gas is therefore dominated by the presence of free electrons, even at a low ionization level. Since the Gladstone-Dale constant of the electron gas is negative, the optical response to density changes, e.g., in the form of the fringe shift in an interferogram, is opposite to the response recorded in the neutral gas. This is of some importance in visualizing a strong shock wave, where the production of free electrons and the increase in neutral density cause fringe shifts of opposite direction, in the worst case actually canceling one another. The preceding analysis of the refractive behavior of gases does not account for the range of frequencies v near an absorption frequency viof the gas. In this case the refractive index n becomes complex, and the imaginary part of n describes the absorption of the transmitted light in the gas. The exact Gladstone-Dale relation in this frequency range must be found from the quantum mechanical solution for the refractive index.8 Close to a resonant wavelength, the gas exhibits anomalous dispersive behavior with (n - 1) being many orders of magnitude greater than in the nonresonant frequency regime (Fig. 1). Since, in addition, the value of the Gladstone-Dale constant is much greater in the resonant regime, the same absolute density change Ap here will produce much stronger alterations in the refractive index of the gas, An, than in the nonresonant case. Bershader8 therefore proposed to improve the sensitivity of optical visualization methods by illuminating a lowdensity gas flow with light of frequency v close to a resonant value u i . A technique appropriate for measuring the gas density in a frequency range close to a resonant value vf is the hook r n e r h ~ dwhich ,~ combines an optical interferometer with a spectrograph in order to record spectral interferograms. After having passed through the test field, the light beam
* D. Bershader, in “Modern Optical Methods in Gas Dynamic Research” (D. S . Dosanjh, ed.), p. 65. Plenum, New York, 1971. M . C. E. Huber, in “Modem Optical Methods in Gas Dynamic Research” (D. S. Dosanjh, ed.), p. 85. Plenum, New York, 1971. @
352
2.
DENSITY SENSITIVE FLOW VISUALIZATION
A ,
I
1, A 2
FIG. 2. Schematic representation of a hook interferogram with a resonant wavelength A, and hooks at positions Al and A*.
which usually forms the interferogram is focused onto the entrance slit of a spectrograph. The interference fringes appear superimposed on the spectrum. In the most often used version of the apparatus, the recorded spectrogram shows parallel oblique fringes. Near the wavelengths where the test gas has absorption lines, the fringes exhibit a hook on each side of the line, and the fringe pattern in between the hooks corresponds to the anomalous dispersion near the absorption line (Fig. 2). The formation of a hook is described by b d(n - l)/dA
=
KH,
(2.2.11)
where b is the width of the flow test section in the direction of the transmitted light; n, the refractive index; and A, the light wavelength. The hook consfanf KHdepends only on the data of the apparatus and must be determined by an appropriate calibration. By inserting the quantum mechanical expression for n into (2.2.1 l), one obtains a quadratic equation in A; the two roots of this equation, Al and A2, describe the position of the hooks in the spectral interferogram. Conversely, by measuring X I , A2, and the resonant wavelength A,, and using values of K H and 6 , one obtains an expression for determining the product ( N h ) , where fr is the oscillator strength; and N 1 ,the number density of the gas atoms or molecules in the lower state of the respective electron transition. With a catalogue of fi values available, the hook method is an appropriate tool for determining densities in high temperature gases. 2.2.2. Deflection and Retardation of Light in a Density Field
After having established that a compressible flow field represents, in optical terms, a phase object with variable index of refraction, the problem now is to investigate the question of how the light transmitted through the flow field is affected by the refractive index variations. The refractive index n is considered to vary as a function of the three spatial coordinates x, y, and z in the compressible flow field, i.e., n = n(x, y, z). An incident light beam initially parallel to the z direction is transmitted through the flow. In a practical arrangement, the flow facility will have viewing windows which are normal to the z direction. The propagation of a single light ray in the phase object is described by Fermat’s principle,
2.2.
REFRACTIVE BEHAVIOR OF FLUIDS
353
which states that the variation of optical path length along a light ray in the object must vanish; hence (2.2.12)
6 / n ( x , y, z) ds = 0,
where s denotes the arc length along the ray, and ds is defined by ds2 = dx2 + dy2 + dz2. As shown by Wey1,’O Eq. (2.2.12) is equivalent to the following set of differential equations: (&x/dz2)
=
[l
+ (dx/dzI2 + (dy/dz)2][(an/ax)
-
(dY/dz)(an/az)l(l/n)
The problem is to determine a solution of this system, x = x(z) and y = y(z), which describes the path of the light ray through the refractive index field. The solution of the system (2.2.13) must be found for a certain initial condition which specifies the particular light ray in the transmitted, parallel beam. This initial condition is given by specifying the coordinates x = and y = -ql, where the ray enters the test volume. Having found the particular solution of (2.2.13), one then determines the coordinates t2and q2of the ray in the exit plane of the flow test volume as well as the respective inclinations of the ray at the exit point. In an optical arrangement, one usually has a recording or photographic plane at a distance 1 from the exit plane of the test volume. In order to predict the observable pattern in the recording plane one has to determine, for each ray, the following of a deflected or three quantities (see Fig. 3): (1) The displacement disturbed ray with respect to an undisturbed ray, i.e., a ray which has passed through a homogeneous field; (2) the deflection angles eZ and ey of the ray at the end of the test volume; (3) the retardation of the disturbed ray with respect to an undisturbed ray. The latter quantity can be expressed by the time difference Ar between the arrival of the two rays in the recording plane. It will be shown later that each of these three quantities corresponds to a particular class of optical visualization methods. The form of Eqs. (2.2.13) is too complicated to permit an overall perspective. It is helpful to simplify the system by making the following assumptions: It is a fact of experience that the deviations from the z direction of a light ray in a compressible gas flow are negligibly small, but that
m*
lo F. J . Weyl, in “Physical Measurements in Gas Dynamics and Combustion” (R.W. Ladenburg, ed.), p. 3. Princeton Univ. Press, Princeton, New Jersey, 1954.
354
2.
DENSITY SENSITIVE FLOW VISUALIZATION
I
recording plane
FIG.3. Deflection of a light ray in an inhomogeneous test object (density field): geometric relations.
the ray may leave the test field with a nonnegligible curvature. Hence, one may assume that the slopes of the ray, dx/dz and dyldz, are very small everywhere as compared with unity; and since in most cases anlax, anlay, and an/& are of the same order of magnitude, the simplified system of equations reads d2x/dz2 = (l/n)(an/dx)
fPy/dzZ = (l/n)(an/ay).
(2.2.14)
Due to the simplifications introduced, the light ray now leaves the test field at the same coordinates 6, 7 where it enters the field, but at an angle which is determined by the line integral of the derivative of the refractive index distribution in the test field. With the aid of the reduced set of equations (2.2.14), one may determine the three aforementioned observable quantities. Expressing the displacement QQ* and the deflection angle E in terms of the respective x and y components, one obtains (see Fig. 3):
(rn*)=1 I"(l/n)(dn/dx) d z , =
i1
(2.2.15a)
(2.2.15b)
At = (l/c)
1"
[n(x, y, z) - n,] dz.
tl
(2.2.15~)
2.2.
REFRACTIVE BEHAVIOR OF FLUIDS
355
In (2.2.15) c is the velocity of light in vacuum; n,, the refractive index of the undisturbed test field in which the reference ray propagates; and and {2 are the z coordinates of the points where a ray enters and leaves the test field. The recording plane is a distance 1 from the exit plane of the test field, and it has been assumed that the gas in the regime between the flow facility and the recording plane has uniform optical behavior. The quantity Ar can be converted into the optical phase difference A p between the disturbed and the undisturbed ray in the recording plane, and one has 4 / 2 =~ ( l / A )
1" LI
[ n ( x , y , Z) -
nm]
dz,
(2.2.16)
where A is the wavelength of the light used. The following sections deal with different optical methods applied to the visualization of compressible flows. The shadowgraph is a technique which visualizes the displacement QQ* as represented by Eq. (2.2.15a). The schlieren system measures the deflection angle E described by (2.2.15b). Optical phase changes experienced by a light ray in the compressible field according to (2.2.16) can be made visible with optical interferometers. As will be shown below, these different classes of visualization method exhibit another systematic behavior: the shadowgraph is sensitive to changes in the second derivative of the gas density p , the schlieren system visualizes changes in the first derivative of p , and with interferometers one may measure absolute density changes. The final results, (2.2.15) and (2.2.16), have been derived from the simplified equations (2.2.14). There are cases where the simplifying assumption that the individual rays follow straight line paths through the compressible flow cannot be applied. An example which requires the solution of the complete set of Eqs. (2.2.13) is the investigation of the turbulent density fluctuations in a compressible gas flow. Taylor" shows that the above simplifying assumption is equivalent to assuming that the region of turbulent flow is thin compared to the distance from the test field to the recording plane. Since this assumption often is not in accord with practice, a complete treatment of the interaction of a plane electromagnetic wave with a turbulent dielectric field is necessary. The assumption might also be violated if one visualizes density variations in a fluid with a higher value of the refractive index than that of gases. A typical example is a stratified fluid made of a solution of salt in water, where the concentration of the salt vanes as a function of space. Empirical relationships are available which describe the refractive index of the fluid as a function of its density. From such relations analogous to the Gladstone-Dale for'I
L. S. Taylor, AIAA J . 8, 1284 (1970).
356
2.
DENSITY SENSITIVE FLOW VISUALIZATION
mula for gases one derives that for the same relative change in density, the variations of refractive index are of the order of 1000 times greater for salt solutions than for air. It is then not permitted to neglect the first derivatives in (2.2.13). Mowbray12has investigated this problem for the special case that the density or the concentration in the fluid depends on the y coordinate only. The system (2.2.13) then reduces to the following form: dLx/dz2 = 0 dLy/dz2 = [l
+ (dx/dz)' + ( d y / d ~ ) ~ ] ( l / n ) ( a n / a y ) .
(2.2.17)
In a practical system, the rays of the incident light beam cannot all be regarded as exactly parallel to the z axis, due to the finite extension of the light source. Mowbray12 has therefore integrated the system (2.2.17) with the initial condition that the rays have a slight inclination x ' ( { ~ )~, ' ( 5 ~ ) when entering the fluid (the prime denotes a differentiation with respect to z). Owing to the relatively large variations of n in the fluid, even small values of the initial inclination will result in a noticeable aberration of the ray from a straight line.
2.3. Visualization by Means of Light Deflection 2.3.1. Shadowgraph Method In its simplest form, the shadowgraph does not need any optical component. The diverging light from a point-shaped light source is transmitted through the flow test field, and the shadow pattern produced by the phase object will be recorded in a vertical plane placed a distance I behind the test field. Instead of this simplest arrangement, a system will be regarded here with parallel light through the test field, which is bounded by plane viewing windows normal to the light (Fig. 4). In order to avoid the use of too great a photographic plate, the recording plane can be focused by means of a camera lens onto a film or plate of reduced size. The contour of a rigid object in the test field appears unfocused on the shadow picture. The sharpness of such objects increases with decreasing diameter of the light source. When passing through the test field under investigation, the individual light rays are refracted and bent out of their original path. A particular deflected light ray should be traced which arrives at a point Q* of the recording plane instead of a point Q (Fig. 3), so that the distribution of light D. E. Mowbray, J . Fluid Mech. 27,
595 (1967).
2.3.
VISUALIZATION BY MEANS OF LIGHT DEFLECTION test facility with viewing windows
357
reference or recordinq plane
J
\
,
I I
I I
I
-
1
'
c
_
FIG.4. Shadowgraph system with parallel light through the test field.
intensity in that plane is altered with respect to the undisturbed case, e.g., Q* receives more light than before, while no light arrives at Q , which can be regarded as the shadow of the respective object point. Assume that I ( x , y ) denotes the light intensity distribution in the recording plane for the undisturbed case, whereas I* (x*, y*) is the intensity for the disturbed case. The intensity I* at point (x*, y*) results from all intensities Zirelated to points ( x i , yi) which are mapped into (x*, y*). In determining the resulting intensity I*one has to take into account that the area illuminated by a particular light beam is deformed due to the mapping of the (n,y) plane into the (x*, y*) plane. The intensity per unit area thereby changes. Each intensity value Ii contributing to the summation has to be divided by a value which accounts for this mapping, and the denominator of this expression is the Jacobian of the mapping function of the system (x, y) into (x*, y*). The intensity resulting in a point (x*, y*) is therefore given by I*(x*, Y*) =
2
[mi,Yr)/la(x*,
Y*)/a(x, Y)ll.
(2.3.1)
i
Assume that only one point (x, y) is mapped into (x*, y*), and that the new coordinates are given by x* = x
+ Ax(x, y),
y* = y
+ Ay(x, y),
where Ax and Ay describe the displacement QX*determined by (2.2.15a). A linearization of the mapping function is introduced by assuming that Ax and Ay are small quantities, and that products and higher powers of Ax and Ay can be neglected. The Jacobian is then la(x*, y*)/a(x, y)I = 1
+ (dAx/ax) + ( ~ A Y / ~ Y ) .
A photographic plate is sensitive to relative changes of the light intensity, which is given in this case by (I*- Z)/I = AI/Z. With the results of the above linearization and Eqs. (2.3.1) and (2.2.15a), one obtains
AI/Z = I
[(az/axz
+ aZ/ayz)(ln n)ldz.
(2.3.2)
358
2.
DENSITY SENSITIVE FLOW VISUALIZATION
By applying the Gladstone-Dale formula (2.2.5a) it becomes evident that the shadowgraph is sensitive to changes in the second derivative of the gas density. From the latter result it follows that the shadowgraph is not a method suitable for quantitative measurements of the gas density, since such an evaluation would require one to perform a double integration of the data field. The errors introduced by such an integration will usually be too large. Furthermore, several simplifications contained in the above analysis might be violated. Owing to its simplicity, however, the shadowgraph remains a convenient method for obtaining a quick survey of a compressible flow pattern. In particular, with the shadowgraph one visualizes easily the geometry of pressure and expansion waves in a supersonic gas stream (Fig. 9, and the turbulent structure of a compressible jet or wake flow (Fig. 6). The trace of the bow shock wave of an axisymmetric body flying at
FIG.5. Shadowgraph of a sphere flying at a Mach number M = 1.7. [Courtesy of Dr. A. Stilp, Emst-Mach-Institut, Freiburg, Germany.]
2.3.
VISUALIZATION BY MEANS OF LIGHT DEFLECTION
359
FIG.6. Shadowgraph of a jet of propane gas exhausting from a nozzle into still air.
supersonic velocity is a dark band bounded on the downstream side by an edge of intense brightness. The incident light is assumed to be a parallel beam oriented normal to the flight axis. The shock-compressed gas region affects the light similarly to the action of a convex lens (Fig. 7). The outer edge of the dark band is therefore the exact position of the shock front in the image, whereas the deflected light may form a caustic which gives rise to the light concentration on the inside edge of the shock. It is possible to relate the observed width of the dark band to the strength of the shock. A similar caustic is generated by an expansion wave in a supersonic flow behind a corner known as a Prandtl-Meyer expansion
3 60
2.
DENSITY SENSITIVE FLOW VISUALIZATION flight direction incident
t
recording plane
FIG. 7. Deflection of light as caused by the shock compressed gas ahead of a sphere flying at supersonic speed.
fan. The exact position of the first expansion wave of the fan is the inner (downstream) side of the bright zone. No completely obscured zone appears in this case, in contrast to the shadow image of the shock wave. A general rule is therefore that a shock wave appears in a shadowgraph in the form of a dark band, whereas the head of an expansion fan, as caused in a supersonic flow by a sharp-edged rearward facing step of the wall contour, will be visualized by a bright band, albeit with a much smaller degree of contrast than the dark shock band. The latter result also applies to shocks and expansion waves occurring in a two-dimensional flow, and with the incident light normal to the flow direction. The incident light is then parallel to the plane of the shock or parallel to the plane of the first expansion wave, and with the laws of geometrical optics one cannot account at all for the formation of a shadow image of these waves, if one regards them as discontinuity surfaces. The formation of a shadow image in these cases is due to the following effects: (1) The light is never actually exactly parallel to the discontinuity surface; (2) the shock is of finite thickness; and (3) the major contribution to the formation of the shadow results from light diffraction at the discontinuity surface. For a plane shock the zeroth diffraction order is a rninim~m,'~ and with white light illumination only this minimum appears in the shadowgraph to form the observed dark band. A further application of the shadowgraph is the visualization of the compressible turbulent flow pattern of free jets and wakes. Shadowgraphs made with short-duration light pulses freeze on the photographic film an instantaneous turbulent pattern with a scale of details much finer than that which the hot-wire technique can resolve. Such a photograph can be evaluated to derive certain statistical properties of the threedimensional turbulent flow .14 Is
H. J. Heifer, H. D. vom Stein, and B. Koch, Pror. Int. Congr. High-speed Photogr.,
9th, 1970 p. 423 (1970). 1'
M. S . Uberoi and L. S. G. Kovasznay, J . Appl. Phys. 26, 19 (1955).
2.3.
VISUALIZATION B Y MEANS OF LIGHT DEFLECTION
361
2.3.2. Schlieren Systems
Schlieren systems serve to measure the amount of light deflection generated by a transparent optical phase object. In the fundamental arrangement, mostly referred to as the Toepler system, a parallel light beam traverses the test object and is focused thereafter by means of a lens or spherical mirror, named the schlieren head (Fig. 8). An image of the light source is formed in the focal point of the schlieren head. A knife edge (the edge being perpendicular to the plane of Fig. 8) is placed in the plane of the light source image to cut off part of the transmitted light. The camera objective focuses the test object onto the recording plane, where one receives a reduced intensity of light, depending on the amount of light cut off by the knife edge. As shown below, an optical disturbance in the test object will produce variations of the recorded light intensity which are a measure of the deflection experienced by the light in the test object. This Toepler schlieren system has found a great variety of modifications, as described in several comprehensive review paper^.'-^.'^-'' 2.3.2.1. Toepler Schlieren System.* The Toepler system, which is representative of all schlieren methods, will now be evaluated quantitatively. The knife edge is positioned to cut off part of the light source image (Fig. 9). The height of that part of the light source image which is not obscured by the knife edge is a , and b designates the width of the light source image in the direction of the knife edge. With a homogeneous test object, the light intensity I arriving at any point (x, y ) of the recording plane is the same, i.e., the recording plane receives uniform illumination which can be described by I ( x , Y ) = const(ab/E),
(2.3.3)
wheref, is the focal length of the camera objective in Fig. 8. The constant factor in Eq. (2.3.3) depends on the original luminance of the light source and on the overall light absorption in the test object. Equation (2.3.3) holds only if geometrical aberrations of the optical system, e.g., coma and astigmatism, can be neglected. With a number of light rays deflected in the disturbed test field by an angle E , the light source image formed by these rays in the plane of the H. Schardin, Ergeb. Exakten Naturwiss. 20, 303 (1942). J . W. Beams, in “Physical Measurrments in Gas Dynamics and Combustion” (R. W. Ladenburg, ed.), p. 26. Princeton Univ. Press, Princeton, New Jersey, 1954. H. Wolter, in “Handbuch der Physik” (S. Fliigge, ed.), Vol. 24, p. 555. SpnngerVerlag, Berlin and New York, 1956. Is
* See also Volume 1 , Section 4.2.3.4.
2.
362
DENSITY SENSITIVE FLOW VISUALIZATION
1
,
knife edge will be shifted by distances Au and Ab normal and parallel to the edge. It is only necessary to consider the vertical shift Au generated by the vertical component E~ of the deflection angle. With small deflection angles, one has
Au
=fi tan ey = E&,
(2.3.4)
where.& is the focal length of the schlieren head. The light intensity in the respective image points (x, y ) of the recording plane is then changed by hl = const(bAu/fi), with the constant factor being the same as in (2.3.3). In observing the recorded schlieren image, the eye discriminates relative intensity changes rather than absolute values. Therefore, with (2.3.3) and (2.3.4), the relative intensity change AZ/I produced by the disturbed test object in a point (x, y) of the recording plane is given by &/I = (Au/u) = ~ u ( f i / ~ ) ,
(2.3.5)
if the knife edge is parallel to the x direction. From Fig. 9 it becomes clear that the shift Ab does not affect the illumination in the recording plane. By inserting (2.2.15b) into ( 2 . 3 3 , one obtains (2.3.6a) The schlieren image can therefore be evaluated to obtain the gradient of the refractive index in the test field. In this case, one measures the gra-
Ab
disturbed image of source slit
-
..-ILo
--I , undisturbed image
knifeedge-
of
FIG. 9. Shift of slit-shaped light source image in the plane of the schlieren knife edge.
2.3.
VISUALIZATION B Y MEANS O F LIGHT DEFLECTION
363
dient in the y direction. Turning the knife edge by 90" delivers the gradient in the x direction. For many gases it is permissible to put (l/n) = 1. Using the Gladstone-Dale relation (2.2.5b), Eq. (2.3.6) then reduces to AZ/I = ( K f i / a )
1"
( d p / d y ) dz.
(2.3.6b)
ZI
This form shows that the schlieren method is sensitive to changes of the density gradient in the compressible flow field. The relative change of light intensity in the recording plane, AZ/Z, is equivalent to the photographic contrast of the recorded schlieren picture. If one supposes that a relative intensity change of 10% is still detectable, the smallest deflection angle which can be measured in such a schlieren system is E,," = O.l(a/fi). Since there is seldom any possibility of altering the focal lengthf,, one concludes that the sensitivity of the schlieren method increases with decreasing value of the aperture a. However, the possible reduction of a and the accompanying increase in schlieren sensitivity are limited for several reasons. The emulsion speed is one factor which determines the lower limit of a for a given exposure time. A second limitation results from the demand that the system should be capable of measuring light deflections for both positive and negative values of E . In the arrangement of Fig. 8, positive E generates an increase of the light intensity; negative E , an intensity loss. The greatest possible intensity change for negative E is absolute extinction. If E,,, designates the greatest deflection angle in a given density field, an optimum situation would be achieved, with A f = - I for E = - E,,,, and from this it follows that the smallest possible value of a is a m i n 2 &Emax. The most severe restriction on reducing the aperture a results from the influence of diffraction. These effects have been investigated extensively by SchardinE5and Suchorukich.'* Due to diffraction, the light source image in the plane of the knife edge is to a certain degree out of focus, and the intensity change observed in the recording plane is not accurately described by Eq. (2.3.6). Diffraction is not included in the theory of geometric optics which has been used above to describe the schlieren principle, and its influence on the formation of the schlieren image must be studied independently. Such an investigation can only refer to a particular test object of given shape and density distribution. Schardin,I5 e.g., investigated the diffractive disturbance of the schlieren image of a circular regime with constant deflection angle inside the circle and with a constant density field outside. If one tolerates that the disturbance is not properly imaged in the recording plane, and if the sole requirement is to generate W.S . Suchorukich, Proc. Int. Congr. High-speed Photogr., 8th, 1968 p. 341 (1%8).
3 64
2.
DENSITY SENSITIVE FLOW VISUALIZATION
the correct contrast AZ/Z at certain points of the image, one derives a minimum value of a which, according to Schardin, is given for this particular case by amln2 (f,A)/2d,where d is the diameter of the circular disturbance. The image of the circular disturbance is then out of focus to an extension twice the size of the undisturbed image. The effects of diffraction can be included directly in the schlieren analysis, if the light modulation in a schlieren system is described by means of Fourier transforms instead of being represented by geometric optics relations. According to Veret,lg the field strength of the undisturbed light which traverses the test object is represented by s = A exp(iot), where A is the real amplitude; o,the circular frequency; and r , the time coordinate. Each light ray propagating at a positio 1 ( x , y) parallel to the z axis experiences a phase shift cp(x, y) in the test field. Behind the test object, the disturbed light is given by s = A exp(i[wt
+ cp(x, y)]} = A exp(icp) exp(iot).
(2.3.7)
The term A exp(icp) can be regarded as the complex amplitude of the disturbed wave. If the schlieren system is operated close to its sensitivity limit, the phase shift cp is small, and the complex amplitude can.be approximated by
b(x, y) = A(cos cp
+ i sin cp)
= A[1
+ icp(x, y)].
In the plane of the schlieren knife edge, the complex amplitude can be described by the Fourier transform of b(x, y):
F [ b ( x , y)] = A[W', Y') + i W ' , ~ ' 1 1 , with x', y' being the coordinates in this plane. The Dirac function S(x', y') represents the undisturbed image of the light source, while t#(x', y') is the Fourier transform of the phase information cp(x, y). Since the light is attenuated by the action of the knife edge, the complex amplitude of the light behind the knife edge can be written b'W,
YO = A [ q Wx', y') + i4(x', y')J,
where the attenuating effect of the knife edge is described by the transmission factor q. In the recording plane, the complex amplitude b"(x",y") can again be found from a Fourier transformation of b'(x', y'). If one assumes a scale factor of one between object and image plane, one has x" = x, y" = y, and
b"(x,y) = F[b'(x', y')] = A [ q + icp(x, y)]. I@ C.
Veret, AGARD Con$ Proc. 38, 257 (1970).
2.3.
VISUALIZATION B Y MEANS OF LIGHT DEFLECTION
365
The intensity distribution in the recording plane is given by and the relative intensity change or contrast of the schlieren picture is
Al/lv=o =
~p'/$.
(2.3.8a)
This equation is equivalent to (2.3.6). The sensitivity or the contrast is seen to become infinite if the transmission q = 0, which is the same as putting a = 0. Diffraction has not yet been included in (2.3.8). Since diffraction causes ,the light source image to be distorted, this image is no longer represented by a Dirac function. A certain amount of diffusive light may reach the recording plane and contribute to the background intensity I,,, of the schlieren picture. The diffusive light contributions can be included in the form of an additive term p, so that the contrast of the schlieren picture is now described by (2.3.8b) and the maximum contrast cannot become infinite, even for 7 = 0. In
FIG. 10. Schlieren photograph of the flow field around a sphere flying at hypersonic speed. [Courtesy of Dr. A. Stilp, Ernst-Mach-Institut, Freiburg, Germany.]
366
2.
DENSITY SENSITIVE FLOW VISUALIZATION
order to find a sensitivity limit, VeretIBproposes a value of Pmin= 5% for a well aligned system. This allows one to determine a smallest detectable phase change &, y) as a measure of the sensitivity. This result is somewhat contradictory, since one knows from (2.3.6) that the schlieren method responds to changes of the density gradient and not to absolute density changes. These two results must be interpreted in such a way that (2.3.6) describes the form of the disturbance necessary for being resolved with the schlieren system, whereas from (2.3.8b) one derives the smallest amplitude of the disturbance which can just be detected. The intensity variations recorded in a schlieren photograph (Fig. 10) can be evaluated quantitatively in order to obtain the gradient of the gas density. A further integration would yield the density distribution in the flow field. However, the measurement of the photographic contrast or of the relative intensity changes A I / I is not very precise, even if one uses a densitometer, and the determination of the gas density from schlieren photographs is associated with severe errors. The schlieren method therefore is mainly used for qualitative purposes, but with a much higher degree of resolution than the shadowgraph. Flow regimes with compression and expansion can be recognized by their opposite intensity variations (Fig. 10). If the flow field is axisymmetric with the axis of symmetry normal to the incident light beam, the same density change appears with opposite change in light intensity on either side of the axis, e.g., the front shock wave in Fig. 10 is dark in the upper part of the picture and it appears with a bright edge in the lower part, i.e., below the flight axis, which here is the axis of symmetry. 2.3.2.2. Modifications of the Toepler System. A great variety of modifications of the Toepler schlieren system has been described for various applications. These modifications involve the general optical setup, changes of the form and the function of the knife edge, the introduction of color, and the combination of the schlieren system with holographic methods. If MI and Mz in the arrangement of Fig. 8 are spherical mirrors instead of lenses, the optical ray system has to be folded, and as a consequence, the light source and its image in the plane of the light source are positioned “off-axis,’’ where the axis is defined as the straight line connecting the center of the mirror and its center of curvature. Two optical aberrations, coma and astigmatism, are thereby introduced into the system, resulting in an uneven illumination of the recording plane, even in the presence of a homogeneous test field. The coma can be compensated by using a 2-shaped arrangement (Fig. 1l), and the astigmatism can be reduced by means of an appropriate set of planocylindrical lenses. A general rule is that these aberrations decrease with decreasing angle of tilt of the spherical mirrors in Fig. 11.
2.3.
VISUALIZATION B Y M E A N S OF LIGHT DEFLECTION
367
light source
S
FIG. 1 1 . Z-shaped schlieren arrangement with two spherical mirrors.
The double-pass schlieren system is applied to increase the schlieren sensitivity. A conical light beam traverses the test field twice, and with an ideal alignment, the result is to double the effective width of the test field (Fig. 12). The divergent-convergent light beam, including the spherical mirror, can be kept entirely in an evacuated tube to protect the system from thermal turbulences in the laboratory room. Several modifications of the schlieren knife edge have been investigated by Stolzenburg.20 The double knife edge is of the same shape and size as the light source image in the focal point of the schlieren head, e.g., if the light source is slit shaped, this knife edge can be a wire, and the result is that positive and negative values of the deflection angle both appear with an increased intensity in the schlieren image. With this arrangement one avoids complete obscuration of certain parts of the recording plane, as would occur with a conventional knife edge for large values of negative deflection angle. By using a light source of circular shape and an equivalent circular knife edge, one may visualize light deflections in any direction with the same contrast. Of course, one loses with the double knife edge the ability to discriminate positive and negative values of E . The latter ability of the schlieren system can be recovered by the introduction of color. If one provides two transparent strips of different colors, one above and one below the aforementioned wire (double knife edge), the regions of positive and negative E in the schlieren image appear with different colors. An additional advantage is that the eye is more sensitive to changes in color than to changes in shades of gray. In many practical arrangements, the knife edge is replaced by a tricolor filter consisting of three colored strips which are parallel to the original knife edge direction." The width of the central transparent filter (e.g., blue) is equal to the width of the image of the light source, which must now be a slit. The colors of the two other filters are chosen to provide a maximum in contrast, e.g., red and yellow. Other color schlieren systems employ an u,
W. A. Stolzenburg, J . SMPTE 74, 654 (1965). Kessler and W. G. Hill, Aeronaut. Astronaut. 4, 38 (1966).
** T. J .
368
2.
DENSITY SENSITIVE FLOW VISUALIZATION
camera lens
object knife edge
be'am splitter light source
mirror
FIG. 12. Double-pass schlieren system using a semireflecting mirror as a beam splitter.
optical dispersing element in front of a white light source, e.g., a dispersion prism22or a diffraction grating.23 A color spectrum is thus displayed in the focal plane of the schlieren head. The knife edge is replaced by a narrow slit which is adjusted to pass only one color, usually yellow, in the undisturbed case. With a variable density gradient in the test field, different colors of light reach the slit. The sequence of the displayed colors is that of the neutral spectrum, and one has no opportunity to select a combination with high contrast as with the tricolor filter. The circular knife edge system, too, has its equivalent among the color methods. A combination of a multicolored source filter in front of a white light source and an appropriate aperture in the focal point of the schlieren head allows one to discriminate different spatial directions of the deflection angle .24 A fundamental problem of all optical visualization systems is determining the three-dimensional density distribution p(x, y, z) from the two-dimensional flow picture. A method for obtaining such threedimensional information is the sharp-focusing schlieren system developed by Kantrowitz and T r i m ~ i .The ~ ~ system enables one to relate the observed schlieren pattern to the density changes in a plane slice of the test volume perpendicular to the incident direction. With an appropriate experimental procedure, one can change the position of the. test slice and thereby the test field in the z direction. The test field must not change its density distribution during the scanning process. Therefore, holography is an important aid for this method, since in a phase hologram one may freeze the instantaneous information of the refractive index field, which can later be reconstructed; unlimited time is then available for the investigation of the reconstructed density field.26 It is obvious that the combination of holography with the schlieren method is helpful also for normal schlieren observations. Figure 13 shows the principal arrangement for taking a hologram of the phase object and for the postponed schlieren ob-
** D. W. Holder and R . J.
*'
North,Nature (London) 169, 466 (1952). A. R. Maddox and R. C. Binder, Appl. Opt. 10,474 (1971). G . S. Settles, AIAA J . 8, 2282 (1970). A. Kantrowitz and R. L. Trimpi, J . Aerosp. Sci. 17, 311 (1950).
R . D. Buzzard, Proc. Int. Congr. High-speed Photogr., 8th. 1968 p. 335 (1969).
2.3.
VISUALIZATION BY MEANS OF LIGHT DEFLECTION
369
laser (a)
object
laser
FIG.13. (a) Arrangement for taking a hologram of a compressible flow field; and (b) postponed visualization of the reconstructed flow field in a schlieren system.
servation of the reconstructed object field.27 The knife edge can now be aligned carefully to obtain a maximum of contrast and sensitivity. The use of laser light, however, reveals diffractive disturbances of the light source image in the plane of the knife edge. In an unsteady flow, the light deflecting object might move across the field of view with a velocity which is apriori unknown. According to the Doppler principle, the light deflected from a moving phase object experiences a frequency shift, which can be related to the object velocity. Schwar and WeinbergZ8have therefore combined a schlieren system with the laser Doppler principle, and this combined system has successfully been applied to measure the velocity of flame frontsz8 and of unsteady shock waves in a shock tubezsfrom the detected Doppler frequency. The frequency signal also carries information on the phase distribution in the moving object; no attempt has yet been made to evaluate this additional information. 2.3.3. Fringe Distortion Methods
The quantitative evaluation of schlieren photographs by means of densitometry is mostly connected with severe error sources. A procedure more appropriate for quantitative evaluation is to measure the distortion
*'
J. D. Trolinger, AGARDOgraph 186 (1974). M. J. R. Schwar and F. J. Weinberg, Pruc. R . Soc. London, Ser. A 311,469 (1%9). 29 W. Merzkirch and W. Erdmann, Appl. Phys. 4, 363 (1974). xa
370
2.
DENSITY SENSITIVE FLOW VlSUALlZATlON
of well-defined fringes, as done with interferometric techniques. Many attempts have been made to modify the shadow as well as the schlieren method to obtain flow pictures with a superimposed network of fringes, whose distortion is a measure of the light deflection in the test object. Weinberg2 therefore has called these techniques dejection mapping methods; they are also referred to often as Ronchi methods.30 The color schlieren system using a tricolor filterz1is a simple form of such a deflection mapping system. Each line separating two colors in the recorded picture is a curve of constant deflection angle E , , if the edges of the filter are parallel to the x direction. Instead of the filter, one may use a grid of alternate transparent and opaque strips of equal width. The spacing of the strips is chosen so that the width of the light source image is equal to or less than one strip width. With a uniform test object, the recording plane appears evenly illuminated. If the test object deflects the transmitted light by small angles only, the system works like a double knife edge arrangement.20 With stronger density gradients in the test field, the rays might be deflected in the plane of the grid by more than one grid period, so that dark and bright fringes appear in the recording plane, each of them a locus of constant range of deflection angles. The precision in determining the deflection angle depends on the focal length of the schlieren head and on the fineness of the strip system. With very narrow strips, however, disturbance due to diffraction increases, and it is advisable not to use a coherent light source. The system described above is characterized by a field of view free of fringes for a uniform test object. When the grid is shifted from the focal point of the schlieren head along the optic axis toward the test section (Fig. 14), an equidistant fringe pattern appears in the recording plane, even for an undisturbed test object. If a disturbance is introduced into the test field, the fringes are no longer straight and parallel. The amount of fringe distortion is proportional to the deviation of the light rays produced by the disturbance. It becomes obvious that the two systems with the grid either in the focal point or out of the focal point of M2 are somehow equivalent to the injinite fringe width and thejinite fringe width alignment of an optical interferometer, particularly if one thinks of shearing interferometers (see Section 2.4.2). If the fringe shift is measured normal to the undisturbed fringe direction, which is assumed to be parallel to the x axis, one may relate the absolute fringe shift As to the deflection angle E, according to Holder and North' by (As), = e,[(fe/g)h+ (g/fi)(l - fill, ao
V. Ronchi, Rev. Opt., Theor. Instrum. 5,436 (1926).
(2.3.9)
2.3.
VISUALIZATION BY MEANS OF LIGHT DEFLECTION object plane
-I
371
schlieren heod
f2
FIG.14. Deflection mapping schlieren system (see text).
where fi and f, are the focal length of the schlieren head and the camera lens, respectively; the camera lens is assumed to be in the focal point of the schlieren head. The distances 1 and g are defined in Fig. 14. The fringe shift does not depend on the spacing of the strips in the grid. With a grid of closely spaced strips one obtains information on the deflection angle at any desired point of the scene. Turning the grid by 90"yields the component eE of the deflection angle. Again it should be noted that the accuracy of the method suffers if one uses coherent laser light, which promotes visibility of diffractive disturbances. The grid or strip system can also be placed on the opposite side of the test object, i.e., into the incident parallel light beam before it enters the test section of the flow facility. One would no longer consider this arrangement to be a modified schlieren system; it is rather a shadowgraph with a superimposed strip pattern (Fig. 15). The strip system is seen through the nonhomogeneous test object and therefore appears distorted. In the recording plane of the simplest shadowgraph system according to Fig. 4, the fringe shift (As), and the deflection angle 8, are related by (ASA, =
(2.3.10)
where 1 is the distance between the test object and the recording plane, and the undisturbed fringes are again assumed to be in the x direction. For the measurement of the deflection angle in arbitrary directions, the linear strip system can be replaced by a more complicated pattern, e.g., a crossed pattern of equally spaced opaque circles or squares, as described by Schardin.15 Other devices have been developed which generate only one single strip across the field of view, and which must be applied to onedimensional test objects, i.e., to a flow field in which the density varies only in one direction. The classic Thovert -Philpot -Svensson crossedslit method (see, e.g., the article by SchardinI5) has been modified by
312
2.
DENSITY SENSITIVE FLOW VISUALIZATION
FIG.15. Candle flame as visualized by a deflection mapping system consisting of a grid of equidistant straight wires.
2.3.
VISUALIZATION BY MEANS OF LIGHT DEFLECTION
373
spheric01 mirror
/
;
of
A < s t i g r n o + i c vertical image -
\
mirror curvature point light source
horizontal image .-camera /
lens
\
FIG. 16. Deflection mapping system described by Knoos.S1
KnOOs31 to obtain a very sensitive deflection mapping system. The system makes use of the astigmatic aberration in an off-axis arrangement containing one spherical mirror (Fig. 16). The astigmatic vertical and horizontal linear images of the point light source are separated by a small distance along the optical axis. An inclined, thin wire is placed in the plane of the first (vertical) light source image, and the shadow of this wire is focused onto the recording plane, where it appears as a black fringe. The refractive light deflections in the test object produce a fringe shift As which, for the case of a one-dimensional density field, is proportional to the deflection angle. The smallest observable fringe shift is of the order of the wire diameter, which must be greater than the thickness of the focal image line. Values of E = rad for the smallest observable deflection angle have been reported. Replacing the single wire by an appropriate grid again yields a parallel fringe system in the field of Another elegant arrangement for producing one single fringe is to place in the focal point of the schlieren head a phase plate which alters the phase of about one half of the light by an angle of Acp = 180°, while the second half of the light passes undisturbed over the edge of this plate. Such dephasing schlieren systems have been developed by Walter,'' and an improved version using laser light has been reported by R ~ y e r With .~~ a uniform test field, the recording plane will remain uniformly obscured due to the interference of the phase shifted light with the undisturbed light. If the phase plate is removed from the focal point in the axial direcS . Knoos, Proc. Inr. Congr. High-speed Photogr.. 8th, 1968 p. 346 (1969). R. Grossin, M. Jannot, and S. Viannav, Appl. Opr. 10,201 (1971). H . Royer, Proc. Int. Congr. High-speed Photogr., 9th, 1970 p. 410 (1970).
3 74
2.
DENSITY SENSITIVE FLOW VISUALIZATION
tion, the condition for light extinction applies only to a few rays passing in the neighborhood of the plate edge, and as a result a dark fringe which is parallel to the edge of the phase plate appears in the field of view. The fringe is distorted by a disturbed test object, and the fringe shift is again proportional to the light deflection in the test field.
2.4. Interferometry As described in Section 2.2.2, a compressible flow field represents an optical disturbance which may change the phase of a transmitted light ray with respect to an undisturbed ray. The development of instruments devised to display such alterations or differences in optical phase has been one of the greatest achievements in optics, and has provided instruments known as interferometers. Indeed, the application of optical interferometers to gasdynamic studies began one century ago, when Ernst Mach used a Jamin interferometer to visualize the compressible flow around a supersonic projectile. Each interferometric system to be discussed in this section can be regarded as a device in which one visualizes the interference of a ray, which has passed through the test object, with a second ray, which travels to the recording plane along a different optical path. Hence, one deals with the class of two-beam interferometers. Though applied to a number of problems, multiple-beam interferometry has not yet found wide application in the field of gas dynamics. The general setup of a two-beam interferometer is chosen to have a parallel beam of light traversing the test field. A coordinate system is introduced with the incident light propagating in the z direction (Fig. 17). Each ray of the incident light beam has a conjugate ray, and the action of the two-beam interferometer is to measure the phase difference between the two conjugate rays after passage of the test field. A preliminary as-
s,
52
z
FIG.17. Principal arrangement of two-beam interferometer with parallel light through the test section.
2.4.
INTERFEROMETRY
375
sumption is that both rays traverse the test field separated by a distance d . Figure 17 is a projection of the y - z plane in which two incident conjugate rays have the coordinates y + d / 2 and y - d / 2 . The entrance and exit planes of the test field are 5, and 52. Behind the object, a lens or a mirror, equivalent to the schlieren head, focuses the light beam; the two conjugate rays will coincide after passing through a certain interferometer unit which is regarded here as a black box for simplicity. A camera lens focuses the test object onto the recording plane. If the condition of optical coherence is fulfilled, the conjugate rays may interfere with one another and produce a certain interference pattern in the recording plane. The plane formed by a pair of conjugate rays is parallel to the y - z plane, i.e., parallel to the plane of Fig. 17. This in no way restricts the following analysis, since the orientation in space of this plane can be changed by rotating the interferometer unit around the z axis. As explained in Section 2.2.2., it can be assumed that deflections of the light ray from the z direction are negligibly small. The difference in optical path length AI between two rays traversing the test field in a plane x = const and at positions y + d / 2 and y - d / 2 is then
A1
=
lL:
n(x, y
+ d / 2 , z ) dz -
n ( x , y - d / 2 , z ) d z , (2.4.1)
where I;, 5;' are the entrance and exit coordinates of the first and second light ray, respectively, while n ( x , y , z ) describes the spatial distribution of the refractive index in the test field. Bright interference fringes appear in the recording plane, where Al/A = 0, f 1, _ t 2 , . . . , and where A is the light wavelength. With the aid of the Gladstone-Dale formula (2.2.5b), the observed interference pattern can then be related to the density variations in the test field. The problem of decoding the information contained in an interferogram is too difficult if one allows a completely arbitrary separation d of the conjugate rays as assumed in Eq. (2.4.1);for the density variations along both rays are unknown in this general case. In an interferometric device one has to provide an appropriate relation between the positions of the conjugate rays, so that one needs to determine the density variations along one light path only. The solution of this problem leads to two extreme situations, which correspond to two classes of optical interferometers. If D designates the diameter of the field of view, this classification of twobeam interferometers can be described by the ratio d / D : With d / D 2 I , one ray of each pair of conjugate rays travels outside of the test object and remains undisturbed. This is the class of reference beam interferometers. For a ratio d / D << 1, both rays traverse the test field, where they are separated or sheared by the small distance d ; such systems often are called shearing interferometers.
2.
376
DENSITY SENSITIVE FLOW VISUALIZATION
2.4.1. Reference Beam Interferometers
As stated above, the second ray of each pair of conjugate rays travels outside of the test field to arrive undisturbed in the recording plane. One may therefore consider the light from the source split up into two beams: one beam passes through the test object; the second, which remains undisturbed, is designated the reference beam. Each ray of the test beam has its conjugate ray in the reference beam. Most interferometers are designed to provide equal geometric lengths for both beams. The optical path of the reference ray or the second integral in Eq. (2.4.1) is then constant, and together with the Gladstone-Dale formula (2.2.5b), Eq. (2.4.1) reduces to (2.4.2)
where po is a reference density, e.g., the constant density of the air outside of the flow facility. The condition for the formation of bright interference fringes, K
[p(x, y , z) - pol dz = iA
(i = 0,
k
1, +2.
. . ).
(2.4.3)
is at the same time the equation of the fringes in a plane z = const, e.g., in the recording plane. For a two-dimensional density field with constant width b = Jft dz in the z direction, the equation of fringes becomes Kb[p(x, y ) - pol = iA
(i
= 0,
+ 1, + 2 , . . . ).
The interference fringes are now curves of constant gas density (Fig. 18). An undisturbed test field of constant density is represented by uniform color in the recording plane. This is called an infinitefringe width alignment of the interferometer. Equation (2.4.2) also shows that reference beam interferometers are sensitive to absolute changes of the gas density. By inserting a plane, transparent wedge into the path of one light beam, one may produce the finite fringe width alignment. For a uniform test field one now obtains a system of parallel, equidistant fringes in the recording plane. The density differences present in the test flow cause this fringe system to be distorted (Fig. 19). With s being the fringe width or spacing of the undisturbed fringes, the relative fringe width As/s is a measure of the density change, namely,
where the notation is the same as in Eq. (2.4.3). By superposing the
2.4.
INTERFEROMETRY
377
FIG.18. Mach-Zehnder interferogram of transonic air flow over a curved surface. The infinite fringe width alignment visualizes lines of constant gas density. A shock wave is formed in the rear part of the profile. [Courtesy of Dr. C. Veret, ONERA, ChHtillon, France.]
undisturbed, parallel fringe system and the distorted system, one produces a moire pattern which is equivalent to the disturbed fringe system of the infinite fringe width alignment. In all practical devices, the parallel equidistant fringes are generated by tilting a mirror or beam splitter (see below), which has the same effect as inserting the plane wedge. Achievement of optical interference between the test beam and the reference beam requires the light of the two beams to be coherent. This coherence is essentially determined by the characteristics of the light source. The effect of both size and spectral impurity of a conventional light source on the fringe pattern of a reference beam interferometer has been investigated by Bennett34 and Tanner.35 A white light source produces one white fringe of zeroth order. The fringes of higher order are more and more colored, due to the overlapping of the maxima and minima of the individual wavelengths. Taking colored interferograms is theresI
sII
F. D. Bennett, J . Appl. Phys. 22, 184 (1951). L. H. Tanner, Proc. Inr. Congr. High-speed Phorogr., Jrd, 1956 p. 14 (1957).
378
2.
DENSITY SENSITIVE FLOW VlSUALlZATlON
FIG. 19. Same as Fig. 18, but with finite fringe width alignment. [Courtesy of Dr. C. Veret, ONERA, Chgtillon, France.]
fore helpful for distinguishing between the individual fringe orders in the interferogram of a complex density field. At the same time, the higher fringe orders are gradually washed out with increasing order. If the spectral range of the emitted light is A 2 AA, the number of fringes obtained on each side of the achromatic fringe (zeroth order) is N = A/2 AA. If the source light consists of only a few sharp spectral lines, a periodic modulation in sharpness is superimposed on the fringe pattern. This is the same phenomenon that is used for interferometric spectroscopy, and it can be useful for determining the identity of a certain fringe behind a shock wave. Practically, by utilizing a laser light source36 one obtains an unlimited number of sharp interference fringes; however, one can no longer distinguish the order of the fringes. Special optical equipment is necessary to expand the laser light to a wide beam and to clean this light of diffractive noise known as laser speckle (see also the section on light sources). The best known reference beam interferometer is the instrument developed independently by Ludwig Mach and L. Zehnder in 1891. This sn A. K. Oppenheim, P. A. Urtiew, and F. J. Weinberg, Froc. R . Soc. London, Ser. A 291, 279 (1966).
2.4.
379
INTERFEROMETRY test section
I ‘
I
camera
I
lens
FIG.20. Mach-Zehnder interferometer.
Mach-Zehnder interferometer3’ combines a wide separation of test beam and reference beam (i.e., d / D > 1) with a relatively large diameter D of the test beam and the field of view. A basic arrangement of this interferometer is shown in Fig. 20. Light from the point source is made parallel with the lens (or spherical mirror) L 1 . The essential components of the interferometer are the plane, fully reflecting mirrors MI and M , , and the plane, semireflecting mirrors (beam splitters) Mi and M i , which are arranged here to form a rectangle. The test section of the flow facility with its two glass windows is brought into the path of the test beam, while two identical glass plates are inserted into the path of the reference beam in order to provide equal distortion and dispersion in both light beams. The second beam splitter M i acts as the aforementioned interferometer unit, which recombines the initially separated test and reference beam so that they coincide and interfere with one another. Such an instrument requires a high precision of material and surface quality of all mirrors, as well as extreme mechanical stability of the frame. Mechanical and optical tolerances are of the order of a wavelength or below. The best quality control is to check the parallelism of the interference fringes. Each of the mirrors is required to permit rotation around a horizontal and a vertical axis for the basic adjustment of the interferometer. With all mirrors being adjusted perfectly parallel, one has the infinite fringe width alignment. By inclining one or several mirrors, one may produce parallel interference fringes of finite width and of any wanted direction. Such an opera3T R. W. Ladenburg and D. Bershader, in “Physical Measurements in Gas Dynamics and Combustion” (R.W . Ladenburg, ed.), p. 47. Princeton Univ. Press, Princeton, New Jersey, 1954.
3 80
2.
DENSITY SENSITIVE FLOW VISUALIZATION
; I camera
loser beam
M1
laser
FIG.21. Mach-Zehnder laser interferometer. M I , M Iare plane mirrors; Mi,Mi,beam splitters.
tion is equivalent to inserting a plane, transparent wedge into one of the beams. A further requirement is that these fringes be localized in the test object, i.e., that both test object and fringe system appear well focused in the recording plane. An instrument which fulfills this condition and which reduces the number of mirrors to be turned for adjustment to one has been developed by Kinder.38 The mirrors of this single-plate controlled interferometer are arranged in form of a rectangle, with M 2twice as far from Mi as M,, and the test section is inserted midway between Mi and M2. When M ;, M 2 ,and Mi (Fig. 20) remain in the basic position under 45" of inclination, only MI has to be rotated to adjust the fringes. The fringes are located in M 1 ,and they appear in focus if the test section is imaged onto the recording plane. A direct modification of this single-plate controlled interferometer has been developed, which uses a laser light source.3g Due to the high degree of coherence of the laser light, the reference beam can be kept small in diameter (< 1 cm) as compared to the test beam (20 cm) (Fig. 21). This allows one to use relatively small mirrors and beam splitters, and to operate the instrument with a large spatial separation between test and reference beam ( d / D >> 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 << 1. One may then assume that two conjugate or interfering rays have in the test object the same entrance and exit coordinate and [2, respectively. Equation (2.4.1)therefore reads now A1
=
1'
[ n ( x ,y
+ d/2, z)
-
n(x, y - d/2, z)] dz,
51
53
S . F. Erdmann, Appl. Sri. R e s . , Sect. B 2, 1 (1951).
(2.4.5)
3 84
2.
DENSITY SENSITIVE FLOW VISUALIZATION
where the notation is the same as for (2.4.1). Developing (2.4.5) into a Taylor series and taking into account only the linear term, since d / 2 is a small quantity, yields for the optical path length difference between two conjugate rays: (2.4.6) J
LI
where the refractive index has been replaced by the gas density by means of the Gladstone-Dale formula (2.2.5b). Equation (2.4.6) is the fundamental relation for all types of shearing i n t e r f e r ~ r n e t e r sand ~ ~ .should ~ ~ be compared with Eq. (2.4.2). Whereas reference beam interferometers react to absolute changes of the gas density, shearing interferometers are sensitive to changes of the density gradient. This also led to the name schlieren interferometer, which is occasionally used for this class of instruments. Indead, the right-hand side of (2.4.6) is equivalent to the right-hand side of Eq. (2.3.6a), which describes the contrast in the recording plane of a schlieren system. One can no longer distinguish between a test and a reference ray, since both interfering rays are affected by the density field in the test object. The equation of the bright interference fringes in a plane z = const (recording plane) is
Kd
ILy
[dp(x, y , z ) / a y ] dz = iA ( i = 0 ,
* 1, 2 2, . . . 1,
(2.4.7)
where A again is the light wavelength. Only in the case of a twodimensional density field of constant width b are the fringes curves of constant density gradient:
Kbd[ap(x, y ) / a y ] = i A
(i = 0,
2
1, k 2 , .
.
.).
A test field of constant density gradient is represented by uniform color in the recording plane (infinitefringe width alignment). Turning .the interferometer unit 90" around the optical axis exhibits the density gradient in the x direction. The finite fringe width alignment can be obtained by inserting a wedge with a parabolic surface into the light beam. In the y - z plane (Fig. 171, one surface of this wedge can be parallel to the x - y plane, whereas the trace of the second surface is a contour z = const 3. One derives then that the fringes are parallel and equidistant curves y = const for the case A. R. Jones,-M. J. R. Schwar, and F. J. Weinberg, Proc. R. SOC.London, Ser. A 322, 119 (1971). W. Merzkirch, Appl. Opt. 13, 409 (1974).
2.4.
385
INTERFEROMETRY
FIG.24. Candle flame as visualized with a Wollaston prism shear interferometer.
of a test field with constant density gradient. Again, a rotation of the interferometer unit may produce a system of parallel, equidistant fringes in any wanted direction. In practice, one uses no parabolic wedge, and the finite fringe width alignment is generated by other manipulations of the optical system. An inhomogeneous density field causes the parallel fringe system to become distorted (Fig. 24). With s being the fringe width of the undisturbed system, the relative fringe shift As/s in a point (x, y ) of the recording plane is related to the density gradient in the test field by =
[a&,
(WW z1
Y , z)/ayl dz,
(2.4.8)
386
2.
DENSITY SENSITIVE FLOW VISUALIZATION
where a 1: 1 scaling between object and recording plane has been assumed. With a two-dimensional test object, As/s is directly proportional to the respective density gradient. It is obvious that a shearing interferogram is more appropriate for properly displaying a density field with relatively steep gradients than a reference beam interferogram. The above linearization as introduced in Eq. (2.4.6) is not permitted if the flow field contains a discontinuity surface of the gas density, e.g., a shock wave. The interference pattern associated with a shock wave is generated only by those pairs of conjugate rays which have one ray passing behind, and the other passing in front of the shock surface. From the exact relation [Eq. (2.4.l)] one derives that the optical path difference of such pairs of rays is given by (2.4.9) if the shock front is assumed to be a plane y = const (Fig. 17). The density jump across the shock is designated by Ap. Equation (2.4.9) has the same form as one would obtain for a reference beam interferometer. However, the respective interference pattern is different for the two types of interferometers. For the geometry mentioned, the shock will be visualized in a shearing interferogram in the form of a band of finite width d , and the fringes are shifted within this band as indicated in Fig. 25. From the same figure it follows that it is more appropriate to align the undisturbed fringes oblique to the shock surface. A further conclusion from Eq. (2.4.9) is that reference beam and shearing interferometers have the same sensitivity when applied to the visualization of a shock wave. The dependence of the quality of the interference fringes on the optical coherence of the light source is the same as described for reference beam type interferograms. In contrast to what has been said in the beginning of Section 2.4.2, it is not necessary that the two beams of the shearing interferometer be separated by means of a beam splitter before entering the test field. With a coherent light source, e.g., a laser, the interferometer
II1Il-1IIIII d
I.
exact position of shock wove
I
--
5
FIG.25. Representation of a plane shock wave with a shearing interferometer at finite fringe width alignment. Direction of fringes is (a) parallel and (b) oblique to the plane of the shock.
2.4.
INTERFEROMETRY
387
unit instead is able to select from the incident light pairs of (conjugate) rays that were separated by the distance d in the test field, and to bring these rays to interfere with one another. Indeed, many of the devices described in the literature are only applicable when operated with a laser light ~ o u r c e . ~ ~ * ~ ~ Many more shearing interferometric arrangements have been described than reference beam interferometers. These systems differ only in the principle of the interferometer unit which combines and superposes the two conjugate rays. The above analysis applies to any interferometer of such type, and an evaluation of the interferograms requires only the knowledge of the amount and direction of the shearing distance d . In the most simple arrangement, a plane glass plate is used as the interferometer nit.^^*^^,^^ The incident, parallel light beam which has traversed the test field is reflected from the plate (Fig. 26). Each ray of the beam undergoes a multiple reflection at the front and the rear surface. When leaving the plate, each reflected portion of one ray (e.g., ray no. 3 in Fig. 26) coincides with several reflected portions from other rays that have passed the test field at distances d , 2d,. . . from the first (here: rays no. 1 and 2). This device is therefore not a true two-beam interferometer, since the interference is generated between more than two rays. However, by a suitable choice of the reflection coefficients and of the angle of incidence, one can make the intensity of the third reflected ray so weak that the recorded interference pattern may be regarded as being generated by the interference of two beams only. The shearing distance d is determined by the inclination and the thickness of the plate. An optical transmission grating may also serve as the interferometer unit5*;and it is possible to generate shearing interferograms by means of h o l ~ g r a p h y . The ~ ~ most common arrangement uses as the interferometer unit a Wollaston prism placed between two crossed polarizers. This method had been originally developed for use in microscopes and has since found wide application in flow s t ~ d i e s . ~ O The - ~ ~ Wollaston prism separates an incident light ray into two components that include the sepaJ . G. Kelley and R . A. Hargreaves, Appl. Opt. 9, 948 (1970). C. J . Wick and S . Winnikow, Appl. Opt. 12, 841 (1973). 58 S. Yokozeki and T. Suzuki, Appl. Opt. 10, 1575 (1971). 5s 0. Bryngdahl, J . Opt. Soc. A m . 59, 142 (1969). Bo R. Chevalerias, Y. Latron, and C . Veret, J . Opt. SOC. A m . 47, 703 (1957). C. Veret, Pror. Int. Congr. High-speed Phologr., 4th, 1958 p. 66 (1959). '* W. Merzkirch, A I A A J . 3, 1974 (1965).
56
63 H . Oertel, in "Kurzzeitphysik" (K. Vollrath and G. Thomer, eds.), p. 759. Springer-Verlag. Berlin and New York, 1967. W. Z . Black and W. W. Cam, Rev. Sci. Instrum. 42, 337 (1971). 65 R. D. Small, V. A . Sernas, and R. H. Page, Appl. Opt. 11, 858 (1972).
388 1
2.
DENSITY SENSITIVE FLOW VISUALIZATION
2
3
\
FIG. 26. Interferometer unit of a reflection plate interferometer with parallel light beam in the test field.
ration angle E , and are polarized perpendicular to one another (Fig. 27). The separation angle E is to a first order approximation independent of the angle of incidence. Behind the prism, each separated component of a ray coincides therefore with the component of another ray that has traversed the test field at the shearing distance d from the first. A polarizer turned at 45" with respect to the optical axes of the Wollaston prism provides equal polarization direction of the coincident rays and allows for interference of these rays. The shearing distance d is determined by the separation angle of the Wollaston prism and by the focal length of the focusing lens (or mirror). A major advantage of this arrangement is that the finite fringe width alignment can easily be realized by a simple shift of the interferometer unit in the z direction. The sensitivity of an optical interferometer is determined by the smallest observable fringe shift in an interferogram. It is common to assume that this lower limit, which can be determined with sufficient accuracy, is (As/s),,,, = 0.1 for a two-beam interferometer, i.e., one-tenth of a fringe width. The smallest detectable difference in optical path length is therefore h/10. Though some authors use still smaller values, one can conWolloston Drisrn
crossed-polorizers
FIG.27. Interferometer unit of Wollaston prism interferometer.
2.4.
INTERFEROMETRY
389
clude that the sensitivity limits for measuring small density changes are of the same order of magnitude as derived for the visualization by means of schlieren photography. S m e e t P has shown that this sensitivity can be improved by two orders of magnitude if one is willing to forego obtaining a complete flow picture, and if one measures the optical path difference between two or more very thin laser beams which all traverse the test flow regime. For this purpose, an extremly thin laser beam is split up into two beams by means of a Wollaston prism, or into several pairs of beams by an appropriate number of prisms. After having passed through the test field, the two beams of each pair are recombined by a second Wollaston prism, and the coinciding beams arrive at the entrance slit of a photomultiplier, which records the signal produced by the interference of the two beams. With this arrangement, one measures the density only a t discrete points of the test field, or along discrete paths through this field; but one may easily record the temporal variation of the density, e.g., as produced by turbulent fluctuations, and the associated sensitivity is high enough to allow for a measurement of optical path differences in the order of A/1000, i.e., one thousandth of a wavelength. 2.4.3. Phase Contrast and Field Absorption A natural step in the development of highly sensitive visualization methods was adapting the phase contrast method to the study of compressible flows.19~87~se The application of this method requires the use of a coherent light source and extreme mechanical stability of the system. The optical layout is similar to that of a schlieren system (Fig. 8). Instead of a knife edge, a phase plate is placed in the focal point of the schlieren head. This very small phase plate coincides with the diffractive image of zeroth order of the light source, and it alters the phase of the light by an angle of 90". The formation of a flow picture by means of the phase contrast method can be explained with optical interferometry: but rather than using optical ray theory, it is now more appropriate to explain the principle of this method with the aid of a vector diagram (Fig. 28). This diagram shows the amplitude and the direction or phase of the light vector for all rays which have traversed the compressible flow field. The radial vector with phase angle cp = 0 designates the incident undisturbed light. The phase shift experienced by the light in the test object is expressed by a certain BB e7
G . Srneets, Shock Tube R e s . , Proc. Int. Shock Tube S y m p . . 8rh. 1970 p. 4511 (1971). R. Bouyer and C. Chartier, Proc. I n t . Conyr. High-speed Photogr., 3rd, 1956 p. 271
(1957).
M. Philbert, Rech. Aerusp. No. 99, p. 39 (1964).
390
2.
DENSITY SENSITIVE FLOW VISUALIZATION
FIG. 28. Vector representation of the phase contrast method. For notation see text.
phase angle cp # 0, which describes at the same time the phase difference between a disturbed and an undisturbed light ray. Since the test object is transparent, the amplitude of the transmitted light vectors are found on a circle of radius 1, if 1 describes the amplitude of the incident, undisturbed light. The phase contrast method is used for the visualization of small density differences, and the disturbed light is therefore assumed to be changed only by small phase angles pi. The vectors of three disturbed light rays 1, 2, and 3 with the respective phase angles cpl, cp2, and cp3 are shown in Fig. 28. It would be impossible to detect the phase differences between these rays from an ordinary two-beam interferogram. Owing to the constant length of all light vectors, the light intensity is the same over the entire recording plane. One may separate each vector in Fig. 28 into two components: one, common to all light vectors contained in the diagram (1, 2, and 3); and a second, individual component. The common vector component is not constructed arbitrarily; its end point A represents the center of gravity of the triangle ‘Formed by the end points of the three light vectors in this particular case, and the “weight” of the three points in this triangle is equal to the fractional area illuminated by the light of the respective vector. The light associated with the common vector component is equivalent to the light of the zeroth diffraction order in the focal point of the schlieren head (see above). An obvious reason for this is that the intensity of the zeroth-order-diffraction light is constant in every point of the recording plane, independent of the light’s phase angle. The (three) individual vector components then represent the light of the higher diffractive orders. With the aid of the above-mentioned phase plate, the phase of the common vector component is turned by 90”. The result is shown in Fig. 28: The new vectors, denoted by l ’ , 2’, 3‘, are now of different
2.4.
391
INTERFEROMETRY
lengths; and as a consequence, the intensity in the recording plane is variable and changes as a function of the phase angle. The optical phase differences are thereby transformed into alterations of amplitude or contrast. Since the phase object had been assumed to generate only weak phase differences, the length of the common vector component (central vector) is not much smaller than the radius of the circle in the vector diagram. The central vector does not differ appreciably from the undisturbed vector (cp = 01, and the lengths of the new vectors, after having turned the central vector by go", are approximately given by 1 + pi,where cpt represents the respective phase angles. Since pi is a small quantity, the new light intensity is 1 + 2 p i ; and the contrast in the recording plane is equal to 2 9 . This contrast can be increased if one provides a certain degree of absorption of the phase plate, so that the length of the turned central vector is additionally decreased to a value a, with 0 < cr < 1. The lengths of the new vectors are now (Y + (pi,and the contrast has been increased to a value Z(cpr/a). The sensitivity of the phase contrast method, or the ability to resolve small density differences in a compressible flow field, is found to exceed the respective value of interferometric visualization methods by a factor of ten.lg*ss The application of the phase contrast method is restricted to flows with weak density variations. If the optical phase is distributed randomly between 0 and 360°, the resulting central vector becomes very short, and a turning of this vector has only a minor effect. For this case Erdmann53 has proposed a modification of the method which allows the visualization of compressible flow fields with an unlimited variation of the gas density or an arbitrary optical phase distribution. The intensity of the zerothdiffraction-order light or the relative length of the central vector is increased with respect to the lengths of the individual vector components. For this purpose, an absorption plate is placed in the focal point of the schlieren head, which leaves the light of the zeroth diffraction order undisturbed, but provides a certain degree of absorption for the higher diffractive orders. The result is shown in the phase diagram of Fig. 29. In
FIG. 29. Vector representation of Erdmann's field absorption method. Original light vectors are denoted by 1 , 2, 3 ; new vectors after lengthening of the central vector are l', 2', 3'. For notation see also Fig. 28.
3'.r,,
1
0
central vector
/
392
2.
DENSITY SENSITIVE FLOW VISUALIZATION
this representation the central vector has been lengthened, while the individual vector components remain unchanged. An optimum situation is achieved if the new origin of the lengthened central vector falls onto the original circle, as shown in Fig. 29. The intensity of the light associated with the new vectors then varies between zero and four. It is obvious that the phase angle is allowed to vary by more than 360”;starting with the new origin, where the respective intensity is zero, the intensity will again be zero after every full period of 360”. This field absorption method therefore generates in the recording plane interference fringes, which are equivalent to those obtained with a reference beam interferometer for the infinite fringe width alignment. The lengthened central vector components, as defined in Figs. 28 and 29, designate the light of the test beam in this class of interferometers.
2.5. Evaluation Procedures The advantage of having available a nondisturbing testing method is associated with great difficulties in evaluating the pattern of a flow photo obtained with an optical method, for the information carried by each light ray which has traversed the compressible flow field is integrated along its path through the object field. The problem is therefore to relate the recorded two-dimensional pattern to the three-dimensional density distribution in the flow. The evaluation is easy if the density in the flow does not change in the z direction, i.e., in the direction of the traversing light beam. These cases of a “two-dimensional” test field have been mentioned in the foregoing sections. In any other case, it is necessary to perform a graphical or numerical procedure which will be discussed in this section. Since for quantitative measurements interferometers are used almost exclusively, the discussion will be restricted to the evaluation of interferograms. The information is available in form of a field of fringe shifts in the interferogram. This should be expressed by the data function D(x, y ) , where x , y are the coordinates in the recording plane. Assuming a 1 : 1 scaling between object and image plane, the data function is related to the gas density p(x, y , z) by
D(x, y) = const
(2.5.1)
where R ( x , y, z ) is a function of the density p . The form of R and the constant factor in (Eq. 2.5.1) are different for a reference beam and for a shearing interferometer [see Eqs. (2.4.4)and (2.4.8)]. Equation (2.5.1)is
2.5.
393
EVALUATION PROCEDURES
an integral equation, since the quantity to be determined, R or p , must be obtained from within the integral. The investigation of an arbitrary three-dimensional flow field requires taking more than one photograph of the object, each at a different viewing angle. If the flow has some kind of symmetry, the number of photographs necessary for evaluation can be reduced. In the case of an axisymmetric field, with the axis normal to the viewing direction, the number of necessary photographs is reduced to one. 2.5.1. Axisymmetric Fields
The test field is assumed to be axisymmetric with respect to the x direction as the axis of symmetry. The z direction remains the direction of observation. The gas density in the test field depends on the coordinates x and r = (y2 + z2)lI2, where r is the coordinate of rotational symmetry. Examples of such axisymmetric density fields are the supersonic flow around a circular cone (without angle of attack), the natural convective flow along a vertical, heated cylinder, or a straight electric arc of circular cross section. The fringe shift and the density distribution will now be investigated for a cross section x = const (Fig. 30). With the line element dz = r dr/(? - y y ,
Eq. (2.5.1) can be written (for a cross section x = const): (2.5.2)
The integration is taken from the coordinate y of the particular ray in the incident light beam to the outer radius rm , which confines the axisymmetric object. It is assumed that the gas density p = pm = const for r 2 ro., i.e., outside of the test object. Equation (2.5.2) and Fig. 30 show that all density values p(r) with r > y contribute to the variation in optical phase
light Y
FIG. 30. Light ray traversing an axisymmetric flow field at position y = const. The x axis normal to the plane of the figure is the axis of rotational symmetry.
Y
-
394
2.
DENSITY SENSITIVE FLOW VISUALIZATION
of the particular light ray. The density function is R = p(r) - pm for a reference beam interferometer, and R = ap/a(r2)for a shearing interferometer; the constant in (2.5.2) is (Kd/A)in the latter case, and ( K / A )in the former case. Owing to the transformation of Eqs. (2.4.4) and (2.4.8) into axisymmetric coordinates, the data function D(y ) also has a different form for the two classes of interferometers. For a reference beam interferometer, D ( y ) is identical with the relative fringe shift A s / s at position y , whereas for a shearing interferometer one has D ( y ) = ( l / y ) ( A s / s ) . For the following analysis it is important to keep in mind this difference in the form of the data function D ( y ) . The integral equation (2.5.2) is of the classical Abel type. The data function D ( y ) is known from experiment; the function to be determined, R(r), is part of the integral, and the integral function has a singularity at the lower limit y = r. The so-called Abel inversion, derived from the theory of Laplace functions, furnishes for the solution of this integral equation: R(r) = -(const/a)[d/d(?)]
Irm r
(Y
D(Y) 1,2 d(yz). r )
(2.5.3)
The integration now has to be performed over the known function D ( y ) . However, D ( y ) is only available in form of a set of discrete data points D(y,), and D ( y ) must therefore be approximated by a suitable analytic function or by a linear combination of such functions which match D(y,) at the discrete experimental points Y , . ~ * * ~ O This procedure has the disadvantage that error sources are introduced due to the integration of the approximated function D ( y ) , and in particular due to the additional differentiation d/d(?) involved in (2.5.3). On the other hand, the advantage of this procedure lies in the fact that an appropriate choice of the approximated form of D ( y ) may enable one to perform the evaluation by analytic means, in contrast to the procedure which will be discussed next. A direct approximation for evaluating the integral in (2.5.2) was described by Schardin,15and this procedure has been exploited numeri~ by G ~ r e n f l o .The ~ ~ circle of radius roois subcally by Bennett et ~ 1 . ' and divided into N zones of width h (Fig. 3 1 ) ; r, designates the outer radius of the pth zone, and one has
0 = ro < rl < r2 < and lrw - rp-ll
=
. . . < rN-l < rN = rm
h. The experimental data points D, = D(y,-,) must be
R. South, AIAA J . 8,2057 (1970). J . -L. Solignac, Rech. Aerosp. No. 125, p. 31 (1968). 71 F. D. Bennett, W. C. Carter, and V. E. Bergdolt, J . Appl. Phys. 23, 453 (1952). R. Gorenflo, Report No. IPP/6/19. Institut fur Plasmaphysik Garching, 1964. 'O
2.5.
395
EVALUATION PROCEDURES
FIG.31. Subdivision of the axisyrnmetric test field into circular zones of width h.
I
known at the edge of each zone. It is assumed that the density function R is constant within each zone, and that it changes discontinuously from one zone to the next. This means that the density is considered as constant within each zone if one evaluates a reference beam interferogram, and that the density gradient is constant in each zone in the case of a shearing interfer~gram.'~The problem now is to determine discrete values of the unknown density function R in the midst of each zone: R, = R(r,-l + h/2). At the discrete points y , , the integral (2.5.2) is approximated by the following summation:
This equation shows that the data point 0, is the result of a summation of all discrete values Ri of the density function, with p < i < N. According to G~renflo,'~ this may be written N
where the coefficients of the summation are determined by solving the integral in (2.5.4): a(p, i ) =
2h{[i" - ( p
- 1)2]1/2
- [ ( i - I ) ~- ( p - I)2]1'z}.
The constant factor in (2.5.4) is still the same as in (2.5.1) or (2.5.2) and will now be designated by c*. The recursive solution of the system of equations for R is
(2.5.5)
7s
W. Merzkirch and W . Erdmann, Appl. Phys. 2, 119 (1973).
396
2.
DENSITY SENSITIVE FLOW VISUALIZATION
From the preceding it becomes clear that this procedure is a step-by-step solution which starts at the outer edge rooof the axisymmetric test field. If the values of the density function Rt with p 1 Ii, 5 N are known, one may determine the value R,. The solution must start with
+
RN = R(rN-1
+ h / 2 ) = D N / ~ Ch(2N * - l)"',
RN-1 = [fh(2N - ~)"'XDN-~/C* - [2(N - 1)"' RN-2 =
-
(2N - ~)"']RN},
...
The value R(r), with r, > 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.
DENSITY SENSITIVE FLOW VISUALIZATION
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.
DENSITY SENSITIVE FLOW VISUALIZATION
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.
DENSITY SENSITIVE FLOW VISUALIZATION
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.
This Page Intentionally Left Blank
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
This Page Intentionally Left Blank
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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