Recently Particle Image Velocimetry (plN, )7,8 has been successfully applied to ...... than the time between pulses, which for a pulse- separation of lms gives a ...
The Development of Particle Image Velocimetry for Water Wave Studies
a thesis by
Callum
Gray
0
for the degreeof
Doctor
of Philosophy
Department of Physics University of Edinburgh 1989
Abstract,
A wave measuring facility based on particle image velocimetry is described. The important field flow the and the subsequent evaluation aspects of recording most is The from theoretical the records considered point view. practical of a and of discussed of error are sources and the performance of the system evaluated main kinematics internal Measurements the and accuracy. of breaker to the shown compare results plunging are made and of an extreme The from a similar wave. model of velocities predicted numerical a well with
in terms of reliability
further is illustrated two the technique applications to vorof with tex shedding behind a cylinder towed through a tank of stationary water and
flexibility
tube. within glass a cylindrical streaming acoustic
Acknowledge
ments
I would like to thank the following people for their help and advice in this work. Dr Clive Greated for his encouragement, assistance and advice at all stages of the project, Dr. Norman Fancey and Dr. Bill, Easson for helpful conversations and useful comments, Dr. John Sharpe and David Skyner for advice and conaboration in the Acoustics and Wave studies respectively. Thanks also to Frank Morris for providing technical support in the design and construction of the wave
flume and who maintained and adjusted the laser. Finally, I would like to thank my Mum and Dad for their support. and encouragement. To both this thesis is dedicated.
Declaration
This thesis has been composedby myself and, except where stated, the work contained is my own.
............... Callum Gray
Contents
5
1 General Introduction 1.1
Experimental techniques in fluid dynamics .............
1.2
Speckle photography in fluid dynamics
5
..
..............
11
Evaluatiou of flow records ..................
2
1.3
Development of the PIV technique .................
16
1.4
Application of PIV to Water Waves
19
Photographic
flow recording
................
for PrV measurement
21
2.1 Seedingconsiderations ....................... 2.1'.1' Particle motion in fluid
2.2
.............
24 i.....
24
2.1.2
Light scattering efficiency ..................
26
2.1.3
Seeding concentration
28
....................
Exposure parameters ......................... 2.2.1
Pulse separation
........................ 1
33 33
2.3
.2.4
2.2.2 2.2.3
Multiple exposures ......................
38
2.2.4
Dynamic range of velocity measurement ..........
39
.........................
41
Mumination systems ......................... 2.3.1
Scanning beam Mumination ................. 1
43
2.3.2
Apparatus
47
...........................
Imaging and Recording
..
......................
50
2.4.1
Lens evaluation ........................
51
2.4.2
Recording medium ......................
57
2.4.3
Refraction effects .......................
62
2.4.4
Three-dimensional flow fields
67
Flow photograph Theory
3.2
System description
PIV
Measurement
4.1
Systematic ýRecording Errors
4.2
Random Errors 4.2.1
................
70
analysis
3.1
72
.................................
83
........................... 1
Accuracy
93
98
...........
100
............................
Experimental measurement of random errors 2
9
37
Exposure time
.......
101
5
4.2.2
Velocity Gradient Uncertainty
4.2.3
Combined errors .....................
Post processing
106
............... ...
114
data
of velocity
116
5.1 Removal of anomalousdata ..................... 5.2
Interpolation
5.3
Vorticity Calculations
Experimental 6.1
6.3
of a deep water breaking
..........................
6.1.2
Wave generation
6.1.3
Two-dimensionality
....................... of waves ................
Measurement procedure ....................... 6.2.1
Seeding
6.2.2
Exposure parameters .....................
Results 6.3.1
7 Additional
124
........................
Laboratory generation of waves ................... Waveflume
6.2
120
.............................
measurement
112
............................
wave
129 131 131 131 134 136 140 140 142
................................. Velocity measurements ....................
149
160
applications of PIV
3
7.1
Surface flow measurements Results 7.1.2
7.2
.....................
165
....
.........................
Discussion
161
1 165
...........................
171
PIV measurement of Acoustic Streaming .............. 7.2.1
Rayleigh Streaming
7.2.2
Experimental Apparatus
7.2.3
Measurements and Results
7.2.4
Discussion
171
.....................
173
................... ..............
....
174 179
. ... .........................
182
DisCUSsion and'Conclusions 8.1
Limitations
184
8.2
Future work .............................
186
Appendix A
... ..
188
..............................
Appendix B: Publications
.......
4
0000000000
190
Chapter
1
General Introduction
E?cperimental
techniques
fluid in
dynamics
Experimental techniques are essential in fluid dynamics research-'The complexity of even the simplest flows make flow measurement necessary for verification of numerical equations of motion as well'as quantitative investigation of flows for which no reliable computational schemes exist. The development of, flow has field techniques into therefore measurement of research a'significant evolved in its own right., Qualitative insight into complex flow situations is made possible by floW'visu'ShadOwgraphi, techniques. Methods Schlieren photography, alisation such as Interferometry and Holography reveal variations in refractive index, within the flow, 'induced by changes of temperature, pressure or density. Velocity, streakline, streamline, and pathfine information is communicated to the experimentalist by means of Particle tracing, smoke wire, direct injection, and spark tracing information large This into is important for insight the *scalestructures methods. detail fine Flow interest., flows in the visualisation also as well as present most of provides guidance in the interpretation of more quantitative point measurements and the design of experiments. The detail and quality that can be achieved with (1984)[120] illustrated books in Van Dyke such methods is such as or JapanSoc. 5
Mech. Eng (1988)[96]. Velocity is typically the most useful measureable quantity and, devices such as Pitot tubes have been widely used to provide point velocity measurements. Howlow. Hot-wire instruments is the temporal these spatial and ever, resolution of hot-film have been anemometers and also used extensively and to a certain ex-
tent overcome,the problem of frequency responseassociatedwith the Pitot tube. However,the measurementsmadeusing thesedevicesis comparative rather than final leading in the to absolute and require calibration considerableuncertainty velocity estimate. Further to this, the physical intrusion of the probe in the fluid flow the that is being measured. affects A class of optical point measuring tools have, developed from the I initial experi I Laser-Dopplerments of Yeh and Cummins, 1964[124]. A cross-beamý_ Anemometer utilises the interaction of suspended microscopic particulate matter with a light pattern set up at a point within the flow. This light pattern is set up within the flow by interference of coherent light from a laser. The evolution of this technique has been rapid and diverse and is now a standard method of obtaining-high precision point velocity measurements. This method its various refinements are well documented in Durst et. al. (1976)[43] and and Durrani and Greated (1977)(42]. Because Laser-Doppler-Anemometry
(LDA) is an optical technique, the limitations of probe interference exhibited by earlier devices are avoided. Sophisticated multicomponent configurations of the LDA have been devised and used to resolve all three components of velocity at a point. In order to determine the spatial flow structure of velocity however, it is necessary to move the LDA probe region throughout the flow [45]. For steady and precisely repeatable flows this is feasible but for random or non-repeatable flows this procedure is limited in its ability to resolve any useful spatial structures. In such cases point measurements have to be interpreted in conjunction with flow visualisations.
The detail present'in' such flow photographs has prompted many workers to 6
images and techniques that have
consider quantitative analysis ofvisualisation Streak-photography-analysis in include tracking the and attention past recieved involve [74,50]., flourescent techniques Both. tracers these xneasimilar and of known interval. in distance time the travelled tracer a surement of of particles Streak photography analysis has probably attracted the most attention [37,33] into in introduced degree Particles an essentially a moderate resulting of success. two-dimensional flow field are illuminated by an intense sheet of light, typically from a laser, and then over a short time interval imaged onto a recording medium (photographic, video etc. ). The analysis of such flow records requires the use highly 1, image in typically techniques of sophisticated only and result analysis moderate measurement accuracy. The main difficulties occur because streaks may lie very close to each other and may even cross. Further problems occur when rapidly moving particles move out of the plane of illumination, resulting in anomalously short particle trajectories. The finite width of the streak must also be taken into account when estimating the travelled distance from a tracer streak length.
"A good generalreview is given by Hesselink(60].
7
x
1.2
Speckle photography
in fluid
dynamics
In 1977-78 a number of workers published results illustrating the application of Scattered Light Speckle Photography (SLSP) to the measurement of flow velocities, [15],[41],[581, [68]. SLSP had previously been used to measure the small [10,26]. distortion The principle of the technique is scale elastic of solid objects quite simple. Coherent light from a laser is used to illuminate the surface of a solid object, or a region within a transparent body that has been seeded with tiny impurities. The scattered coherent illumination is then imagedO'nto photographic film where the random phase relationship of the scattered light leads to the formation of a speckle pattern [52]. Movement of the scattering object will cause the speckle pattern to move in space and so, as long as the motion of the
is object perpendicular to the wds of the imaging lens, two similar but displaced , specklepatterns can be recorded photographically. This photographic record of the in-plane displacementof the object is called a "specklegram". Quantitative analysis of the specklegramis possible either by point analysis-With a-narrow laser beam [23] or by a full field probe spatial filtering technique which displays displacement [24]. of equal contours The local analysis techni que utilises optical, processing of small regions of, the ' film where displacement is assumed to be constant., A narrow laser beam passed through the film generates speckle displacement or Youngs fringes in, the back focal plane of a lens (see figure 1.1). The orientation, and periodicity of. these fringes can then be related back to the displacement speckle pattern recorded. and thus the object motion. The full field technique, first described by Celaya, Jonathan and Mallick[24] utilises, spatial filtering in the, Fourier plane of the specklegram to reveal fringes of equal displacement in the second image plane. In the preliminary applications of SLSP to fltu*d dynamics, a ýsurfice was generated within the fluid by illuminating a heavily seeded flow with an intense sheet of laser light. The speckle field generated by the random interference of scattered
8
Youngs fringes specklegram/PIV
negative
ýII ý" i probe laser beam Fourier transform leas
Fourier plant
Figure 1.1: Optical processing of a "Specklegran3!' to generate Youngs fringes e
laser light was recorded photographically two or more timeS2 (see figure 1.2) to generate a specklegram that could be evaluated using either of the methods described above. Barker and Fourney[15] measured the laminar velocity distribution across the the diameter of a circular pipe., Illumination was provided by 13 laser Q-spoiled by the the Youngs ruby and resulting a specklegram analysed fringe method to yield the radial velocity distribution of the pipe. It is suggested in this paper that decorrelation of the speckle pattern due to out-of-plane motion lead turbulence to loss of measurement and a more sensitive speckle would or interferometry technique (SLSI) is-suggested as a solution '. Application's of the speckle photography technique for measuring velocities along a circular tube Dudderar[41] by Simpkins two and measured also groups. are reported other the velocities by the Young's fringe method whereas Grousson and Mallick[58] in full field filtering their commeasurement spatial as well as point employed 21wata, Hakoshima. and Nagata imaged resolved particle images rather than generating a speckle pattern from overlapping images. They also used multiple exposures to artificially increase the seeding density within their flow. SApplication of the SLSI technique to solid body motion had, been published [82] but no been found. has fluids this technique to of reported application
9
m
cylindrical lens
laser beam modulator
lens + tter Particle imagesat sPecklepattern
I
camers.
Figure 1.2: Arrang Iement for recording in-plane displacement of flow field parison with simple theory. The first applications of this technique cited above but illustrated flows the potential of to were simple one-dimensioual slow steady the technique for measuring the instantaneous distribution of velocities in fluids. Further applications of Speckle photography to fluid flows were reported soon DudSimpkins introduction dynamics. fluid the the technique to and after of derar followed up their initial work with an application to unsteady convection [113]. Binard Nagata[68] Iwata, Hakoshima a, measured velocities within cell and irnageS4. within ý convective cell using multiple exposures and resolved particle
An important point that should be made here concernsthe nature of the random pattern recorded from the light scattered from within the flow. In most initial image the the was a classical speckle pattern. experiments recorded of Small out-of-plane motion of the scattering particles betweenilluminating pulses ffirn intensity the the photographic and ulpattern recordedon changes random timately leads to decorrelation of the separatedspecklepatterns and therefore loss of measurementat that point. This problem is avoided by reducing the in in images, density a pattern of resolved particle which case resulting seeding 4Iwata et al also measured the instantaneous flow velocity distribution holographic technique[681. double exposure using 10
0
within a similar cell
lp
the particles have to move completely out Of the light sheet before total decorrelation occurs. Consequently, adopt the particle
fluid dynamics applications most
image mode of operation.
Further
approach include improved optical penetration, fluid the risk of altering characteristics generate a speckle pattern. this method.
of this technique
advantages of this latter
lower seeding cost, and reduced
by the large seeding density required to
This factor has altered the terminology
Laser Speckle Velocimetry
(LSV)
concerning
has been used because of the
technique's close links with SLSP but a more popular name for the technique is , Particle image velocimetry (PIV) [97]. 1 will use PIV unless explicitly referring
to a speckle technique.
1.2.1
Evaluation
of'flow
rýcorcls
To date, the development of speckle fluid been flow has. to photography applied considerable. Measurements are no longer confined to slow flows or small areas [79,111]. Automatic analysis systems have been developed that are capable of point-by-point evaluation of the double exposure flow field record over thousands [7,93,86,13]. These systems process the photographic flow record by of points different variety of a numerical processing techniques using flow data from the Fourier plane of the film in the case of Youngs,fringes and the image plane when the speckle pattern or particle image data, is used directly.
Fýurier
i3lane analysis
Initial processing schemeswere intended more for the analysis of solid body displacement specklegram (87,66,73,80] than for fluid dynamics applications. The motivation for such work is the limited accuracy, resolution and speed associated with a manual analysis of fringe patterns generated from specklegram or PIV negatives. Automatic processing of the Younis fringes presents the possibility of a rapid and accurate evaluation of the velocity information contained on the
11
film and the -subsequent calculation of derivative quantities such as strain or vorticity.
I
Ineichen et. al. [66] proposed a semi-automatic system where the speckle displacelines fringes the to of aVideo camera and are manually aligned parallel ment then digitally integrated along, their length. This step compresses the amount is data for further that, the noise suspeckle of required analysis and smooths is fringes fringes. bias The DC the the sit estimated on perimposed on which by integration perpendicular to the fringes and then subtracted from the 1-D
fringe data. The autocorrelation of this de-biasedsignal is then calculated to further increasethe signal to noise ratio for subsequentmeasurementof fringe periodicity. Estimated accuracy attainable with this system is 1% and 1deg for fringe separation and orientation respectively. Kaufmann-et. al. [72] employ a sin3ilar 1-D integration technique but perform the integration optically using a cylindrical lens arrangement. Detection of the resulting 1-D fringe signal is performed using a linear photodiode array. Orientation of the fringes is achieved manually and measured using a sin-cos potentiometer to better than 0.5deg for good visibility fringes. Fringe periodicity is measured from the Fourier transform fringe the'de-biased of signal and accuracy of 11im is quoted for good visibility fringes. Meynart [91,89,92] in his work on Rayleigh Binard flow uses the speckle photography technique to draw isovelocity contours by spatial filtering. Quantitative analysis of the flow record is also performed using a 1-D fringe analysis technique fringe processing technique similar to, those described above. Additionally, -a based upon calculation of the average autocc?rreUation in a number of different directions is discussed. This technique gives the fringe frequency in a number directions, different of and combined with the knowledge that the fringes are straight, allows the fringe direction as well as the fringe periodicity to be-cal. culated without user intervention. This fully automated system is used in an application of the speckle technique to study vortex pairing in a low-Reynolds (Re=2300), jet Meynart (1983) [93]. This is a much cited publication number and represents a key piece of work in the development of PIV. It illustrates the 12
instantaneous for the technique of measurement of velocities precise potential flow. Similar and calculation of vorticities over extended regions of an unsteady based discussed by, Ertechniques on multidirectional are systems correlation [47], Reynolds et al [1041, and Robinson [105]. Robinson also describes a -beck fully automatic'l-D integration system where the direction of integration is determined by comparing the sum of pixel values in a number of radial directions about the centre of the fringe* pattern. ' The main limitation of such systems is that for low fringe density (less than five, fringes) ýthe accuracy and -reliability with which-the fringe spacing and direction can be determined is significantly This -Problem is compounded by the noise and low visibility. of fringes reduced. typical of flow displacement records. -A more natural approach to, the evaluatiOn of fringe periodicity and direction, in the presence of noise is transformation of the fringe intensity distribution, to the matial frequencv domain, -The enintensity fringe distribution. the of ergy associated with the fringes is thus concentrated'about two symmetrically adjacent points in the freqi,.ency plane while the speckle noise is distributed randomly, This'approach is suggested,by Robinson [106], [6] Adrian but, they and emphasise the comput a,tionally expensive nature of the -calculation. I-P
.I;,,
A fully automatic fringe analysis system based upon 2-15 Fourier analysis of fringe intensities is described by Huntley-(1986) (64]. Digitisation ofthe diffraction patterns is performed-using a mechanically scanned linear photodip_de array with digital processing of the intensity data on a microcomputer. Theperformance of the system is analysed,by recording the accuracy and reliability with which fringes of known frequency and visibilityare evaluated. It is shov!n that for good, visibility fri n'ges the displacement of speckle and orientation patterns . determined be to within 0.1 Am and O.Ideg respectively. Reliability is 100% can for fringes down to 7% visibility where uncertainties are increased to 0.3 Am for displacement 0.4deg and and orientation respectively. This, accuracy, and is impressive but, as noted above, requires considerable reliability computation compared with 17D-integration and multi 1-D correlation techniques. Commercially available hardware is available for the rapid calculation of. 2-dimensional Fourier transforms. and can transform large (512*512) arrays in under a second.
13
fringes high for are not visibility This suggests that many applications where be technique transform preferred. two-dimensional the will probably guaranteed
image
plane
analysis
Evaluation of PIV negatives and specklegramsare not confined to the analyfrom data intensity 2-D fringe Numerical Youngs processingof patterns. sis of the'image plane of, displacement records has been achieved using a variety of techniques. Adrian and Yao (1984)[123]performed particle displacement mealocal directly from image the regions of a pairs over small surements particle PIV negative by integration of the digitised image intensity distribution in two directions. then 1-D The numerically are records orthogonal resulting pair of processedby correlation and Fourier transform techniquesto resolve the mean [21] image displacement Particle tracking in directions. the z and y particle has been used to locate possible particle images within a local area after filtering and thresholding of the digitised flow record. Statistical techniques are then used to link the centres of the located particle images to give the flow direction and magnitude. These techniques assumethat there is a reasonable information flow direction local interrogation to the amount of per spot relating low due is In to to the noise ratio and magnitude. certain caseswhere signal low contrast images, decorrelation between images etc. ma.-drnurn use of the full data 2-D image be In autocorrelation of cases available must such made. the local image plane intensity data is most appropriate (6,9]. and produces two displacementcorrelation peaks,symmetrically displacedabout a central self in information This displacement the to the same peak. correlation appears put form as 2-D Fourier analysis of Youngs fringes discussedabove. The Young's fringe intensity pattern, detected in the back focal plane of a biconvex lens, is the power spectrum of the photographic density distribution within the region by beam. The film illuminated laser interrogating the power spectrum and of the autocorrelation function form a Fourier transform pair by the autocorrela. tion theorem/Wiener-Khintchin,
Fourier [22] 2-D and so analysis of ý relation
the fringes and autocorrelation in the image plane are actually equivalent. 14
An alternative method for the analysisof flow recordsproposedby Collicott and Hesselink (1986)[28] utilises displacement information from a line on the photograph rather than a point. The technique described as "Anamorphic optical processing"performs a 1-D optical Fourier transform on a coherently illuminated direction film dimensional in imaging the the perpendicular to of and strip one this using an arrangement of cylindrical lenses. This allows simultaneous mea, surement of the velocity component perpendicular to the illuminating strip along the whole length of the analysis area. Processing of the generally curved fringes produced in this manner is achieved by multiple 1-D correlation.
15
1.3
Development
th&-PIV of
technique
-
Present work in PIV is directed at improving the accuracy, resolution and applicability of the technique. Velocity measurements should be of sufficiently high derivative to accuracy and resolution quantities such as permit calculation of vorticity, and streamlines and so afford a complete description of the flow field being investigated. In pursuit of this goal many different aspects of the operation been have PIV investigated and reported in the literature. of
Work by Adrian and Yao (1984)[5,4] has consideredihe scattering power and concentration of seed materials neccessaryfor optimal operation of a Particle image velocimetry system. Seedingis discussedin terms of the opposing requirements of increasing size for photographic detectability and decreasingsize for maidmilm positional accuracy and resolution. A two step photographic process is described by Pickering and Halliwell[102,101,100] illustrates the reduction in speckle noise in ýYoung's fringes that can be and achieved by contact printing a negative flow record. , ý!
II-"?,
-, ý%"
I fringes has The influence of the pedestal function in the evaluation of Youn gs been considered by a number of workers. It has been shown that this bias function causes a shift in the positions of the fringe mamma and in fringe patterns of low visibility a similar but reduced shift in fringe minima. Kaufmann (1981) [71] showed that this effect was less fringe when processing was perpronounced formed by Fourier transform techniques.
the of pedestal -ýubtraction ,function prior to processing of fringes is usually adopted to reduce the influence bias this of and' extend the lower limit of measurable fringe density (98,99,641,
It can be seenthat the evaluation techniquesdescribedabove allow the magnitude and direction of the flow to be determined acrossa PIV negative. However, there is no information present on the film from which the senseof the flow can be determined. Particle tracking and flourescenttracer techniquescan label' the direction of the flow unambiguously by meansof coded illumination and tracer 16
/
from be inferred direction flows For the a-priori can tails respectively. many flows. Two for impossible difficult if but is this information more complex not [8,81,7] Adrian have been different solutions to this problem et. al. proposed. imaged (1979)[48] the by Ewan first describe a technique whereby suggested flow field is subjected to a shift between exposures so that the largest negaThus, displacement. but displacement positive tive positive appears as a small below this displacements in and above measured and negative velocities result "pedestal" displacement. Negative and zero velocities are then resolved when in is that is technique is known This to the used a subtracted. shift analagous Shifter is to Phase Laser Doppler Anemometry superused where a crossbeam impose a known positive velocity on all the measured velocities. An alternative Coupby been has directional to this reported approach resolving, ambiguity land et al (1987)[32,311 where Image plane holography is used to label the first first images the in flow. Reconstruction the and secof and second recorded independently, images by illumination of the flow record with two separate ond bedetected beams, to image the two allows reference positions sets of particle displacement determine then to cross-correlated and an unambiguous particle A further advantage of this approach is the added immunity of ihe'resulting flow information from anomalous random connections when less than three complete image pairs are present within the analysis area. particle An implicit assumption in the pointwise analysis of Particle image flow records is that the velocity and therefore particle pair separation does not vary significantly within the local analysis area. For most practical applications of PIV this assumption is only partially satisfied. Real flows have velocity gradients both lead turbulence, to a range of particle separations within and of which will the probe region. When fringe analysis is adopted in the evaluation of such flow fringe fringe by, the the the caused visibility pattern. will vary across records image decorrelation It but displaced the two patterns. of particle slight similar has been shown that, excluding all. other sources of fringe decorrelation and a is it image possible to relate the complete set of particle statistically separations, fringe visibility distribution to the velocity distribution within the probe region.
17
(1984)[61] have Hinsch (1979)[48] Ewan et al commented on this p6ssibility and the fringe investigated to the velocity probability measure visibility use of since density distribution. Further work by Arnold et. al. [11] has experimentally investigated this principle by taking double exposure photographs of a seeded degree turbulow in turbulence tunnel of controlled with a small wind a airflow lence. The resulting flow records were analysed using a 20MM diameter probe beam, to ensure a statistically complete set of particle pair separations and the intensities Turbulent fringe distribution the pattern. measured across visibility were then shown to compare well with simultaneous measurements made with a hot wire probe. Real time recording and processing of double exposure flow records have been achieved using a reusable. photorefractive material to record a double exposure displacement pattern with instant analysis of the flow record using a conventional [29]. (1988) beam laser Hessellink to Youngs Collicott fringes, probe generate and The photorefractive variations
material
in the refractive
M12Ge02o
(BGO)
images the as spatial stores
index and can be quickly
This offers the possibility
erased and used again.
of accurate real time measurement high speed digital processing techniques. combined with
of velocities when
The possibility of measuring three ýomponents of velocity has als6ýbeen 'con. sidered using stereo photography, De Almeida Dias Delgado, (1986) [361'and Holographically, Royer, (1988)'[1071.
The possibility of very high speed processing of PIV films using purely optical techniques has been realised. Photorefractive materials such as Bismuth silicon (BSO) has been [125] 2-D to the used optically compute autocorrelation oxide so for based digital Optical the avoiding need slow correlation on the use analysis. of a spatial light modulator as a frequency plane filter has also shown potential as a method of rapidly evaluating PIV flow records [126] with the accuracy and reliability associated with the autocorrelation approach to film analysis.
18
1.4 "Application
PIV of
to' Water
Waves
It can be seen that the Laser Speckle/Particle image technique has considerable data A fluid in preof experimental standard mechanics. experimental potential limitations to the measurement point conventional of clue unattainable viously techniques has been shown to be available in a number of preliminary experilight laser intensity high increasing the sources and affordability of ments and this to has novel apply prompted a number of researchers powerful computers technique to a wide variety of flow situations.
The work discussedin this ,thesis describesthe"'developmentand evaluation of laboratory for PIV kinematics internal the generated of a system measuring breaking water waves. Previous experimental studies based on more conventional measuring techniques have had a number of difficulties and limitations associatedwith them. The main difficulty is due to the transient nature of the breaking be limited the that and wave can number of siLultaneous measurements made with propeller meters, hot-wire or LDA probes etc. In the past.it has been ' individual to make necessary measurementsat different positions within succesbuid to up a completevelocity map[45]. This requires that individual sive waves be measurements made over very short time intervals (,- 1ma) and assumesthat the wave is completely repeatable. Sophisticated signal gating and processing techniquesallow the velocity within a short instant to be made accurately and individual that timing to measurementsare systems are used ensure elaborate However, the other positions. same phase made at position as measurements at individual doubtful is although advanced wave genof waves exact repeatability do Successful fair degree techniques ensure application eration a of repeatability. of the PIV technique would avoid such experimental complications and most imbe independant portantly would of wave repeatability allowing random events to be investigated. The following chapters consider in detail the theoretical and practical aspects of PIV a wave measuring facility that has been developed by the author. Chapters 2
19
flow in PIV 3 the records a general way recording and evaluation of and consider that is not particular to wave measurement but introduces several novel features that hav e been developed in the application of PIV to waves. Uncertainty and identified both the recording and analysis procedures are errors associated with in Chapter 4. The most significant sources of error are considered in some detail and permit an estimate of the accuracy associated with the final velocity data be Chapter 5 to the and resulting velocity quoted. examines measurements describes procedures used to remove anomalous data values, interpolate missing lnA'calculate 'K *c6iaýlete description of the wave generation points' vorticity. facility, flow recording procedure and film processing is described in Chapter 6. Wave measurements are made from a series of negatives that were taken at equal time intervals through the breaking of a typical deep water wave and the results are shown to be consistent with what is known about such waves. Two further applications of, the PIV technique are described in Chapter, 7. Firstly the use incoherent illumination system for the measurement of surface flows and of an secondly the measurement of Acoustic streaming using PIV. Finally, Chapter 8 identifies some of the advantages and limitations of the PIV technique and in the light of these considers the direction that future work should take. .
20
Chapter
2
Photographic
flow recording
for PIV
measurement
The measurementof velocity inescapablyinvolvesthe measurementof a travelled distance ina known time interval'or vice versa. In thecase of particle image instantaneous by the is flow velocimetry velocity of a region within a estimated. photographically recording, over a short time interval, the travelled distance of have been introduced into the flow. The arrangement shown particles which in figure 2.1 illustrates the basic principle of recording process where,the Ithe I the tracer is imag motion of particles ed onto a recording medium as as a random distribution of particle image pairs. The motion of a particle represents the flow'as seen from a La'grangian viewpoint' (z, its initial function the is time"t the and position of y, z) of where a particle position
(x', y', z') at t=0
ie.
x=f
(x', y, z' : t), Y=f
(x', Y's Z' : t), z=
f (XI
t). The velocity as a function of position x1y, z and time t, ie. : IYIjz1 jL(ojyjz : t) represents the Eulerian velocity and is the most commonly used fluid dynan3ics'where, in viewpoint +V2j
+ V3k
directions in the the x, y, z respec-ýV3represent componentsof velocity tively. The relationship, between the Lagrangian,particle position. and a single and'VI,
V2 ,
21
11 I Pulsed light sheet
,.
e....
OKI 01.
"., -.
90
Figure 2.1: photographic recording flow field in'-plane the_ of a motion of component of, the Eulerian velocity is embodied in the expression, LXfil
Vi = dt = lira tl; --+t2
--i(ti
)-
xj(t2)
tl - t2
(2.2)
Zi(t2) are ýthe' coordinates ýin the x, y orýz direction, of a, single particle at tj-, and t2. For a time interval ti - t2 that tends to zero the ratio of the travelled distance over the time of travel will closely approximate the instantaneous Eulerian velocity. In-practice, the-time interval need only be small compared to typical Eulerian time scales of the flow for the. measured whereXi(ti),
velocity to closely 'approximate the instantaneous velocity.
A double exposed flow record obtained from a seededflow will therefore be distribution a random of particle image pairs where the local instantaneous (tI t is 't2)/2 at-tiM'e velocity proporional to the siparation of any particle ý-image pairs that may be found at that point. Beyond the approidmation betweenthe instantaneousEulerian velocity and the measuredLagrangian motion of the tracer particles, the precision with which 22
by limited be the be accuracy with instantaneousvelocities can measuredwill be between interval displacement the time can fluid exposures the -and which determined. Typically,, the time intervalT between pulses can be measuredto in degree the high resides uncertainty of source major while accuracy of a very flow. the fluid displacement the measurementof the at a particular point within An ideal recording'systemwould produce a double exposureimage of the seeded flow that had a very denseconcentration of perfectly recorded very small paxticle images. This would give an essentially continuous distribution of velocity information acrossthe flow record where, no matter how small a local area of the film was considered,there would always be particle image pairs relating to the motion of the recorded flow. Small particles will respond to the flow better This'would images therecording medium., and will produce'smaller'Particle on both temporal increasing the to'be" thus permit'smaller"displacements resolved and spatial resolution of the flow recor& Additionally, the po'sition,'and theref6re separation, of particle images will'be more accurately"measuviableif they are small. It can be seentherefore, that practical recording'systeM'Sshould aim to record as high'a concentration of small paxticle images'asis possible. Listed below are the main factors that control how close to, this ideal practical measurements can come
Flow characteristics:
index, area max/min velocity, refractive
I Seeding: m aterial,dimensionsconcentration, light scattering efficiency Recording Light
medium:
sensitivityresolution,
noise, size
source: power, wavelength, directionality
Recording Exposure
optics:
resolutionjmage distortion, magnification, aberations
parameters:
'pulse duration, pulse separation, number of pulses
The choiceof Componentsand parametersfor recording a flow field will often be limited the the of resolution and recording sensitivity a compromisereflecting 23
form light the the the scattering efficiency of power and of source, xnedIUM, tracers, financial limitations etc. The final configuration of the flow recording be dependent thus upon the resourcesavailable andthe priorities of system will the user concerning accuracy,measurementarea and resolution. The following in detail details the the consider more. sections parameters outlined above,and flow facility the that has been developedfor the measurementof recording of velocities under breaking waves.
2.1
Seeding considerations
Particle Image Velocimetry like Laser Doppler Velocimetry relies upon the interaction of light with particulate matter suspended within, and assumed to follow the fluid motion. In Laser Doppler measurements the light scattered from one or more particles within a small -intense spatial pattern of light is monitored to determine the time dependent velocity of the flow in one direction at that In contrast PIV measurements rely upon capturing spatial information point. over a very short time interval throughout an extended region of a flow field and typically encompasses many thousands of particles. This raises questions concerning the precision with which the particle motion matches that of the fluid, the detectability of particles during the small timescales and large areas involved, and the optimum seeding concentration required for good resolution and accuracy.
2.1.1
Particle
motion in fluid
An implicit assumption in the recording of PIV photographs is that the particles introduced into the flow faithfully follow theýmotion of the fluid and that the flow is not perturbed by the presence of the tracer particles. Consideration of the relative motion between particles and flow has been given in experimental studies utilising Laser Doppler Anemometry and a number of different, effects 24
have been identified.
ýI
fluid the density the A difference in the will Inertial effects particles and of This fluid delayed to in the acceleration. particle responseof a result high large is difference density is the or when effect only significant when frequency fluctuations of the flow are important. In liquid flows with be is to densities this a problem. unlikely reasonably matched seeding Large differencesin density are most commonly found in air flows and this For the become high frequencies. particles small apparent at effect may flow is to the fluid the power spectrum power spectrum related of [42], 44(w) in the of particle velocity an approidmate manner ja 12ýý.(W) ..
41).(w)
(2.3)
where
K,, a ..
)1/2 + (K, 2, W2
(2.4)
and K"
36yo T2-pp+ po)dp2
(2.5)
flow the velocity is varying sinusoidally , and w,, is the frequency of where 4 is the particle diameter, yo is the fluid viscosity and pp and oscillation, po are the particle and fluid densities respectively. The complex amplitude of the particle oscillation is a, It can be seen from the expressions above (aa, *n)1/2 will decrease rapidly at high that the real amplitude'lal_= frequencies. Differences in density between the seeding and the fluid can also cause settling, or upwards drift of the particles. For small particles the velocity
Gravity
drift is by this the expression, given of V
'CP Pg (PP PO) 181, to
(2.6)
is due to gravity. -Again, unless pp and po differ acceleration where g However, be the settling velocity will negligible. a settling significantly flow is but the that velocities, not significant with velocity compared which 25
in the difficulties detected, be required maintaining cause may can still duration for density the of an experiment. ,-i, seeding . is diameter This turlýulence Spatial averaging of will occur'ifthe particle " I filtýringfluid, in dimensions the the out than a causing 'larger a of eddies a1l for important ' the'majorThis will noiý'be scale motions of very s'm 'of 'expeniments but may become significant'in small scale studie's ity of turbulence.
Other effects such as the Magnus effect and Shear,floW liftingforce, are due.to but in large the particle spin and effect of motion velocity gradients respectively their effect 4 small in non-extreme flows even compared-to Inertial and,gravity diamFor, the in measurements effects., waves pollen particles with an average eter of 5014mwere used. When introduced into the flow, the particles remained for hours several. suspended without, visible settling. This suggests that the rein the the relatively low accelerations in waves will be very particles of sponse. good.
2.1.2
Light
scattering
efficiency
Light scattered from particles moving within the illuminated region of the flow is The imaged the camera optics and recording medium. mean collected using onto film intensity image is 17 is the B Tt the mean plane and exposure on where =. , t is the exposure time. The time t. should be short enough to freeze the motion E flow that the the the within must particles exceed required while exposure of to generate an, image density, after development, that is greater than the gross fog level of the film [341. Thus, , successful recording of the flow field-will be dependent upon the intensity of the particle images and the threshold exposure Bý required to record an image. The light intensity I reaching the film will be illumination intensity, the function the the of particles, efficiency of scattering a the solid angle through which light is collected (lens f number) and any light losses associated with propagation through the fluid and past flow boundaries. 26
laser due limited the is to The illumination intensity cost and capabilities of light sources and is further reduced when the beam is expanded to form a sheet.' The scattering efficiency of particles is very low perpendicular to the incident light, Kerker,, 1969[75]'and is dependent upon the shape, size, and material of important is Selection therefore the seeding. very of seeding size and material if the scattered light is to be sufficient to expose the photographic emulsion desired be image is the to forming that consistent with small enough an while accuracy., A study of the scattering power of spherical seeding particles in a 0.2 to 15 um has by Yao, following been Adrian diameters 1984[5] and simipublished range of lar studies for Laser Doppler measurement. Calculations of scattering efficiencies based Mie droplets, were on scattering calculations of polystyrene, glass and oil intensity increased for that the indicated mean rapidly scattering particles and diameter
-3,pairs existing within probe area as a function of displacement image image density. and particle
36
lower Ior D diameter less the than particle be especially probe significantly will image concentrations. A further consideration when deciding upon the pulse separation T is the maximum resolvable spatial frequency for a probe diameter D and a transforming lens of focal length fj[85]. Figure 1.1 illustrates the generation of displacement fringes in the Fourier plane fringes by, is flow The the given record. separation of of a multi exposed
A. f, df = MvmasT
(2.12)
A. is fj is focal length the illuminating beam, the the the wavelength of of where lens, M is the magnification of the image recording optics, and V,,.. the largest The fringe diffraction larger df be than the velocity. expected period must limited point responseof the optical system which is given by, do =
3.83Aft
(2.13)
rD
If df ý: d, then TMv,,,.: 50.8D
(2.14)
indicating that the ma](imum particle image separation should be less than 0.8 times the probe diameter. The Probability of obtaining more than three particle times 50.8 the diameter is 0.13,0.45 and 0.73'for an average of probe at : pairs 5,10 and 15 particle images per interrogation spot respectively. Thus, for particle less 0.8D, diffraction fringes be in limited than the resolved a can separations but image the 3 probability of complete obtaining particle or more pairs system is low. In most situations therefore, the maximum measureable particle image be less between ie. 0.5-0.8D. 0.8D, than significantly will separation
2.2.2
Exposure
time
The exposuretime used for recording particle imagesis chosenso that the mo. tion of the particles in that time is very small. This ensuresthat the particle is 37
that been blurring. has It velocity measureshown recorded precisely without in but-results long times[85] particle exposure ments are possible with relatively images that are streaked in the direction of the flow. Such measurements will be due be to precision limited flow the the complicated will and P analysis record of of the blurred particle images[111]. The use of a pulsed laser avoids any difficulties (30-80ns) is that the the time this pulse motion of as sort sufficiently short of high Only but is the time. velocity particles pulse extremely all arrested within laser is CW does The become the time variable used a parameter. pulse when a laser discussed CW is from illumination in the next a generation of pulsed planar beam is described. This that a system a scanning and method employs section of illumination allows a very short pulse time; zý31is while maintaining a high intensity. Pulse times of this duration are typically short enough to freeze particles travelling at up to several m/s.
illumination
2.2.3
Multiple
exposures
When more-than two exposures are used to record the flow the correlation in the direction of the flow is significantly increased. Thisis seen most vividly in the increased visibility of the Youngs' fringes that can be produced from PIV flow photographs which have more than two exposures. (see figure 2.8). This fringes the of comes about because of the presence of harmonic terms sharpening from the correlation between the first and third image, the second and generated fifth image etc. and also because of the density of image pairs recorded on the This aspect of the recording process is considered in greater detail in increased 3 the chapter where correlation in the direction of the flow is shown to result in increased reliability and accuracy. film.
38
Figure 2-8: Youngs fringes generated from a multiply exposed flow phot9graph.
2.2.4
Dynamic
range of velocity
measurement
In most applications it is important that the measurement process is capable of evaluating an extended range of velocities. The velocity dynamic range is defined as, V, nam -
Vinin
(2.15)
VMin
According to the discussion above the maximum measureable velocity may be be, to estimated 0.6D (2.16) jW-Tand the minimum measureable velocity is that which corresponds to a particle by ie. being image and separated centres are separated one particle no more pair image diameter d,. Vinin
d. "ýýI VT-
(2.17)
dynamic defined is the range as above and so (0.6D AV ,:ts d, 39
(2.18)
Thus, for a probe diametýr of 0.75mm and particle image diameter of 20,um Av ý- 20.
40
2.3
Illumination
systems
The scattering efficiency of seeding particles perpendicular to the illumination is typically quite poor and combined with the limited sensitivity of high resolution high intensity light source to photographic emulsions necessitates'the use of a flow field. light Lasers have been the record a most common source succesfiAly because highly intense beam in directional the the they propast of and used duce. However, even with the light power available from a laser the scattered light reaching the recording medium is still of very low intensity. Expansion of the laser beam into a sheet of sufficient width to illuminate the region of interflow the further within will est reduce the intensity by two or three orders of CW laser is used the useful light is further diminished and when magnitude, a by the short pulse times required to record the particles without blurring.
The following section considers the expansion and modulation CW light of a for illumination of a region covering several thousandc7n2 source and velocities 'm. The 9-1. use of pulsed laser souceshas been avoided in the work of several; describedhere due to the difficulties encounteredin their use. Specifically, alignment of optics, safety, and the requirement that a single light source should be fleidble be to used for a large range of velocities, and measurement sufficiently areas. For the measurementsdescribed in Chapter 6 it Was necessaryto illuminate an area.within the wave tank 50-70cm wide. -The most commonly employed method of producing such a light sheet is to use an arrangement of cylindrical lensesto expand the beam in one dimension. Temporal modulation of the beam is performed either by a mechanicalchopper or an electro-optical device. Figure 2.9 illustrates a typical beam expansion system incorporating these methods. The limitations of such a system become apparent when high velocities, large be both to are recorded. areasor A 25jum particle moving at 1 m/s will require a,pulse time of less than 10 lis to
41,
Figure 2.9: Beam expansion and modulation using cylindrical lenses and a me. chankal chopper
42
times 100 is This diameter. its less half its about to than motion own confine Providing shorter than the fastest shutter speed available on most cameras. It high. is intensity light B= It then requires that the very adequate exposure limited light be that the source of seen maximum measurement area, with a can increases. is in that the region reduced as maximum expected velocity power, Also, the chopping of the laser beam at a low mark-to-space ratio means that fraction An being is light the alternative, used. of available only a small power is field flow illuminating a'planar region within a more efficientý method of described below.
2.3.1
Scanning
beam illumination
The illumination system describedhere was developedprimarily for the recording of flow fields under laboratory generatedwater waves. However, the principle , flows be illuminated. to technique the a wide range permit of will efficiently of High. intensity, short duration light pulses are imparted to the flow field by interest. laser beam The seedingparti. through the of region rapidly scanninga des within the plane of interest are illuminated by the very high intensity laser beam for the time that it takes the finite width of the beam to pass a single Figure 2.10 illustrates the configuration of such a system set up to particle. illuminate water waves. The scanning frequency is the time between pulses for the chopped system. Assuming that the exposure time t is short enough to freeze the motion of the 'region flow the exposure for a-chopping system s=**Iar'to particles within the that iflustrated in fig. 2.9 is given by. CIt
(2.19)
is intensity I the average where of the expanded sheet of laser light, t is the C is time and a constant which accounts for the fraction of the incident exposure light that is scattered and imaged for a given magnification and lens aperture. The exposure from a scanning system illuminating 43
the same area with the same
0
Figure 2.10: Scanning beam system incorporating rotating octagonal mirror and dish parabolic colUmating
44
is by, given pulse sepaxation
EI = CIlt'
(2.20)
is beam that intensity the time the t' F isihe of unexpanded and average where it takes the beam to pass any point within the measurement area. The intensity F is by, laser beam given of the unexpanded
I'I;: e.Uld.
(2.21)
L is the width oi the measurementarea, and d, is the diameter of the scanning beam. t've- Td. 1L
(2.22)
T is the scanning period and the time between pulses for the chopped system Rt where R is the ratio of the pulse separation to pulse duration ie. ie. T E' can be expressed as R= Tlt. _Therefore
E'::.,,,CI(Lld, )t(d. IL)R = CItR
1(2.23)
So, for the same laser power and the same area of illumination the exposure value is increased by R over that provided ky a modulated, expanded beam system. Meynart and Lourenco, (1985)[94] quote 20 as being a minimum value for R. Therefore, the scanning system offers the equivalent of at least an order of maglight illumination power over useful a conventional chopped more nitude system. The effective exposure time is also reduced in most cases ie. if (Lld. ) >R then for laser has the the option of measuring over Therefore, t. t' < same power one higher higher larger the velocities as as well permitting and/or area, use of a film thus smaller and more accurately measureable paxticles. and resolution
The most obvious drawback of this system of illumination is the extended time taken to record the whole flow fiel4. The scanning method of illumination is focal different to image the plane shutter a where Parts of are exposed analogous depending horizontal different times in illumination the the upon position at large When the the is of speed shutter compared to the speedof the not area. be in final the there stretching a will or present compression recorded object image. The sameis true for the scanning beam. If the scanningfrequency is too 45
low then the image of the flow field will be stretched or compressedcompared to the flow itself, depending upon the direction of the scanning beam relative to the direction of the flow. The expression that relates the positions of the illuminations particles at consecutive
becomes, XI(tl)
V,, =
-
tl -
X2(t2)
t2
(2.24)
(XI
t2=
X2)1 andX2(t2) 21to the + tj LY ti L&, +Iand to are x x1(ti) where = -Iilluminated the is to the time within particle area, an measurement of of positions the beam initially entering the measurementarea at x=0 and V is the scanning (2)
frequency. The mean measurementtime t =(tl + t2)/2 = to + -L WI+WI 2v + -L Lv horizontal the to the The position measurement within according varies area. between for taken time the time beam the to cross exposures and approximate the'flow field is 11v =T and so it can be seen that this effect is going to be be timescales T is to to typical small compared chosen of the flow. The small as time taken to doubly expose the whole flow with a scanning beam is 2T. So, does flow in 2T difference long the the between not change significantly as as measurementsusing a conventional system and a scanning system will not be significant. It can I also be seen that the time between exposures t2 - tj 02) depends upon the horizontal distance travelled by the particle between exposure's (01 - 02). The horizontal component of motion of the flow field will AX enlarge distance beam therefore time the the has to travel to arrive that and or reduce back at the same particle for a second exposure At
L
Ax Lv
(2.25)
by, is Ax here given vy
Ax = V.At
(2.26)
is the in the the time between exposures average velocity of v. particle where At. Therefore,
At =
v(l + v. lLv)
46
(2.27)
The horizontal and vertical components of velocity are given as, Axv Ax Tit =V = (1 AxIL)' -, a,nd
AY At
AYV VV=
(1
.AX + (L - Ax)')
(2.28)
(2.29)
if Ax known both is in be calculated exactly It can be seen that v. and v. can final imposed the However, the direction. upon error velocity and magnitude is is For* if the very not calculated small. velocity a corrected measurement length horizontal 60 the 1.5Tnm displacement scanning a over of cm, of particle is 0.25%. the < error magnitude of
2.3.2
Apparatus
is implemented beam using a rotating mirror and a parabolic The scanning system Physics laser is Spectra 171 Argon The Ion which used a mirror. recollimating directed beam is The 15 onto the rotating watts. has a mwdmum. power of laser beam through the an accurately reflects which repeat. mirror octagonal beam is The then the into the deg 90 sweep of angular reflected up arc. able by the its the focus mirror with rotating parabolic mirror area at measurement beam the within velocity scanning measurement area. constant to produce a , (Lincoln Lasers octagonal is a commercially purchased unit The rotating mirror by low dc its a vibration/high replaced motor speed relatively slow mirror) with The rotational speed of the motor and therefore scanning 2.11). (figure motor by input digital the is altering voltage and measured using a controlled rate from the a phototransistor receiving a signal positioned at oscilloscope storage (sýe figure 2.12). the mirror parabolic end of The parabolic mirror is milled from a solid piece Of aluminium. and given a by, bonding length long highly reflective coating of perspex that has been front a The is by Aluminium. surface generated silvered evaporation of silvered with Aluminium, within an evacuated evaporation chamber and produces a reflec47
I
F%b-
Figure 2.11: Octagonal rotating mirror mounted below the wave tank and at the focus of the parabolic collimating mirror
Figure 2.12: Oscilloscope trace showing the phototransistor signal from which the scan rate v is measured. 48
tive surface of approximately 91-92% reflectivity for wavelengths of 450-550 Mn Kingslake, 1965[76].
49
2.4
Imaging
Recording and
Accurate measurement of particle displacements rely upon precise imaging and lens the The image formed by be ideal of particle positions. would recording an an exact point-by-point reproduction of the object which in our case is a random distribution of tracer particles moving in the plane of a narrow measurement vol(figure 2-1). Deviation ume of the actual particle image shapes or positions from the ideal axe due to diffraction of light at the lens aperture, aberations due to the limitations of lens design and manufacture [76], or errors in lens construction. The precision with which the diffraction and aberation degraded image is recorded is further compromised by the limited spatial frequency response, nonlinearity and noise associated with the recording medium.
PIV measurementsare generally performed using a photographic emulsion with either a celluloid backing for most 35mm or 120 formats or glass when a plate camera is used. Photographic recording is used in preferenceto other media such as thermoplastic films or Video (Vidicon or CCD) as it offers the best between compromise sensitivity, r6solution, and cost. However, a fundamental disadvantageassociatedwith photographic recording is the time imposed penalty by the developmentprocess.The useof a Video camerawould provide the spatial information in a form that can be digitised for processing on computer very but limited is in quickly resolution (typically 512*512pixels) restricting its useto Films areas. very small suchas Kodak Holographiccan resolveapprox. 1250line pairs/mm while remaining relatively sensitive. However, short exposure times illumination limited intensities generally makes the use of more sensitive and films necessary. This is not such a limitation as it may seem since modem films such as T-MAX 100/400 provide good sensitivity 100400 ASýLwith good lines/mm, 125-200 depending resolution on contrast.
50
2.4.1
Lens evaluation
The suitability of a lens for recording the positions of point images such as seed. ing particles is determined by its spread function (the luminance blur associated (deviation image distortion image the the of a point source) and of overall with the geometry of the image compared to the flat object field). The point spread function represents the fundamental limit on the ability of the lens to reproduce fine detail and for high quality lenses is ultimately limited by the effect of diffraction at the lens aperture. The use of larger apertures reduces the effect diffraction for lens in and a smaller point source image. will result of a perfect However, the limitations of the optical components become significant in the non-wdal regions of the lens used at small f nos. and aberations will begin to distort and widen the Point Spread Function (PSF). A comprehensive overview of procedures used to test objectives such as camera lenses is given by Shannon, 1969[76] where the effect of chromatic, astigmatic and spherical abberations on the point response of a lens are illustrated and techniques capable of identifying them described. A detailed analysis of the effect of lens ab'eration', is not taken but it is here important to recognise that lens :!. on win compromise ns -r-%';., the accuracy of velocity measurementsmade with PIV.
Particle
imaging
considerations
The accuracywith which the separationof a particle image pair can be measured is limited by the sizeof the particle imagesthemselves.In generalthe position of image is less large particle a precisely measurablethan that of a small image. It is important therefore that the imaging lens system and photographic emulsion the is to particle record positions capable of resolving particles that are used be to desired the consistent accuracy. Diffraction at the with small enough lens aperture and aberations associatedwith the lens will tend to spread small features of the image over a larger area. The level of aberations present in the image of a given lens will tend to be a function of its cost. Sophisticated design
51
that optical components and manufacturing processes are capable of producing lens the limited diffraction the most aperture represents and so are very nearly fundamental limit on the resolving power of an optical imaging system. by (OTF) factor the function is Optical transfer The -applied weighting complex a lens system to the spatial frequencies present in the object plane and decreases lens does beyond frequency, limiting the to zero, past a not transfer any which information. This is consistent with the blurring of small image detail as higher lost due image in frequencies form the to plane are sharp edges required spatial to the upper limit of the OTF. The decreasing response of the lens system used
for the measurementsdescribedin Chapter 6 is shown in figure 2.13. This shows the Modulation Transfer Factor, which is the magnitude of the OTF, for three different spatial frequenciesacrossthe image plane 6 and shows a clear decrease in the responseof the lens to higher spatial frequencies. The effect of limited frequencyresponseof a lens in the imaging of small point sourcessuch as seeding be can more easily seenby consideringthe point spread function (PSF) particles is Fourier the transform of the OTF and represents the responseof the which imaging system to a point sourcein the object plane (ie. impulse response), T(k., ky) = JI(S(z y)) 0
(2.30)
) T(k., k, is OTF the where of the lens system, S(z, y) is the point spread func. tion and .17represents a Fourier transform.
diffiaction limited is lens PSF Airy inten. the the a aperture with a circular For distribution, sity I(r)OC
(2.31)
r is I(r) intensity image the distance from the the of a centre where of a point r first is J, length Bessel focal f is lens, funcion, the A the the a order of source. light lens the the of and a aperture. radius wavelength of For a point source in the object plane the intensity distribution in the image diffraction limited lens be to that in figure of a similar represented will plane 6TheMTF is shownto vary with imageheightie. atance from the imagecentre. 52
Modulation transfer T as a function of image height u Slit orientation tangentialsagittal T f-number k 1.0.
2.8
0.8
Wpile light Spatial frequencies R-
0.6
19 cycleS/mm 20 cycles/nun 40 cycles/mm
1. MTF Diagrams The Image height u- reckoned from the Image center - is entered in mm on the horizontal axis of the graph. The modulation transfer T (MTF - Modulation Transfer Factor) Is entered on the vertical axis. Parameters of the graph are the spatial frequencies R In cycles (line pairs) per mm given at the top right hand above the diagrams. The lowest spatial frequency corresponds to the upper pair of curves, the highest spatial frequency to the lower pair. Above each graph the (-number k Is given for which the measurement was made. "White* light means that the measurement was made with a subject Oumination having the approximate spectral distribution of daylight.
OA 0.2
10
20
30
40 u [MM]
Figure 2.13: Modulation transfer function for Zeiss Planar T* f2.8 80mm lens
53
\
-----
Figure 2.14: Airy intensity distribution or point spread function for a diffraction limited lens
54
it Therefore, image the 95% the central spot. 2.14 where resides within of be function to diffraction the the is reasonable to consider spread radius of This the first between the distance the when occurs minimum. the centre and limited imaging diffraction For function= 3.832. Bessel the system a argument of is dd diffraction diameter M the the given spot of a magnifcation with operating by[71, (2.32) dd = 2.44(l + M)f #A
f#=f where
/a is the f number of the lens.
Any real image, which can be considered to consists of a large collection of point function. Point light, be the image ideal the with spread of will convolved sources The particles used to seed the flow are small and so the PSF will affect the final image relatively more than the imaging of a larger object. This win enlarge the imaged particle and the diameter of a particle image can be approximated by[7], d.
FM
2 U2d!2+d dd2, + p
(2.33)
dp is the diameter. particle where
Smaller seeding particles will begin to closely represent point sources of light and consequently reducing the size of the seeding much past the dimensions of the PSF will not reduce the diameter of the image but will only reducr. the average intensity of the irtiage.
Image fleld distortion The illuminated region over which velocity measurements are made consists of imaged lens the thin which using a with a seeding particles arc itab within a depth of field that extends through the thickness of the measuring volume. The from be image ideally to the such related a set-up should measurement recorded factor M In is by the the of recording which a scale magnification optics. area is by image distortion. between the complicated and relationship object practice
55
Figure 2.15: Image distortionjeft,
baml positive pincushion, right, negative
If one considers a light beam entering a lens obliquely from an object point to one side of the lens axis the principal ray of the beam will pass through the image plane at height H' (where image height is defined as distance from the For imperfect lens this will differ from the ideal image height axis). an -optical Hol. The difference between HI and HO'is a measure of distortion usually given HO' is of and often expressed as a power series. as a percentage Ho' = aII03 + bHo'6+---
(2.34)
0
A positive value of distortion leads to an image of a square that is pincushion barrel distortion large, too produces a whereas negative shaped and everywhere is (figure 2.15). too everywhere small shape which Point measurements made from a film which exhibits distortion will not corimage desired Al if it to the is that and object relates point assumed respond fields. If the precise nature of the distortion is known then a scheme to relate from film the the to measurement area within measurements points exact point This devised. PIV be the analysis of will records considerably complicate can lens has low is to to the uge a which acceptable solution a problem and a more between image/object distortion level that the points can of mismatch enough be neglected. 56
V Distorlionin % of imageheightu 2.0
1.0
0
3. Distortion Here again the Image height u is entered on the horizontalaxis in mm. The vertical axis gives the distortion V in % of the relevant image height. A positive value for V means that the actual image point is further from the Image center than with perfectly distortion-free Imaging (pincushion distorlion), a negative V indicatesbarrel distortion.
1.0 .
-2.0 -
10
20
30
40 u [MMI
Figure 2.16: Distortion in % of image height
The measurementsdescribedin Chapter 6 were made using a Zeiaj Planar T* f2.8,80mm lens. Figure 2.16 showsthe distortion in % of image height[3] and showsthat the furthermost comers of the image plane for a 120 negativesize is approidmately -1.5% of the image height ie. 39.95mm*-1.5%=-0.6mm.The % distortion reducessmoothly to zero at the centre of the image so, for central image but be for the the positional accuracycan good enough all very regionsof high accuracy measurements.This lens also has an extremely flat image field loss focus image. the the of avoids of which at outer regions
2.4.2
Recording
medium
The photographic emulsion used to make the wave measurements was T-MAX (56.5*56.5 This film is film format 120 loorl] modern characa mm2). on L terised by its good sensitivity (100 ASA) and resolution (200 lines/mm at 1:1000 for PIV is highly medium and as such measure. a recording suited as contrast)[1) high imaged is for resolution required precise recording of particles ments where images to that are successfully recorded ensure and good sensitivity required despite low image intensities and short exposure times. 57
.
KODAK T-MAX 100 Prolosslonal
Film 16052
lis
1.16
is
I ag
-1.16
-LIS -AVILINOTH mQuirso I* omduce so"*"
deowly
Figure 2.17: Spectral sensitivity of T-MAX 100 emulsion (Taken from Kodak T-MAX technical specifications). Emulsion
sensitivity
The Argon Ion laser used for illumination in the scanning syst'em described above has a maximum ukeable power output of 15 Watts over 8 lines from 514.5 nm457.9nm. Figure 2.17 shows this to be well within the main region of spectral in it is difficult the As this to of chapter sensitivity emulsion. mentioned earlier calculate the scattering efficiency of the randomly shaped and sized pollen grains illumination intensity the so required to give well recorded particle images was found experimentally by exposing a number of frames at different illumination intensities and then inspecting the developed images under a microscope -to determine the particle images with the best contrast. It should be noted here that the correct illumination intensity for successfull recording of images also depends upon the scanning rate of the illumination system and the image object Thus, for different two these of configurations variables it is magnification.
58
KODAK T-MAX 100 Professional
Film I SCIS2
150
Modulatlon-Transfer Function
100
Thesephotographic modulation-transfervalueswere determinedby using a method similar to that describedin ANSI Standard P112.39-1977(RI984). The film wasexposed with the specifiedilluminant to spatially varying sinusoidal test patternswith an aerial imagemodulation of a nominal 35 percentat the Imageplane. with processingas Indicated. In most cases,thesephotographicmodulation-transfer valuesare influenced by developmentadjacencyeffects,and are not equivalentto the true optical modulation-transfer curve of the emulsion layer in the particular photographic product.
50 30 20
I
10
3 2 123
45
IF-W
60
100
200 300
SPATIAL FAEOUENCY (cyclowmmi
Figure 2.18: Modulation Transfer Function for T-MAX 100 (taken from Kodak
data sheet). for in to illumination intensity the to these adjust account variation neccessary 0
parameters. $,
i,
Spatial resolution 0
for different image differs film the contrasts, of being somewhere about 200 lines/mm for contrasts above 1: 1000. It
The exact spatial resolution the limit
like lens from figure 2.18 film in that the MTF be the seen shown can -there is an upper limit on the detail that is resolvable and which will produce an impulse response function similar to that exhibited by the lens. It is important therefore to consider the effects of the limited combination.
resolutionýof
the lens/emulsion
It has been proposed [9] that an approidmate value for the recorded
59
image diameter may be calculated from, di
(M2
dP2
+
d2 d
+d
2)112 f,
(2.35)
be function diameter the d, is the the may of emulsion and point spread of where (i. 51im). line is just 1/200mm by that the = resolved e. width of a approximated This is an approximate relationship and the contrast dependant, resolution of the further. images broaden to tend particle emulsion will
Adjacency
effects
A further limiting factor on the accuracy with which partical images may be is due due Adjacency to uneven to effects arise recorded adjacency effects. development of the latent image stored on the exposed emulsion and is caused by the development inhibiting agents produced as the developer becomes"spent" Obviously, developer will become exhausted more rapidly in regions up. or used latent image have been highly exposed and if new developer does not the that of diffwse into these regions development will slow down compared to those regions image that low-exposure film. be from to It are close of seen can regions of figure 2.19 that in the case of two closely situated particle images the region between the images has strong development followed by less rapid development is developer cqnsurned. The outside edges of the two images are close to as developing low development at a greater rate. regions of and so will continue The net result of uneven development in this particular situation would be to from drift image the away centres each other as cause an apparent of particle strong development on the outside of the pair produces more silver grains and therefore greater photographic density. The obvious solution to this problem is to ensure that the developing solution is constantly in motion about the film developing be is This by tank the achieved constant agitation of which can during development. This speeds up the overall development of the film and be in development for the total has to compensated with reduction a slight so time. Normal procedure is, repeated inversion of the tank for 5 seconds every 30 secondsfor 8 minutes with the developer at 20deg C. Experience has shown 60
uneven development
fresh developer
exVauslid d velo,
fresh developer
W. 4 CD ". 4 0
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144
Figure 6.8: Frame "2", 10.425 seconds.
145
Figure 6.9: Frame "4", 10.475 seconds.
1-16
Figure 6.10: Frame "6", 10.525 seconds.
147
Figure 6.11: Frame "8", 10-575 seconds.
148
6.3.1
Velocity
measurements
Each negative was analysed point-by-point over an area 49mm horizontally and 35mm vertically on a 1mm 2 grid. The resulting values were scaled using M displacements film from T the the to give velocities and then measured on and Chapter Missing in 5, to the erroneous velocity values. as remove all edited, data interpolated then the values where possible and values were replaced with transfered to the mainframe to be plotted.
149
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difficult became displacements more the the particle, smallest wave of regions to measure reliably. The pulse separation used to make the measurements was the in the were wave, that to of upper region velocities ensure sufficiently small the lower in This the particle that wave of-the regions means measureable. displacements will not be sufficiently large to allow the two particle images to be resolved from each other., Above the still water level velocities were reliably measured right up to the A the useful number of velocities were also measured water. upper surface of light that It is the the sufficient was present surprising wave. spout of within laser. illumination had to thewater the the this exit since wave region of within just ahead of the wave and then re-enter the water, within the spout without becoming diffused or refected in other directions, Iný the later sets of measurements (frames 6,7,8, figures 6,.18,19,20) it can be ,, just the is lacking, that in measurements. the region ahead of wave crest seen Inspection of the corresponding photographs of the negatives from which the measurements were made show this region, to be obscured by reflected light. This may be due to reflection from the upper surface of the wave which is directly. above this region. It is possible that repositioning of the camera so that the region of interest within the wave is in the centre of the frame may solve this Alternatively, the use of a polarising filter may allow such extraneous problem.
be to reflections excluded from the film.
-
It is possible that greater detail may be obtained within the crest region of the by larger using a wave magnification. However, this would exclude most of the the of wave. For the measuring system in its present form-it would be up rest to the user to decide, based on the nature of the velocity data required,. what magnification and therfore area of the wave is to be recorded.
159
Comparison
of results with
numerical
model
A numerical model of wave breaking was available for comparison with exis fully The non-linear, timenumerical scheme a perimetal measurements. stepping model developed by Dold and Peregrine[38] at Bristol University and Skyner hardware Edinburgh, to et al, 1989[114]. Inviscid, adapted run on at incompressible and irrotational flow is assumed in the model and calculation of the fluid motion evaluated at discrete points along the water surface. Comparing the results of the experiment with the numerical model is mainly intended to show that PIV can give sensible measurements through an extreme difficulty The fundamental in such a comparison is achieving an wave profile. between laboratory the match numerical and accurate wave through breaking. The numerical model requires the wave height and velocity potential to be defined in space at a particular time while the experimental wave is generated by a drive timeseries fed to the wavemaker which is 2 meters from the measurements Relating the two waves linearly is difficult as the wavefield is highly region. in its non-linear propagation upto and through breaking. The development of for scheme a accurate matching of numerical and physically generated waves is beyond the scope of this thesis but will form a large part of future work.
Hereit Will sufficeýo showthat the velocitiesmeasuredfrom the laboratory wave and calculatedfrom a similax numerical waveare reasonablyclosein magnitude form. and The numerical model was run in a simulated water depth 53cm of which is the laboratory the as same wave and given a starting surfaceprofile derived from the wave gauge record shown in figure 6.2. The numerical wave was then allowedto propagateuntil it reachedthe limit beyondwhich the calculationsbreak down. This was done severaltimes, adjusting the componentsof the numerical field wave eachtime, so that the numerical.wave would not break prematurely and would develop a wave profile that looked similar to the laboratory wave. The complexnon-linearity of the breaking processmeant that, using this rough 160
0.00
139.
-------
62.00-
---
....
-------
--- -----------------------. ........ .. ..... .......
........
I
.........
........... .................
.........
-93.00-
........ ........ ........ ......
......... ......... ....................... ........
ST 170. . ...
-248.0(Jj
........... ....................... ....................... ....................... 2051.00
.............. ..........
.
..
..
.
. . .. . . .. . . ..
.. .. ..
. . .
......... ....... ..... ...........
2163.50
.
2276.00
I.....
2388. SO
. . ... . . ... . . ... 2501.00
Figure 6.21: Surface profile and internal velocities generated from time stepping
numerical model. it scheme, was only possible to get a very approximate temporal and spatial between the waves. Figure 6.21 shows the velocity distribution and surmatch face profile of the numerical wave at the point at which it most closely resembles
one of the measuredframes (frame 7).
161
261'
230.00
E 150.00
A HExperimental
values
70.00
still
water. level
-10.00
-90.00
numericaI
scheme
-170.00
-250.00' 0.00
11 0.4&0
Horizontal
0.60
1.20
component
--- -IIII 1.80
2.00
of velocity
2.40
M/S
L
Figure 6.22: Horizontal components of velocity for laboratory wave (frame 7, ) and numerical wave on a vertical line behind the wave spout. Horizontal components of velocity vertically down through the wave just behind the spout were plotted for the laboratory and numerical waves and are shown in figure 6.22. The match between the two sets of data is surprisingly good, especially around the still water level. The magnitude of the velocities through the wave profiles is consistent with each other and the form of the velocity distributions are close but with a noticeable deviation in the upper regions of 162
the wave. Inspection of the laboratory wave measurements show a slight amount of randomness but the error baxs, determined from the considerations discussed in Chapter 4, do not look unrealistic when compared. Further comment is hardly justified due to the loose criteria used to match the waves. These initial results indicate that there is considerable potential in the use of PIV for water wave studies. The large area over which the velocity measurements were made will ensure that measurements that can be realistically scaled to offshore dimensions. jhe accuracy, of the technique appears to be good, both from the estimated error bounds and the smooth progression of the measured in figure is hoped 6.22. It that a more precise match between shown velocities laboratory waves and numerical waves may be achieved so that the numerical be-verified scheme can more fully. This would allow velocity measurements, from laboratory obtained waves beyond breaking, to be used as a starting point for extending theoretical schemesbeyond the limit at which they presently break down and ceaseto give realistic results.
0,
163
Chapter
Additional
7
applications
PIV of
The main interest in PIV, within this project, was its application to water waves. One of the major features of this technique however, is its versatility. The same flow field the to under water waves can be easily adapted equipment used record to record the motion of other flows. It is merely a matter of re-assessing the recording parameters such as illumination intensity, pulse separation, duration and the optical magnification so that they satisfy the required accuracy and resolution sought by the user. All PIV negatives can be evaluated in a similar manner on the analysis system described in Chapter 3.
This flexibility is illustrated in this two descriptions brief other of chapter with applications of PIV (Vortex shedding and Acoustic streaming). The work is prelin-ýnary in nature and therefore limited in Howexperimental sophistication. ever, the potential for development is illustrated by the results and lays the
for groundwork more detailed work.
164
7.1
Surface
flow measurements
One of the limitations associated with PIV is the need for a high intensity light source (usually a laser). The Argon Ion laser used in the wave studies is easily the most expensive component of the system. This section describes nonlaser recording of two-dimensional vortex shedding from a cylinder being towed through a tank of stationary water. The light source in this caseis a pair of ordinary photographic flash lamps which behind illuminate the cylinder. A clean, film to the the water surface of are used free, water surface ensures that the surface motion of the water represents the 2-D motion through the water tank. Seeding of the surface then allows a PIV form flow be to the the to taken need record of without an intense laser sheet for illumination of a plane within the fluid. The axrangement used is shown in :figure 7.1. The cylinder is supported from the trolley and mounted vertically in the water. The trolley itself runs on six sets of sealed precision bearings which carry it along two polished sets of rails mounted along either side of the tank. This ensures the trolley moves along the tank smoothly. The camera is mounted on the trolley and is fitted with a motor wind so that a number of frames can be taken on each run.
This whole approachassumesthat the vortex wake behind the cylinder is twodimensional in the plane perpendicular to the cylinder and thatthe surface is motion the same as the motion of the water within the body'of the-fluid further down the cylinder. Studies by Saouthi and Gerrard; 1981 [115) showthat the vortex-wake behind the cylinder can be rendered almost completely two-dimensionalby minimising end effectsat the lower end of the cylinder next to the base of the tank and where the cylinder emergesfrom the water. If the between the bottom of the cylinder and the base of the tank is not less gap than approximately 1/6 of the cylinder diameter the vortex wake becomesthree dimensional and the vortices taper off towards that end. The dirt film that collects on still water after several hours also affects the vortex wake near the
165
Stepper motor and gearing
C&mera 7f Fo-rr Cylinder mount Pulley
puey
Trolley
LIS
A Aluminium
foil reflectors
2iv-
rIA
B
Cylinder
rain hole
Figure 7.1: Towing tank with surface flow measuring arrangement mounted Flash trolley. lamps on are mounted below and to one side. The light above burst is reflectedand diffused from the crumpled Aluminiurn foil along the sides Cleaning detergent tank. is by introducing the the of water surface achieved of A floating Steady then detergent dirt B. motion of at end off and excess at end the trolley is ensuredby the spring damped steppermotor drive.
166
0
Aluminium foil refle
Figure 7.2: Configuration of towed cylinder, camera'and flash guns. The cylinder is moving into the page.
f
11
167
dirt film the This the the surface motion of at constrains water water surface. and will ultimately
damp all motion completely.
Both of these problems can be solved fairly easily. Precise positioning of the limits bottom bottom is the the gap, small and end near effects cylinder ensures by introducing detergent Surface the the onto, minimised cylinder. effects are of dirt floating the the tank off at the other end through of and water at one end "clean hours before holes. This for surface will persist a state" several overflow inhibits forms film the tank. the the water of at of and motion surface surface The water surface is seededwith the same pollen grains used in the wave studies. Seeding is distributed evenly over the surface by sprinkling with a fine sieve to distribution The light the even on water. and of extent to which particles give a the seeding affects the surface motion is not known and remains an area for
further research. The results presentedbelow show that the measuredsurface is is known with consistent what and suggests that the influence of the motion dirt film. is the minor compared extremely with pollen Background motion of the water is minimised before an experimental run by for to hour the water settle an or so after cleaning the surface. There allowing is always a certain amount of background motion in the tank due to convective These be the heating but it is currents can with reduced use of wires currents. felt necessary at this stage. a complicated procedure and was not The illumination system, which is based on dual flashlamps, is triggered by the delay box. The initial electronic via an shutter contact is used to trigger camera the first flash with the second flash triggered by the timing electronics. The flash guns are mounted on the trolley and point across the water surface to the side of the tank where crumpled aluininium foil is used to diffuse the light burst to illumination more even a of the seeded water surface, figure 7.2. In order give final base the image the the contrast of maximise of the tank was painted matt black and the flashguns aimed at an acute angle to the water surface.
168
7.1.1
Results
The cylinder diameter used in the experiment was 9.5mm and towed at a speed The 76. double Reynolds 8mms-1 number of exposure photograph gives a of figure 7.3 0.203 in taken was with magnification of a and a pulse separation shown by The 120ms. estimated magnification was accurately photographing a piece of floated transparent photocopied onto acetate sheet paper and carefully of graph below The is the the approximate camera. pulse separation surface water on indicated on the pulse delay box and a precise measure of the delay obtained digital for the and storage oscilloscope phototransistor arrangement used using 4
the wave measurements. The PIV negative was analysed on the analysis rig and the resulting velocities in figure The 7.4. is arrows shown vortex vector as structure not evident plotted from this representation as the flow was photographed from the trolley. The
behind the that travelling different shed are cylinder are at a vortices velocity to the trolley and in order to reveal the vortex structure it is necessary to subtract the difference between the velocity of the trolley and the vortices. This is shown in figure 7.5 where the regions of positive and negative rotation can be seenquite clearly. Calculating the vorticity distribution from the velocities shown above in figures 4 and 5 allows the vortex structure to be seen clearly.
7.1.2
Discussion
The completeness of the data shown in figure 7.4 above shows that extremely flow be from the surface flow arrangephotographs quality can obtained good For here the there were only 18 spurious values velocities of array shown ment. identified illustrating the extremely high quality of photographs that can be taken using this technique. As to the validity of the measurements however, it is less easy to say. The structure of the flow clearly shows the vortex street that 169
Figure 7.3: PIV negative of the flow behind the towed cylinder. left to right
and the cylinder
is just outside
1-10
the photograph
The flow is from
on the left hand side.
--*
----*
----
----
----
----
----
---
b --.
0 ---+
---+
---+
-+
--b
---+
---. + ----+
---+
-.
+ --b
--4
K ---+
---+
---qb
---b
. 001 ooo,
om
--4
ý --+
---)
11-4
--
lomm. lomms-1 N
Figure 7.4: Velocity measurementsfrom PIV floýv negative of the vortex wake behing a 9.5!nm diameter cylinder towed through water at 19.5degC at 8mms-I (Re =76).
,
171
---+
-0
o.
00,
t
lomm
--------------------lomms-I
L
Figure 7.5: Velocity distribution
figure but in 7A difference the with shown
between the velocity of the trolley and the vortices subtracted in order to reveal the vortex wake.
172
---. b
i
$ ...
0 ---4
---+
----#
---
---4
--+
---+
---4
--+
---4
--f
+
--*
--4
---+
---4
--4
-+
--+
-ý-+
--b
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----
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--4
--4
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--4
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+ --4
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-+
-4
Figure 7.6: Vorticity distribution derived from the velocity measurements above. The vorticity distribution is independant of the frame of reference and would be the same if the velocities from figures 4 or 5 were used.
173
---$
--#
--+
---*
is well known for Reynolds numbers in this range. However, the relationship between these measurements and the vortex structure within the body of the fluid cannot be known at this stage. This is an area that wiH require further flow be if is this two-dimensional to taken a method of recording any research further. If further research shows that the motion of the surface in such flows is reprefluid flow body the the this technique could provide a the within of sentative of lot of valuable experimetal data. The high quality of the photographs that can be taken and the reduced cost of putting together a measuring system without incentives for in PIV this laser context. good using are a
I
174
7.2
PIV
measurement
Acoustic of
Streaming
Acoustic streaming is the generation of non-zero mean flows by a sound field. The phenomenon typically arises due to attenuation or dissipation of a sound field in its medium of propagation or, when a solid wall is present, in the boundfor [83]. The layer the transferring attenuation mechanism momenprovides ary tum into the body of the fluid and thus generates the streaming.
Severalarticles considerstreaming [?,20] but as a whole the subject has been intensities before because is high This are neccessary possibly acoustic neglected. Additionally, difficult it is to takes place. particularily measurewith streaming its instruments due the to type measuring creating own measuring probe probe boundary layer and upsetting the flow under investigation. Consequentlymost [20]. is based flow on visualization experimentalwork The work that is described in this section is the result of collaboration within the Fluid Dynamics group at Edinburgh and is more fully documented from [109]. dynamics in Sharpe, 1988 fluid thesis the point of view appropriate a . The Streaming under investigation is known as Rayleigh streaming and was because it is be relatively understood, well can easily generated and considered the streaming velocities independently estimated by measuring the pressure of the sound field[109].
7.2.1
Rayleigh
Streaming
This form of streaming was first explained by Rayleigh himself in his theory of Sound [1031and occurs due to the viscous attenuation of the sound field in the boundary layer when a solid wall is present. The most common configuration in which to observe this effect is when a standing wave is set up in a circular tube. The form of the streaming is evident from the PIV photograph in figure 7.7 with the direct current, along the walls, always pointing towards the velocity 175
Figure 7.7: PIV negative of Rayleigh streaming within
a glass tube
nodes and the return current, along the tube centre, towards the antinodes. For a standing wave of the form
U(x, t) = U(x) sin wt
(7.86)
The volocitv -it the wall of the tube is given by Rayleigh's Law as,
dU(x) U. (x) = -3 W-lu(x) dx 4
(7.87)
It is then possible to calculate the velocity at any point of the tube perpendicular
to the axisi.
(1 U.
2)
2,9
(1
a2
Where a Is the tube radius 'This
neglects the boundary
distance is the and s layer.
176
from
the tube centre.
-88)
Figure 7.8:, Arrangement used to record Acoustic streaming.
7.2.2
Experimental
Apparatus
The figure is in 7.8. for the shown streaming The experimental setup recording laser 32 high that low and a power was not neccessary very velocities meant very (A flow. In illuminate the this the to 633nm) laser case He-Ne was used = mW by form then to laser from chopping the pulsed a sheet and beam was expanded & focus 1. A beam of the with a rotating mechanical chopper positioned at C2 lenses C1 lens I two to cylindrical are used and and spherical of combination into introduced the tube light was approximately form the which, when sheet behind 1 An 1/2 thick. high shutter controlled positioned 2cm external mm and frame than the and was used camera rather exposures the number of on a given film The the T to in at plane. period minirnise pulse was vibration order shutter the scope and storage arrangement. usual photodiode using measured The camera used to record the flow was a Nikon 35mm. SLR with a 55mm flat 177
focus lens. Image distortion due to the lens was minimal comared to that due to the curved tube walls and this was estimated by photographing a regular grid been had introduced into Subsequent the tube. measurement with a travwhich distortion from that negligible showed was apart aproximately elling microscope imm from the tube wall. Flare from the wall precluded measurement in this it introduce to was not neccessary any corrections. region and so The film used to make the measurements was T-MAX 400 which has adelaser for the power used and also exhibits good resolution quate sensitivity (; ý-,1001ines/mm depending on contrast). Recording the flow field was always done in complete darkness to maximise the contrast and therefore resolution in the final image.
7.2.3
Measurements
and Results
With the apparatus arranged as in figure 7.8 sound of frequency 2460 Hz was introduced into the tube (length 450 rnrn, internal diameter 23.3 mm) using. a horn loudspeaker. The tube was terminated with a rubber bung having a metal firmly it. With to this arrangement the sound field corresponded attatched plate to the 7th normal mode of the air column.
An independant check of the streaming velocities was provided by means -of pressuremeasurementswhich were made using -a probe microphone inserted through the rigid end of the tube (Bruel & Kjaer type 4166). Tobacco smoke was used as seeding and was introduced into the tube through the hole normally occupied by the probe microphone. The sound field was switched on and the streaming left to develop and mix the seeding. In this case It was possible to estimate the maximum velocity by eye'which allowed the pulse duration of the chopper to to be set to suitable values (0..114 and separation and 0.0057 seconds respectively). The flow was photographed with four pulses near a velocity node. Individual smoke particles can be seen quite clearly and are not 178
This is because but images the vibrational as point as streaks. rather recorded large in to the quite used generate are streaming and places are as amplitudes large as the particle displacements themselves. Certain dificulties were encountered in the analysis of the flow negative due to the large vibrational amplitudes mentioned above. Each particle image on the flow negative is recorded over 15 or so periods of the sound field and because the positional probability distribution of a particle in the sinusoidal sound wave is double peaked the particle images are dumbell shaped. The length of each dumbell image is proportional to the amplitude of the sound field which varies down the length of the tube ie. large at the antinodes and small at the nodes. This produces a non-circular diffraction halo in the Youngl fringe pattern, figure 7.9, whose ovality depends strongly upon the vertical position of the PIV negSubtraction interrogated. halo from individual being an average shape of ative fringe patterns prior to transformation to the autocorrelation plane was found in the to the desired fashion. Instead, it central correlation suppress peak not found halo introduced that the subtraction of average actually unwanted was figure R(s), into 7.10. This problem was solved in rather an ad hoc way. peaks The analysis program was instructed to aquire its "average halo" for a given from fringe that measurements patterns themselves. Each fringe row of row of in had different fringe frequency Tow, general, within a a pattern and orientation but a similar halo function and so by averaging all the halo's in one row it was possible to come up with a relatively smooth halo shape. Performing function that this then allowed the displacealong subtraction with row pedistal be high detected to peak at a proportion of the measurement ment correlation points[1111. Erroneous measurements were identified and removed and the missing values interpolated to form the smooth flow field shown in figure 7.11. The measurements look slightly asymmetric with the velocities in the up per left vortex seeming to , sweep over to the right. It is possible that this was caused by outside air curhave to which were observed rents quite a profound effect on the stability of the 179
Figure 7.9: Typical fringe pattern from the flow negative in figure 7. streaming.
It was neccessary to lag the sound tube with insulating
that warm or cold draughts would not upset the streaming.
material
so
Even the warmth
of a nearby hand had an observable effect. From equation 7.88 it can be seen that the axial velocity should be parabolic across any section of the tube with the maximum velocity given by equation 7.87 and zeroes at distances 0.707a from the axis of the tube.
Axial velocities were taken from the data for three
lines separate across the tube a few mrn from the velocity node (see figure 7.12). The solid parabolae were fitted using the measured velocity theoretical
zero points.
maxima
and the
The fits are quite good though the velocities do show
deviation some near the left crossover point, reflecting the assymetry mentioned The maximum
7.87 calculated using equation was and a earlier. slip velocity be This dB 6.5rnm/s. 151 for to the : ýý would make value of maxiinuin pressure the maximum axial velocity a few min from the velocity node 3-4rams-1 which is in aggreement with figure 7.12. The cross sectional velocity measurements agree with the theoretical
curves to within
about 10% in the region of the central
return velocity but deteriorate in the outer regions.
180
I
Figure 7.10: The two central peaks in the autocorrelation plane shown above from from fringe diffracti the subtraction average a on'halO pattern of an resulted halo in different wa-s significantly shape. whose
181
% 4t
0.6
Figure 7.11: Velocity map of the streaming at standing wave node.
182
7.2.4
Discussion
It has been shown that PIV can be used to successfully measure acoustic streaming. This would not have been possible with conventional probe devices and Laser be Doppler Anemometry the that made with unlike measurements could PIV measurements were made "instantaneously" over a whole field. To obtain an it be have LDA to would a twoneccessary equivalent set of measurements with it LDA traversing to system move across the system with a precise component flow field and a carefully controlled environment to prevent external conditions from upsetting the streaming over the time required to make the measurements. There are difficulties and limitations associated with the PIV technique however. A large out-of-plane component present in the flow being measured can result in
being introduced into the in measurements errors significant and extreme cases causing such decorrelation in the final flow record that no measurementsare More importantly in this casethough is the effect of the vibrational possible. displacementon the velocity displacementmeasurements.The streaking of the particle imageslimit the smallestparticle displacementsthat can be resolvedand so reducethe -rangeof velociesthat can be measured; The analysisof the flow photograph was also complicated considerablyby the elongation of the particle images. A way around this problem would be to pulse the laser illumination at the same phase position of the sound field for a sufficiently short time to freeze the motion of the particle. For the frequency of the sound field used here this would require lasting a pulse only a fraction of a millisecond and would then require laser power of several watts. Successfull,implementation of this idea would depend upon precise synchronisation of the illumination system with the sound field. This is not inipractical and this research is likely to be pursued for sound fields at ultrasonic frequencies.
183
Figure 7.12: Experimentally measured a3dalvelocities in the tube (symbols) and theoretical curves.
Diagram 7.12 taken from Sharpe et al, 1988,[1111-
184
Chapter
8
Discussion
Conclusions and
PIV the investigate to this the The main aim of project was possibility of using technique to make full field velocity measurements*under breaking waves. This has been done and the main components of this work are summarised below. 1. An initial literature search gave an indication of the extent to which par-, tide image velocimetry has been developed and put into practice. A brief, resume of the most important pieces of work is given in Chapter 1. , 2. From background reading and experience based on initial experiments the photographic recording process is considered with regard to the flow char* acteristics, seeding, illumination, recording medium, optics and exposure parameters. A novel PIV illumination system based on a rapidly scanned laser beam is introduced be is to significantly more efficient and shown than the more conventional pulsed illumination arrangement based on exbeam. laser CW pansion and modulation of a 3. Evaluation of PIV negatives is discussedin Chapter 3. The automatic analby basis the PIV point is point a over on area ysis of negatives achieved the local that mean velocity is flow It within the each shown of negative. , from the extracted be reliably autocorrelaanalysis area can accurately and distribution. The intensity image local autocorrelation tion function of the digital techniques. optical and function is obtained efficiently using 185
4. A 'preliminary investigation of the accuracy associated with the above methods of recording and evaluating the flow field is given in Chapter 4. The factors that limit this accuracy are identified and discussed. It is individual from that shown an example wave record can be made with accuracy approaching 1%.
velocity measurements
5. The evaluation of the velocity data from the, analysis rig is considered. The identification of spurious data values is discussed and the procedure for the filtering out of these values described. Interpolation and calculation is also considered. vorticity of 6. A wave measuring facility. base&on the foregoing techniques is described deepto to applied Water when a plunging give sensible results and shown breaker. The versatility of PIV is illustrated with two further applications flow to the technique surface and acoustic streaming. of
From the results of the studies that are summarised above it is believed that PIV has been shown to have considerable potential for future studies of water wave breaking. The wave recording and analysis system that has been developed in the course of this project has the potential of providing the basis for an ongoing program of work that would concentrate'on both the "application of the measurement syIstem t6 various wave types and the continuing development of Additionally, it has been'shown that as well having be'applied for PIV significant as potential wave measurements can in .a It is wider context. anticipated that further development and commercialisation
the measuring techniqu*e itself.
imaging lead it being in the to widely technique particle will academic of used industrial research. However, the n'eed'for'good quality optical access to and the flow being studied and high power laser light sources will necessarily confine PIV to being a research tool for use in a laboratory environment.
186
8.1
Limitations
It is apparent, from research described in this thesis and from the published limitations field, there in that this associated are certain other work of workers for limitations is One fundamental PIV. the the a certain need of most with flow before information the serious measurements can amount of a-priori about be made.
1. It is neccessary to know what the maximum velocity will be. 2. If no facility to resolve directional ambiguity is in corpor ated into the measurement system then the basic structure of the flow must be known so that the velocity vectors may be interpreted correctly. 3. The smallest scales of motion present within the flow must be known so that a suitable combination of photographic magnification and probe beam diameter can be used to resolve these small flow structures in the measurements.
Associated with the latter
of these points is the limited
spatial resolution
of
measurements made from a flow negative. It is neccessary that, for a successful measurement, thb probe spot covers three or more complete particle pairs so that the flow particle displacements can be unambiguously
identified
among all
the random connections between particle images. Increasing the seeding dendecreasing the particle displacement would allow a smaller probe beam sity and diameter to be used leading to greater spatial resolution. However, the upper limit of particle concentration
before speckle is formed in the image plane' and
the smallest measureable particle separation are themselves limited
by the di-
ameter of the particle images. Smaller particles will form smaller images down lin-,Lit of the camera lens and the recording medium. These factors therefore represent the practical limit on the resolution and accuracy to the resolution
'see Chapter 2
187
diffraction lower be lens the Large to PIV. used apertures can achievable with limit of the lens system but their effect is constrained by the limitations of lens design and manufacture. (Holographic,
Higher resolution photographic emulsions are available but are of very low sensitivity (1 ASA). This then 1250 lines/r=)
requires higher illumination
intensities and although extremely high power lasers
hazards begin to prohibit the associated cost and are not uncommon Reducing the total measurement
their use.
increasing the magnification and area
of the
imaging system will perrrdt an increase in the measurement resolution and accube but the over which measurements neccessarily reduces area can made. It racy be for therefore, that seen a given configuration can
lens/recording of
resolution,
be there will power a range of measurement areas be but that scales will and of motion measureable which cannot be maidmised seeding size and illumination
simultaneously.
A further limitation I that was mentioned earlier is the requirement that there be optical accessto the flow for introducing the illumination sheet and imaging the measurement area. Thus, certain measurements will require the construction of special transparent models e.g. acoustic streaming in musical instruments. Furthermore, the measurements described within this thesis are limited to. two dimensions and although sterio-photography techniques have been used [36] the third component is limited in accuracy.
188
8.2
Future ,
work
Particle image velocimetry
is still at a relatively
early stage in its development
its is likely future it 'How'that cover all aspects operation. will of and research in in'the this project that would there undertaken are research ever, several areas benefit significantly
from some further work.
The analysis rig described in Chapter 3 could be improved in several respects. 1. An improved method of removing the central peak in the autocorrelation plane of the image density at each probe position., At, present, incomplete cancellation of this central, peak limits the smallest measureable particle displacement and compromises the reliability of the search for the' displacement. correlation peak. It is possible that subtraction of the central correlation, peak directly from the autocorrelation plane may lead to a more robust method of resolving small particle displacements. 2. The approach to identifying the displacement correlation peak could be improved. This peak is located with a simple ma, 3dma search but could be improved by a more sophisticated search algorithm that considers neigh. bouring values as well as the averageýbackground value.
I An overall improvementin the efficiencyof the softwarecould lead to a significant decreasein computation time per measurementpoint. A floating point co-procesiorand translation of the presentinterpreted languageinto a compiledlanguagecould easily result in an'Orderof magnitudereduction in analysistime. ' 4. Registration of individual frames is, at present, done manually, and is rel. ativly crude compared to the rest of the film interrogation process. An automatic film registration system which could level and position the negative at its origin could be developed which would form the basis for an analysis rig capable of analysing whole films without user intervention. Combined with an increase in the overall computation speed it would be 189
flow to 'ensemble or automatically make averages',of a compute possible flow. a of measurements at a number of phase positions
In, Chapter 5 it was shown how spurious data points are identified and removed by means of an interactive vector editing program. In order to remove the predjudicial influence of a user and to speed up the operation for very large data for identifying be formahse it the to anomalous procedure will neccessary sets data points. Local comparison routines are uselesswhen there is a large proportion of erroneous values. It is felt that a more reliable approach based on the identification of the most probable coherent structure from the data present and then removal of those that are inconsistent with this structure would mimick the by Additionally, human data is the there the correlaof a observer. evaluatý6n tion strength Cf which is shown to be related to the confidence attributable to a measurement and which could be used as a further indication of the reliability of a given measurement. Two fundamental sources of random error were identified in Chapter 4.
Errors due to random correlation noise. IIe Errors du to the discrete and random manner Iin which the continuous' flow field is. represented by particle image pairs on the PIV negative.
A formal treatment of the uncertainty associatedwith random correlation noise is neededso that the random errors associated,with the flow recording process be Additionally, the measurementerrors due to the finite range evaluated. can displacements further the of particle probe area require work. This treatwithin ment would relate the randomnessof the measurementsto the seedingdensity and sizeand the nature of the flow within the probe region. The simplest caseof dimensional Chapter is 4 the a one velocity gradient consideredat and it end of is proposedthat more complexsituations such as higher non-linear flow profiles and turbulence be treated in a similar manner. 190
.1
Appendix
A: Youngs
fringes
Consider a region of film with N particle image pairs recorded randomly over it be identical images to the and circularly symmetrical. where particle are assumed The photographic
density distribution
be by, can represented
N
[8(r - rl) + 6(r - r, - dj)]
h(r) g(r) =
(. 89)
h(r) represents a single particle image, 8 is the Dirac delta function and * where is a convolution operator. The particles originally at rl have been_displacedby di where all the values of d, for the N particles within the region have a small amount of, scatter about a mean value due to random particle positions over a spatially varying flow field ie. turbulence or velocity gradient. With the film positioned in front of a converging lens and illuminated over the region of interest with a beam of spatially coherent light (from a laser) the optical Fourier transform of g(r), will be present in the back focal plane of the lens [51] (see figure 3.6). and is given by, N
+ exp-jq(pl("I+dl))
1
(. 90)
where F. T. denotes a Fourier transform, A is the wavelength of the coherent illumination, and rf the position in the Fourier plane of the lens. This expression neglects a phase factor, but is of no consequencewhen the intensity is detected. Thus if k= ! intensity detected the in Fourier the is by, Mrf, plane given Xf . JýF.T. fg(r)]
12
giving N
S(k)
(exp-j"I
+ exp
-jk(ri+di))
I:
N=,
m
(exp jkr,.
+ expik(rm+dln))
(. 92)
where the first and second bracketed terms are also denoted by BI and B,*,, 12 [h(r)] S(k) 1, T. finally F. respectively and which gives, = I(k) = S(k)
(EI=1(2
+2 cos kd, ) 191
+EN
(BIB, * + BI*B, 1
