flow visualization and image analysis

FLOW VISUALIZATION AND IMAGE ANALYSIS

FLUID MECHANICS AND ITS APPLICATIONS

Volume 14

Series Editor: R. MOREAU MADYLAM Ecole Nationale Superieure d'Hydraulique de Grenoble Boite Postale 95 38402 Saint Martin d' Heres Cedex, France

Aims and Scope of the Series

The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role.

As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modelling techniques.

It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advancement. Fluids have the ability to transport matter and its properties as well as transmit force, therefore fluid mechanics is a subject that is particulary open to cross fertilisation with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains.

The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of a field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list oJ related mechanics titles, see Jinal pages.

Flow Visualization and Image Analysis

edited Ьу

F. Т. М. NIEUWSTADT Laboratory [or Aero and Hydromeclzanics, Technical University о[ Del[t, The Netllerlands

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Library ofCongress Cata1oging-in-Publication Data

Flow visualization and image analys1s I ed1ted Ьу F.T.M. Nieuwstadt. р. ст. -- (Flu1d mechan1cs and its applications ; v. 14)

ISBN 978-94-010-5191-0 ISBN 978-94-011-2690-8 (eBook) DOI 10.1007/978-94-011-2690-8 1. Flow visualization. 2. Image processing. 3. Fluid mechanics

-Data process;ng. 1. Nieuwstadt, F. Т. М. (Frans Т. М.), 1946-II. Ser1es. TA357.F544 1992 681' .2--dc20

ISBN 978-94-010-5191-0

Printed оп acid-free paper

АН Rights Reserved © 1993 Springer Science+Business Media Dordrecht

Origina11y published Ьу Кluwer Academic Publishers in 1993 Softcover reprint ofthe hardcover 1st edition 1993

92-32223

No рзrt of the materia1 protected Ьу this copyright notice тау Ье reproduced or utilized in апу form or Ьу апу means, e1ectronic or mechanica1, including photocopying, recording or Ьу anу information storage and retrieva1 system, without written permission from the copyright owner.

Table of Contents

Preface vii

R.D. Keane and R.I. Adrian Theory of cross-correlation analysis of PIV images

S.B. Dalziel Decay of rotating turbulence: Some particle tracking experiments 27

H. Stapountzis, J. Westerweel, J.M. Bessem, A Westendorp and F.T.M. Nieuwstadt Measurement of product concentration of two parallel reactive jets using digital image processing 55

H.A Siller, R.J. Perkins and G. Janke Image analysis of oil film interferometry - a method of measuring wall shear stress distributions 71

L. Lourenco Recent advances in LSV, PIV and PTV 81

P. Guibert, Q.C. Duan, M. Murat and J. Julien Development of particle image velocimetry: A new computation method with direc-tional resolution 101

A Chavez and F. Mayinger Algorithms for automatic measurements of size and velocity of spray droplets from holography reconstructions 117

J. Massons, Jna. Gavalda, J. Escoda, X. Ruiz and F. Diaz Characterization of Savonius rotor wake using image analysis processing techniques 143

V. Baier, W. Bechteler and S. Hartmann An application of image processing methods to determine the critical shear stress in sewer systems 159

M.P. Arroyo and C.A Greated A three dimensional particle image velocimetry system and its application to the measurement of acoustic streaming 167

e. Hugi and A. Mueller A camera for measuring density, size and velocity of rising air bubbles and water velocity in a bubble plume 189

D.R. McCluskey, e. Elgaard, W.I. Easson and e.A. Greated The application of PIV to turbulent two-phase flow 207

P.A. Quinn, D.I. Skyner, e. Gray, C.A Greated and W.J. Easson A critical analysis of the particle image velocimetry technique as applied to waves 227

J. Stefanini, G. Cognet, J.e. Vila, B. Merite and Y. Brenier A colored method for PIV technique 247

AK. Hind Digital PIV applied to flows around artificial heart valves: Analysis by autocorrelation 259

PREFACE Image analysis as measuring technique in flows

Progress in fluid mechanics depends heavily on the availability of good experimental data which can inspire new ideas and concepts but which are also necessary to check and validate theories and numerical calculations. The usual experimental probes, such as the pitot tube, the hot wire but also the more advanced laser-doppler equipment, can be characterized as point measurements, i.e. they only give us information in a single spatial point in the flow field. Although useful, such information is at the same time quite limited especially when flow phenomena are dominated by spatial structures. A primary example of the latter case is turbulent flow. One may even venture to say that these limitations of experimental tools have hampered the progress in this field.

With the advent of new recording and image analysis techniques new and promising eX'perimental methods in fluid flows have presented themselves ,vhich are able to obtain spatial information. In some aspects these techniques are almost oldfashioned because they use the well established methods of flow visualization. Howev~r, the progress lies in the fact that now we are able to obtain quantitative information from these visualizations by applying the methods of image analysis. Examples are the rather newly developed techniques such as particle tracking velocimetry (PTV), particle image velocimetry (PlV) and laser induced fluorescence (LIF).

Within the regular series of Euromech colloq uia a meeting was organized to discuss the rapid developments in this ne\v field of the application of image analysis techniques to flow measurements. The meeting was held at the Delft University of Technology from July 2 until July 5, 1991 under the title Euromech 279, Image Analysis as Measuring Technique in Flows.

At this colloquium invited lectures were given by experts in the field on various specific topics. In addition shorter presentations were given by the participants on their recent results. We feel that some of these presentations are interesting for a wider audience, because they offer up to date information on this rapidly developing new experimental field. Therefore, participants at the colloquium were invited to submit their presentations for publication.

All submissions were subjected to the usual reviewing procedure. As an outcome of this procedure four papers have been selected to be published in an issue of Applied Scientific Research. These papers are also thought to represent characteristic examples of various techniques. The other papers, which certainly contain interesting and new material on various aspects of image analysis, have been collected together. In this proceedings the four articles mentioned above have also been reprinted so that a rather complete overview is given of the present state of the art on the use of image analysis techniques in flow measurements.

vii

F.T.M. NIEUWSTADT

Editor

Fig.7. Representative steps of the image processing of a single pulsed holo

gram of the RIl3 spray.

1) Original image, 2) smoothing and gradient extraction and 3) hi·

narization. The right picture column shows an enlargement Al of

the droplet zone of picture (1), its noise filtering and the final droplet

identification. (p. 134)

posilon downstream from grid/rum

.... ,.UoI\1 .. bo •• ,~ ru, _ ,~ ru. _ ,~

~ - ,." =- ~ ~- = =- ~ 0.01 - 0.G3

-om - om -0.03 - -om -0.06 - -{I.03 -{1m - -{I.06 ~ -~

-{Ill - -o.oIiI -{I.IS - -{I.U -o.~ - -0.13

.OM.., I(1 t..low -Ol~

Figure 6: Vorticity map of a the flow field shown in Figure 5. (p. ~ 16)

TRIPLETS PHOTOGRAPH

TRIPLETS PHOTOGRAPH

Ifg.7a (p . 256)

fig. Sa

(p. ~57)

Theory of cross-correlation analysis of PIV images

RICHARD D. KEANE & RONALD J. ADRIAN Department of Theoretical and Applied Mechanics. University of Illinois at Urbana-Champaign, 216 Talbot Lab., 104 S. Wright Street, Urbana, IL61801-1793, U.S.A.

Abstract. To improve the performance of particle image velocimetry in measuring instantaneous velocity fields, direct cross-correlation of image fields can be used in place of auto-correlation methods of interrogation of double- or multiple-exposure recordings. With improved speed of photographic recording and increased resolution of video array detectors, cross-correlation methods of interrogation of successive single-exposure frames can be used to measure the separation of pairs of particle images between successive frames. By knowing the extent of image shifting used in a multiple-exposure and by a priori knowledge of the mean flow-field, the cross-correlation of different sized interrogation spots with known separation can be optimized in terms of spatial resolution, detection rate, accuracy and reliability.

For the direct cross-correlation method of single-exposure, double-frame systems which model video array detector interrogation and of double-exposure single-frame systems which generalize earlier direct auto-correlation methods of interrogation of photographic recordings, optimal system parameters are recommended for a range of velocity fields in order to eliminate signal bias and to minimize loss of signal strength. The signal bias resulting from velocity gradients in auto-correlation analysis can be eliminated in cross-correlation interrogation by appropriate choice of the optimal parameters. Resolution, detection rate, accuracy and reliability are compared with direct auto-correlation methods for double- and multiple-pulsed systems.

Key words: PlY, cross-correlation, auto-correlation

1. Introduction

Particle Image Velocimetry (PIV) uses images of marker particles in a fluid flow to measure instantaneous velocity fields in experimental fluid mechanics. In general, the particles are illuminated by pulsed sheets of light at precise time intervals to produce images that are recorded on photographic film or on a video camera array. The analysis of these images to measure the particles displacements is a key element in the PIV technique and many approaches have been proposed and explored experimentally. In the high image density case, the concentration of particles is large so that it is natural to measure the average displacement of local groups of particles, as opposed to tracking individual particles. This case will be the principal topic of the present work. The preferred forms of analysis of high image density images are based on correlation techniques that are performed numerically on digitized image data, numerically on the digitized Young's fringe patterns that result from optical Fourier transforming of the image data, or in a purely optical correlation processor. Each of these methods is basically similar so it suffices to discuss the first of the three.

In previous work, two variations of the PIV technique have been studied theoretically and by numerical simulation. Adrian [1] and Keane and Adrian [6] analyzed double-exposure single-frame PI V, in which an individual frame of photographic film or of a video camera array is exposed to a double light pulse. This is the

F. T. M. Nieuwstadt (ed.). Flow Visualization and Image Analysis, 1-25. © 1993 Kluwer Academic Publishers.

2 R. D. KEANE AND R. 1. ADRIAN

most commonly used PlY recording technique. Keane and Adrian [7J analyzed autocorrelation of multiple exposures on a single frame. Examples of this approach are the experiments on fluid flow behind an impulsively started circular cylinder by Lourenco and Krothapalli [l1J and Meynart, Simpkins and Dudderar's [12J study of an unsteady descending plume in a convection cell. More recently, Cenedese and Paglialunga [4J used mUltiple exposures to study vertical structures in a mixer and Arroyo, Yonte, Quintanilla and Saviron [3J investigated Rayleigh-Benard convection using a number of exposures ranging from 13 to 40.

An alternative to single frame recording using either double or multiple pulses is to record images on multiple frames using video cameras or high speed cinematography. Although cinematic recording offers higher resolution than digitized video camera data recording, the precision needed for frame registration and the relative ease of video camera recording have made video recording an attractive alternative. Other methods of obtaining single images include the use of two different laser wavelengths and two color recording of consecutive exposures of a double-pulsed system onto color film and the subsequent use of color filters to retrieve first and second images [5]. Adrian and Zoltani [2J obtained separate image frames on adjacent halves of a single video array by changing the polarization of successive illuminating light sheets and using a polarization sensitive prism assembly to separate the image fields. In these cases, it is natural to evaluate the image displacement between pairs of frames by cross-correlation. Spatial cross-correlations of pairs of single-exposure frames have been used by Kimura and Takemori [9J to analyze flow around a circular cylinder, and Willert and Gharib [15J have applied a similar procedure to analysis of a vortex ring experiment. Goss et al. [5J also used cross-correlation of pairs of frames of colorfiltered images to study a jet diffusion flame.

Furthermore, cross-correlation can be applied with some advantage to the analysis of double- or multiple-exposure single-frame images by cross-correlating two different regions on the same frame. Turbulent open channel flow has been investigated using this approach by Utami et al. [14].

In multiple-exposure, single-frame photographic recording, a number of pulsed illuminating beams forms light sheets of variable center and thickness within the flow field in order to record multiple images of the particles in the light sheets, on a single photographic recording. Knowledge of the flow field enables the locations and thicknesses of the illuminating sheets to be chosen to enhance successive particle image recording. In a study of the turbulent characteristics of droplets injected into a pipe flow, Lee et al. [lOJ located multiple illuminating light sheets perpendicular to the direction of mean flow with spatial separations based on the axial velocity of the droplets in order to record multiple droplet images on a single exposure. The interrogation is then carried out by illuminating two small interrogation spots on the single frame, centered at Xl and X2 and with diameters dl and d2 , with interrogation beams of intensity 111 (X) and I dX) respectively as shown in Fig. 1. With singleexposure multiple-frame PIV, the images of seeded particles within successive light sheets are recorded on each frame. The interrogation spots, identical in location and

ANALYSIS OF PIV IMAGES 3

~ I LIGHT SHEETS

Illl .. ··· ... ······< lJ .. : .:.~~~~..... MEASUREMENT VOLUMES

2

IMAGE PLANE

Fig. 1. Light sheets and image recording system for planar pulsed laser velocimetry.

size to those above, are then illuminated by interrogation beams. The displacement of the particle image AX(X l ) is determined and scanning Xl and X2 over each frame produces displacements over the entire image plane.

If the images are recorded on a digitized video detector array, instead of a photograph, the interrogation spots can be defined by choosing digital window functions with shapes corresponding to III and 112 , to multiply the data in the array detector. The image displacements then yield the measured in-plane velocity for each interrogation spot centered at Xl'

The present work is an extension of that reported by Keane et al. [8] and Keane and Adrian [7]. Its purposes are, firstly, to develop a theoretical description of the mean value of the cross-correlation function of particle image fields for the case of locally linear fluid velocity variation. Secondly, the properties of the cross-correlation function as a means of estimating the particle image displacement are explored using Monte Carlo simulation. As with earlier auto-correlation methods, the performance of this PIV method is determined by the spatial resolution, the detection probability

4 R. D. KEANE AND R. J. ADRIAN

and the accuracy of the in-plane velocity measurements. Unlike auto-correlation analysis, the spatial resolution is limited by the maximum spatial separation of recorded particle images as well as the size of the first interrogation spot and is assumed here to be the greater of these two values. The detection probability is defined as the fraction of interrogation spots that prod uce velocity measurements that satisfy certain interrogation criteria; and the accuracy is defined as the error of all detections that satisfy the interrogation criteria, whether they are valid or not, with respect to the true velocity field. These three parameters are affected by the experimental configuration, the recording medium, the interrogation procedure and its detection criteria so that optimization of PIV performance necessitates an understanding of all these facets of the technique. Finally, the cross-correlation procedure is compared to the auto-correlation procedure for double-pulsed and mUltiple-pulsed single-exposure recording.

2. Interrogation by spatial cross-correlation

Following the notation and theory developed by Adrian [1], the transmissivity of a distortion-free nearly paraxial photographic recording of a single exposed flow field with illuminating light sheet intensity 101 (x), sampled at time t is

(1)

where To is the transmissivity of a particle image and where x;(t) is the location of the ith particle in the flow field at time t. The transmitted light of the photograph after interrogation by a light beam of intensity 111 (X - X 1) centered at Xl with diameter d 1 IS

(2)

Similarly, for a second recording of the flow field sampled at time t + At, we have

(3)

and

(4)

For a double exposure of the flow field, with linear, unsaturated recording, the transmissivity of the photographic recording is additive:

(5)

ANALYSIS OF PlY IMAGES 5

Then the transmitted light intensities after interrogation, I I(X), 12(X), are given by equations (2) and (4) with r replacing r I and r 2 respectively.

In PIV the two-dimensional spatial convolution of I I and I z with separation vector s

R(s) = fII(X)Iz{X+S)dX (6)

is used to determine the image displacement. It approximates a true spatial crosscorrelation to the extent that the area integral corresponds to the ensemble average of I I (X)I 2(X + s). For brevity, it is conventional to refer to the estimator in (6) as the cross-correlation, and we shall continue that practice here.

If the images are recorded on a video detector array instead of a photograph, the interrogation spots can be defined by choosing digital window functions, with shapes corresponding to 111 and I l2 to multiply the data in the detector arrays. Thus, the formula defined above applies to both photographic recording and to video detector array recording.

Following previous work [1,6], it is convenient to decompose the estimator for cross-correlation of single-exposure frames into three components

(7)

where Rc is the convolution of the mean intensIties, RF is the fluctuating noise component of the correlation estimator and RD is the displacement correlation which gives the image displacement. The equations for these components are presented in the Appendix. In contrast, either the cross-correlation estimator or the autocorrelation estimator of double-exposure frames consist of five components: Re, R F , RD+, RD- and Rp where RD+ and R D- are displacement peaks which are reflectionally symmetric for auto-correlation analysis and Rp is the self-correlation peak or "pedestal" (Adrian [1]). When using cross-correlation of single-exposure frames there is no necessity to employ image shifting to resolve directional ambiguity, and there is no correlation of particle images in either exposure with themselves. Thus the Rp component which occurs in the correlation of multiple-exposure frames is absent from (7). The dynamic range of the cross-correlation estimator of singleexposure frames is larger than that of the auto-correlation estimator if the latter is formed without image shifting. Furthermore, as this cross-correlation estimator does not include any contribution from particle images in a given frame correlating with other images in the same frame, the number of random noise peaks in crosscorrelation analysis from (Rc + RF ) in single-exposure frames is approximately half of the number of peaks produced in cross-correlation and auto-correlation analysis of double-exposure frames.

The components of the spatial cross-correlation function for a pair of singleexposure interrogation spots with 10 I = Ioz, 111 = h2' Xl = Xz and d I = d I = d2 are


R

(b)

ANALYSIS OF PIV IMAGES

, I I

I II .11 iii :11

7

R

(e)

Fig. 2. (a) Cross-correlation function R of image transmissivity I(X) for single-exposure frames where N/ = 15, /';.X/d/ = (0.10,0.10). (b) Auto-correlation function R of image transmissivity I(X) where N / = 15, /';.X/d/ = (0.10,0.10). (c) Cross-correlation function R of image transmissivity I(X) for double-exposure frames with optimal window displacement /';.X = X z - Xl where N/ = 15, /';.X/d/ = (0.10,0.10).

illustrated in Fig 2a for a constant in-plane velocity field and for which each interrogation spot contains randomly located particles for which N[ = 15. For the corresponding double-pulsed exposure, the auto-correlation function and the crosscorrelation function with X2 = Xl + AX are illustrated in Figs 2b and 2c, respectively.

As for auto-correlation function analysis, the mean image displacement across a given interrogation spot is determined by locating the centroid of R D , namely

(8)

from which the mean velocity is estimated as f.1D/MAt, (M is the magnification).


It can be seen from Fig. 2 that the approximation of locating the peak in RD(S) by locating the peak of R in the s-plane of the cross-correlation function of singleexposure frames will be more accurate than a similar procedure for the autocorrelation function as the random noise peaks from (Re + R F) are approximately halved in single-exposure analysis and Rp is absent there also. In addition, the crosscorrelation function for the double-exposure frames contains a taller peak of RD + than auto-correlation when the displacement between frames is optimized, as in Fig. 2c.

3. Mean cross-correlation

To compare the cross-correlation function R for single-exposure double-pulsed systems with the auto-correlation functions for both double- and multiple-exposure systems which have been analyzed in earlier works [6,7J, the statistical properties of the cross-correlation function need to be determined. An ensemble of pairs of single-exposure recordings and corresponding double-exposure recordings of identical velocity fields is used in which each realization contains different sets of randomly located particles. For a given velocity field u(x), the conditional average <R(s)lu) calculated over random particle locations measures the mean behavior of R and its components, given the flow field u.

The conditional average of R for single-exposure systems is determined by conditionally averaging equation (7). It has the form

<R(s)lu) = <Rds)lu) + <RD(S)lu). (9)

The form of the conditional average for the cross-correlation function for doubleexposure systems is identical to that for autocorrelation, namely

<R(s)lu) = <Rds)lu) + <Rp(s)lu) + <RD+(s)lu) + <RD-(s)lu). (10)

In each case <RF(s)lu) = O. To facilitate the study of the important parameters of <R(s)lu) and to compare the

behavior of these parameters to those of double- and multiple-pulsed systems using auto-correlation, identical specific models of transmissivity functions, light intensities and particle image properties have been chosen. Thus, the intensities of the light sheet pulses 10i are modelled as either top hat functions equal to 10 within the light sheets and zero outside of them, or as Gaussian functions centered on the light sheets with width L1zoi measured at the e - 2 points. The interrogation beam intensities are modelled as Gaussian functions centered at Xl and X2 with diameters d1 and d2

respectively, while all particle image transmissivities are Gaussian functions centered at the image center with image diameter dr. Thus we have

I o;(z - zo;) 10 Iz - zoi 1 < L1z,,;/2 or 10 exp( - 8(z - zo;)2 / L1z~J,

o otherwise (11 )

ANALYSIS OF PlY IMAGES

and

1I1(X - Xl) = Al exp(-8IX - Xllljdi),

IdX - Xl) = Al exp(-8IX - Xllljd~)

8'00 1 1 'o(X) = -dl exp( -81XI jdr),

n r

9

(12)

(13)

(14)

where '00 = S '0 dX is determined by the photographic process and the development process.

The components of the cross-correlation of single-exposure images in (9) for a general displacement field Llx(x, t) become

1 1 1 (Nj)l ( 4 1) <RC(s)lu) =;- IoAIA1'00~exp - d; Is + Xl - Xli , (15)

<RD(s)lu) = c f dxlol(z)I01(z + Llz) f dX AIAl exp( -81X - Xllljdi)

x exp( - 81 X - Xl + sll jdD X (8'010)1 exp( - 81 X - Mxll jd;) ndr

x exp( -81X - Mx + s - MLlxlljd;). (16)

For the cross-correlation of a double-exposure image, the components in (10) are

1 1 1 1 (Nj)l ( 4 1) <Rds)lu) =;- IO(AI + Al ) 'Oo~exp - d; Is + Xl - Xli , (17)

<RD+(s)l) = <RD(s)lu), (18)

(19)

The average interrogation spot size d* is defined by

(20)

and the average particle image density Nt is defined consequently as a weighted average of the image densities in each window, (NI)I, (NIb

(21)

assuming circular interrogation spots.


These results are valid for arbitrary velocity field u through the displacements Ax, and they generalize earlier results on the auto-correlation function for double-pulsed PlY in [6]. From these earlier results on double-pulsed PlY, [6], it can be seen that the mean intensity correlation, (Rc(s)lu) in equation (15) is one quarter of the corresponding mean intensity correlation for auto-correlation of double-exposure images and the displacement correlation, (RD(s)lu) in equation (16) is identical to that for auto-correlation of double-exposure images for arbitrary velocity vector fields u where Al = A2 = J [0, d1 = d2 = d[ and Xl = X2 . Thus, with identical interrogation spots for comparison, spatial cross-correlation of single-exposure double-frame PlY, like auto-correlation analysis of double-exposure single-frame PlY, will be limited in signal strength by loss-of-pairs effects due to in-plane and out-of-plane motion. However, as seen in Fig. 2, the height of the spurious noise correlation peak in singleexposure double-frame PlY is approximately half that in double-exposure singleframe PlY. In general, the mean intensity correlation is a poor estimator for the peak of the noise correlation, being too low for the seeding densities considered here.

In the presence of a velocity field which varies sufficiently slowly that it can be approximated by a locally constant velocity field within an interrogation spot, the mean displacement component of both single- and double-exposure cross-correlation functions can be written more simply as

xexp{ -41s - MAx121d;}

= -'- A1A216 'd620 {NJ Fo(Az)F[Cs)} exp{ -41s - MAx121d;}, n ,

(23)

where F[ is the normalized correlation of the interrogation intensity across the interrogation spots, generalized from Adrian [1], namely

(24)

Likewise, F o(Az) is the normalized correlation of the intensities of two successive light pulses in terms of the out-of-plane displacement Az between successive pulses,

(25)

These terms reduce (RD{S) due to loss-of-pairs as a consequence of in-plane motion and out-of-plane motion, respectively. Nt FIFo represents the mean number of particle image pairs, weighted by the light intensity distribution and is the mean effective particle image density contributing to (RD(S) 1 u).


In order to minimize the loss-of-pairs effect, the interrogation spot shift, X2 - Xl, can be chosen so that X2 - Xl = M!lx, and the second light sheet thickness, !lZ02,

and relative position (Z02 - ZOl), can be chosen to ensure that all particles whose first images lie in the first light sheet will have second images in the second light sheet. In addition, by selecting d2 sufficiently larger than d l all particles with images in the first spot have images again in the second spot so that F 0 = FJ = 1 and the effective image density becomes Nt with the signal strength maximized to be

(26)

when s = M !lx, as illustrated for cross-correlation of a double-exposure system in Fig.2c.

In the case where the velocity field varies sufficiently rapidly that velocity variations within the interrogation volume cannot be ignored, spatial cross-correlation analysis of single-exposure and double-exposure double-pulse PIV can reduce and eliminate gradient bias, which, in previous double- and multiple-pulse PIV systems, biased the location of the centroid of (R D ) towards the displacement of lower speed particles. Choosing adequate size and reasonably correct relative displacement of the second interrogation window ensures that all first images are captured again in the second window, and it eliminates in-plane gradient bias because no images of more rapidly moving particles are lost. However, the distribution of image displacements reduces the amplitude and increases the width of the displacement correlation peak reducing the probability of a valid measurement by producing more undetectable peaks in regions of large velocity gradient. This detection bias is independent of the size or relative displacement of the second interrogation window and reduces the apparent velocity gradient for a given realization by measuring noise peaks in Rc + RF rather than true correlation peaks. The detection bias depends upon the random instantaneous noise peaks in Rc + RF which, in single-exposure systems, have less than half the amplitude of those in comparable double-exposure PIV systems that use either cross-correlation or auto-correlation methods.

Thus gradient bias can be eliminated by appropriate selection of interrogation windows and detection bias occurs less frequently in spatial cross-correlation analysis of single-exposure PIV as will be shown in the next section when individual realizations are examined to compare possible PIV alternative systems.

4. Comparative cross-correlation performance

4.J. Procedure

The comparative performance of the spatial cross-correlation interrogation method of measuring image displacements for a single-exposure or a double-exposure double-


pulse PIV system as an alternative to auto-correlation analysis of double or mUltiplepulsed systems is measured in terms of three criteria, which were developed previously. These are the spatial resolution, the detection probability and the accuracy of the velocity measurements that are considered to be valid. Random fluctuations of the spatial cross-correlation function for specified dimensionless parameters listed above affect the performance of the interrogation method. Specific details of the interrogation procedure will similarly affect the method's performance.

The interrogation procedure for a single-exposure double-frame PlV system is identical to that of double- and multiple-pulsed systems using auto-correlation methods, but it is more simple as there is no self-correlation component Rp to consider. It locates the highest peak in R(s) and, in assuming this peak corresponds to R D , compares it to the second tallest peak, thereby determining the detectability, D, as the ratio of the tallest peak to the second tallest peak. As there is no self correlation component, R p , in the cross-correlation function and as there is an arbitrary displacement, X2 - Xl, between interrogation windows, the peak search is conducted over the whole cross-correlation plane. On the other hand, although the crosscorrelation function for a double-exposure may contain a self-correlation component Rp, the peak location of Rp is known to be Xl - X2 and the peak search is then conducted over the plane to the right of (Xl - X2). The centroid of the crosscorrelation peak is determined by integrating a small area around the peak, as described in [6] in order to calculate the measured velocity.

4.2. Monte Carlo simulation

As the analysis of the mean cross-correlation function does not directly address questions concerning probabilities of measurement and the fluctuation in the statistics of individual velocity measurements, the numerical simulation of the PIV image field and the interrogation analysis by cross-correlation methods of both single-exposure and double-exposure double-pulse systems have been modified from the multiplepulse system. The details of the simulation can be found in [6]. The simulation has been performed for three dimensional velocity fields over a range of velocity gradients. Figure 3 shows a simulated region of two interrogation frames in which a constant velocity causes a mean particle displacement with MAx/d l = (0.10,0.10,0.10) where X2 = Xl + MAx,(N[)1 = 12 and d2 is chosen larger than dl to ensure no loss-of pairs due to in-plane motion. Although d2 need be no larger than dl in this example which illustrates an optimal relative displacement and a constant velocity field, in general, lack of precise a priori knowledge of the velocity field and its gradients necessitates larger second spots to ensure that all particles with first images possess second images in the second spot. Out-of-plane motion within the flow field reduces the number of paired images within the light sheet, as shown. In general, the location and thickness of the light sheets and the size and relative displacement of the interrogation frames can be optimized to ensure that no such loss of paired images will occur.


b o

o ----;.- - - - - - - - - "l)- - - - - l

o

0

~

d1

o 0

0

0

d2 -_. 0

o o I

I

0

X 1 " 0 a

0 b

0 0' c , 0

0 0

0

0

0

X2

X1/ 0 a

0 0 0

0

o

o c

13

-

-

Fig. 3. A simulated pair of interrogation frames for a single-exposure double-frame system in which (N I ), = 12, M/>"x/d , = (0.1,0.1,0.1), X, = X, + M/>,.x and FI = 1.

The detection probability and the valid detection probability for a given ensemble of realizations of experimental parameters are defined in [6].

4.3. Comparative performance

Comparisons of the mean cross-correlation function to previous results for doublepulsed and multiple-pulsed systems, show that the detection probability can be related to the mean effective number of particle image pairs within a pair of interrogation spots, namely Nj F[F 0 for cross-correlation analysis and N pF[F 0 for multiple-pulse auto-correlation analysis, where N p = (n - l)N[ for an n-pulse system.

For the case of a constant velocity field, it can be seen from equation (22) that if the interrogation spots are chosen so that their relative separation, X2 - Xl' satisfies

s = (X2 - Xd = M!J.x (27)


then Fr = 1 from equation (24) and there is no loss-of-pairs due to in-plane displacements. Similarly by choosing appropriate light sheet thicknesses and relative positions, out-of-plane pair losses can be eliminated and F 0 = 1. This technique has been used for example by Lee et al. [10]. In this case, for a given particle seeding density in the fluid C, the strength of the displacement correlation peak is maximized.

More generally, in the case of a constant or variable velocity field, loss-of-pairs due to in-plane displacements can be eliminated by choosing the relative separation, X2 - Xl' to be approximately equal to the local mean image displacement and then choosing d2 sufficiently larger than d I so that all particles with images in the first window have images in the second window. In this case, Fr = 1. Out-of-plane pair losses can be eliminated in a similar way to ensure F 0 = 1 as well. That is, the second light sheet can be made thicker than the first so as to capture all particles illuminated by the first light pulse. Thus, the effective particle image density is determined by the number of particles in the first window, (N[ll = N r = N[F[Fo. The larger second window ensures optimal use of the chosen seeding density.

A comparison of the valid detection probability as a function of the mean effective number of particle image pairs can be made for the cross-correlation analysis of single-exposure and double-exposure images and for the auto-correlation analysis of double-pulse and triple-pulse systems considered in previous work ([6], [7]) using a detectability criterion of Do = 1.2, for a range of image displacements IdXl/d[ of constant three-dimensional velocity fields.

In addition to cross-correlation analysis using identical interrogation windows initially and image displacements considered in earlier work, a more general range of larger image displacements, IdXl/dl , has been used in the cross-correlation analysis, as the relative window separation, X2 - Xl' and d2 have been chosen then to ensure that there is no aliasing of the measurements and no loss-of-pairs, due to in-plane motion. From Figs 2a, b, c and from [7], it can be seen that the non-signal noise peak for cross-correlation analysis of single-exposure systems is generally about half of the corresponding peak for cross-correlation and auto-correlation analysis of doubleexposure systems. Furthermore, increasing the second interrogation spot diameter d2

relative to d1 within reasonable limits does not increase the non-signal noise peak enough to affect the valid detection probability when Do = 1.2. A range of(d2ld l ) from 1.0 to 4.0 was chosen to analyze the effect of superfluous noise peaks, but there was no significant decrease in valid detection probability for the latter case (figures 4a, b). However, a choice of d2 which is unnecessarily large will reduce the pixel resolution of the first window in the digitization process leading to bias errors as discussed in Prasad et al. [13].

Figure 4 shows the relationship between valid detection probability and the mean effective number of particle image pairs in the interrogation spot for cross-correlation analysis of single-exposure and double-exposure images and auto-correlation analysis of double- and triple-pulse systems. To ensure a 95% valid detection rate for crosscorrelation of single-exposure and double-exposure images, it is sufficient to choose


+ -~ 0 Frame 1 Frame 2 -

..ci 80 \ 0 D D 0 .... ) . a..

t: 60 \ 0 FI <1.0,FO =1.0 0

t) I. FI =1.0,FO <1.0 Q) 0 - 0 Q)

ot! • a 40 + A2/A,=2

[iJ "t:I i Cij 0, 0 A2/A,=4 ~ > 20 IT • A2/A, =9

0 • A/A,=16

0 0 5 N F F 10 1 5

I I 0

(a)

100 a fi;I~ Uioj -• - j gB

~ ~ ~ 80 I ..ci

0

6 0 .... a.. A

60 6

t: 6

0 6 In Plane FI = Fa = 1 t) • 0 Out of Plane FI =1 ,Fa <1 Q) t -Q) 40 6 • Out of Plane FI = Fa = 1 a "t:I I cu > 20

0 0 5 N F F 1 0 1 5

I I Q

(b)

Fig. 4. Comparison of valid detection probability for (a) single-exposure double-frame cross-correlation analysis, (b) double-exposure single-frame cross-correlation analysis, (c) double-pulse autocorrelation analysis and (d) triple-pulse auto-correlation analysis as a function of effective particle image density, N[F[F O' where Do = 1.2.

16

.0 o ... C. 60 t: o -t.)

.s Q)

Cl "0

m >

-~ o

.0 o ... C.

t: o -t.)

~ Q)

Cl "0

m >

40

20

0

c ~: . II

c c.

c 0

Be

0

40 c c

20 0

o

•

R. D. KEANE AND R. 1. ADRIAN

c In Plane (2P)

• Out of Plane (2P)

5 10 1 5

C In Plane (3P) • Out of Plane (3P)

O~--~-----.----~-----r----~--~

o 5 10 1 5 N F F

I I 0

(d) Fig. 4 (Continued)

(N [)1 > 6 and (N [) I > 8 respectively, and then select the relative displacement X2 - Xl and d2 to ensure that there is no loss-of-pairs. It is no longer necessary therefore to restrict IAXI to be less than 0.30 d1 as F[ = 1 and (N[)1 becomes the main parameter for constant velocity flow fields. This result for single-exposure images is a significant improvement over auto-correlation of a double-pulse system as shown


earlier [6] where NrFrFo > 8 requires N r > 15 for a reasonable range of image displacements to ensure a 95% valid detection probability. Further, it is a moderate improvement over the method of auto-correlation of triple-pulsed systems for which it is necessary to have N] > 8 in order to ensure N]FrF 0> 4 for an acceptable range of in-plane and out-of-plane displacements to obtain a 95% valid detection probability. The mean effective pair density, N pFrF 0 for this triple-pulse system then satisfies N pFrF 0 > 8. However, quadruple-pulse systems, interrogated by auto-correlation methods, require N] ~ 5 to maintain a 95% valid detection probability over a range of image displacements [7]. They provide an alternative to single- and double-pulse systems using cross-correlation methods when the seeding density is low and velocity gradients are small.

The result, (N r)1 > 8, for double-exposure images is very similar to that obtained earlier [6] for auto-correlation methods with small improvements caused by reductions in peak noise correlation when optimal use of image pairs enables smaller first interrogation windows to be used for a given seeding density, C. Thus, although the mean effective pair density is not significantly decreased for cross-correlation analysis compared to auto-correlation analysis, the elimination of image pair losses allows lower seeding densities while maintaining acceptable detection probabilities. Alternatively, for a given seeding density, C, this more efficient use of particle image pairs permits a higher spatial resolution by reducing the size of the first interrogation spot needed to achieve a required number of effective image pairs. The spatial resolution is limited by the spot size to obtain this number of effective pairs and by the smallest scales of recorded images.

In the limit of small N r, the probability of detection is equal to the probability of finding precisely one image for a given particle within each interrogation spot. In both auto-correlation of double-pulsed systems and cross-correlation of single-exposure and double-exposure systems, this probability is that of finding one particle in that portion of the interrogation volume from which the particle could be displaced and still remain within the interrogation volume at the time of the second exposure. From the arguments above, the average number of effective image pairs is NrFrF 0, and from a Poisson distribution it follows that

P{exactly one pair ofimages}=NrFrFoexp(-NrFrFo)exp(-2NrFo(1- Fr)).

(28)

Figure 5 confirms the correlation between detection probability and the single parameter, N r F r F 0 when the image density is small and shows that equation (28) is valid for NrF]Fo less than about 0.3. Comparative results are shown for double- and triple-pulsed systems using auto-correlation analysis for a range of image displacements as well as a similar model for a triple-pulsed system. Although these results are independent of Do due to the lack of noise peaks in R(s) for very low NrFrF 0 < 0.3, it can be seen that triple-pulse analysis, with twice as many image pairs, is superior to cross-correlation analysis of single exposures. The latter, in turn, is superior to double-

18

80 I-"Xl/d D 0.1 Cross-corr. -~ 0 0 0.2 - a 0.3

.0 60 M 0.1 Auto-corr.(2P) 0 .... + 0.2 a.

)( 0.3 c 0 6. 0.1 Auto-corr.(3P) - 40 A 0.2 (J cp " 0.3 -Q)

Cl "C

(ij 20 >

A6.

16 A 6. .. .. D IJ

A .. d!I ~ ¥ +Ma 0 '+ aM a

c

0.5 N F F

I I 0

)(

+ M

b

R. D. KEANE AND R. 1. ADRIAN

A 6.

A6.

"" D De + + a

• .. .. )( a d

e

1.0

Fig. 5. Valid detection probability for low image density interrogation windows in terms of the effective image density N[F[Fo for detect ability Do = 1.0. Full curves (a), (b), (c) illustrate equation (20) for doublepulse auto-correlation and cross-correlation analysis while (d), (e), (f) model triple-pulse auto-correlation.

pulse auto-correlation analysis as there are fewer noise peaks and the choice of interrogation windows can ensure F[ = 1.0. In these low image density cases, alternative simpler methods such as one dimensional orthogonal image compression [16J can be used as the valid detection probability depends principally upon the probability of finding a true pair of images.

While the above comparative performance is based upon constant velocity fields, the presence of a velocity gradient in the flow field determines both the size of the two interrogation windows and their separation. Velocity gradients diminish and broaden the mean cross-correlation peak (RD > necessitating higher particle image densities to overcome peak splintering in individual realizations. However, gradient bias in the location of the mean cross-correlation peak (RD > can be eliminated by selection of dl , d2 and X2 - Xl to guarantee no loss of pairs due to in-plane motion and by selection of AZOI and AZo2 to guarantee no loss of pairs due to out-of-plane motion.

In order to compare the cross-correlation method with auto-correlation analysis of double- and multiple-pulsed systems, the simplest case of a velocity gradient in which (au/ay) #- 0 has been examined for variable interrogation window sizes. This determines the effect of velocity gradient upon valid detection probability by pcak spreading, upon gradient bias due to loss of image pairs and upon individual realization variation in measured velocities, due to random particle locations within an interrogation spot.

The dimensionless velocity variations, such as MAuyAt/d, in the case of simple shear, affect the amplitude of (Rv > and the detection probability because the degree


of peak spreading due to velocity variation must be measured in terms of particle image diameters. However, in cross-correlation analysis of velocity fields with a velocity gradient, smaller interrogation windows can be chosen than for autocorrelation for a given seeding density C due to the maximum utilization of images pairs in cross-correlation. Then, in order to compare cross-correlation with varying window sizes with auto-correlation methods, more useful non-dimensional velocity variations to consider are of the form M!1u/1tjd 1 • These variations are independent of the window size d1 or dI for a given velocity gradient, as !1uy = (8uj8y).(dd2).

With a constant seeding density for cross-correlation and auto-correlation chosen to yield NI = 15 from earlier work, Fig. 6 shows that for single-exposure crosscorrelation the first interrogation spot can be halved in area in cross-correlation to maintain no loss-of-pairs with (N1)1 = 7.5 and achieve an acceptable valid detection probability of95% provided that MI!1ul!1tjd 1 < 3%. For a given velocity gradient, a range of first interrogation spot sizes is considered as M !1Uy !1tjd 1 is constant for a fixed u. This is an improvement over auto-correlation methods as the spatial resolution can be increased for a given seeding density of a varying velocity field without loss of detection probability. However, the valid detection probability can become unacceptably low, if the effective image density is reduced by further increases in the spatial resolution. Figure 6 illustrates this loss of detection probability when the first interrogation spot is halved in area again yielding (N1)1 = 3.75.

However, in comparing cross-correlation analysis of single-exposure frames and

100 ~::" I!I N =7.5 Single Exp

~ ~

-.;;. ""'" .'" 0 N =3.75 .. .. "1!10.:;

"" ":'\. + N =7.5 Double Exp ;

.0 + '1 11 N >10 .. .. + :I!!i.:

0 ~

Co + =*= ~ e !

c: 0 - 80 () Q) -Q)

C "0

ctI >

+ Single Exp Range • \~ lor N >7.5

\ :~

+ ) + '\

., \

'\l

t \ ~\ lIf'

" m: =*= :::~

~[ 8 :'\" Double Exp Range ~; A :.,! ,'\.

:",,-lor N >10

~;! " ~ '" 60 o 2 4 6 8 1 0

M~u Mid (%) y 1

Fig. 6. Comparison of valid detection probability for a plane shearing velocity in terms ofrelative variation of image displacements, M/1.u/'1tjdI , in cross-correlation analysis for both single-exposure and double-exposure imaging.


double-exposure frames, Fig. 6 also shows that for double-exposure frames crosscorrelation analysis requires a higher particle image density to achieve an acceptable valid detection probability of95% for a given velocity gradient, due to the doubling of noise peaks from a single-exposure analysis to a double-exposure analysis of the same realization. For the latter case, (N[)1 must be larger than 10 to achieve the same acceptable valid detection probability of 95% with the same velocity gradient and an identical range of first interrogation spot sizes. Thus, there is less spatial resolution possible with a double-exposure image for a given seeded fluid flow as the interrogation spot sizes cannot be reduced to the same extent as above for crosscorrelation of single-exposure frames.

For the above parameters in cross-correlation analysis, velocity gradient bias, which is present in all correlation analysis when loss-of-pairs occurs, can be removed by choosing dz sufficiently large that F[ = 1. Figure 7 shows that in-plane velocity gradient bias is substantially reduced and finally eliminated whenever loss-of-pairs is reduced and similarly eliminated by choosing d2 > d1. This is compared to the theoretical result from auto-correlation methods in which the velocity gradient bias was shown to be a linear function of the velocity variation tensor, dij = duddXj, in equation (35) from [7].

Finally, as for double- and mUltiple-pulsed systems, the random particle locations in individual realizations of a variable velocity field cause the centroid of RD(S) to be located over a finite range of displacements, causing random variations in valid measurements. Figure 8 shows that variation in measured velocity in terms of the

B !l ~ I I II • 100.0 • •

6 .6. !! 0 • • 0

.6. g • •

A • t

6

~ .6.

~ .6.

• Single Exp F, =1.0 .6. .2 95.0 '" .6.

E • .. F, =0.38 ::J .6.

.6. .. F, =0.83 Autocorrelation Theory where

0 Double Exp F, =1.0 F, = 0.75

0 " F, =0.98

6 " F, =0.83

90.0 0 2 4 6 8 1 0

Mt"u t1t/d (%) y

Fig. 7. Relative measured mean velocity in terms of relative variation of image displacements illustrating no velocity gradient bias for optimal choice of cross-correlation parameters.

ANALYSIS OF PlY IMAGES

0.02 ~----------------------.

~ <iE b::J 0.01 :E

I!I N=7.S Single Exp

6 N=7.S Double Exp

o N=10.0

o N=1S.0

I 6 Il

0.00 ~2:::::;=-_~_~---r-_~_~ 0.0 0.5 1.0 1.5

M~u Mid y "C

21

Fig. 8. Variation in measured velocity for a plane shearing velocity in terms of the relative image displacement variation M l1uyM/d, for both single-exposure and double-exposure cross-correlation analysis.

relative image displacement variation M AUyM/dr is higher than that obtained from auto-correlation analysis of double- or multiple-pulsed systems and is independent of the mean image displacement. The variation is greater because the choice of interrogation windows guarantees no loss of image pairs whereas in auto-correlation analysis, the variation is restricted by the loss of larger displacement pairs. In order to restrict this variation to 1% and obtain an acceptably high detection probability, the velocity gradient must satisfy MAuyM/dr < 1 and M~uyAt/dl < 3%.

Thus, in single exposure, double-frame systems using cross-correlation methods, the choices of d1 and M are determined with criteria for adequate spatial resolution, valid detection probability and variation in velocity measurement about a measured mean. The maximum velocity variation, IAul that can be tolerated for successful interrogation is no longer dependent upon u but is bounded by the above restrictions for valid detection probability and acceptable variation in velocity measurements.

5. Summary and conclusions

Cross-correlation methods of interrogation of single-exposure and double-exposure double-pulse PlY systems have been studied as alternatives to auto-correlation analysis of double- or multiple-pulse PlY systems. Theoretical analysis of full 2-D spatial cross-correlation demonstrates the similarity and versatility of the method compared to that of auto-correlation of double- or multiple-pulse systems.


Both single-exposure and double-exposure double-pulse PIV systems which use cross-correlation methods of interrogation are superior to double-pulse PIV systems using auto-correlation analysis for a range of velocity fields for a numbcr of reasons. The versatility in selecting the size and location of successive interrogation spots permits a higher level of spatial resolution by more efficicnt pairing of particle images. As there are ideally no pair losses in the optimal implementation of the crosscorrelation method, the spatial resolution can be improved or the seeding density can be reduced while maintaining acceptable detection probabilities. Furthermore, in cross-correlation of single-exposure systems, image shifting is not required to resolve directional ambiguity. Without this need for image shifting the dynamic range of velocity measurements is doubled. (Other researchers have claimed much greater increases in dynamic range from cross-correlation but this is not the case when image shifting is implemented in auto-correlation.)

In experimental fluid flows where high seeding densities are not possible, multiplepulse PIV systems with n ~ 4 can maintain acceptable detection probabilities at lower seeding densities than single- or double-exposure PIV systems using cross-correlation methods. Triple-pulse systems, on the other hand, while being superior to doubleexposure PIV systems using cross-correlation analysis require a higher image density than single-exposure images to yield acceptable detection probabilities.

In fluid flows which have significant velocity gradients, the above choice of interrogation spots and light sheet thicknesses in the cross-correlation method removes the gradient bias in velocity measurements which occurs in auto-correlation methods. It has been shown that the size of the second interrogation window is not critical provided that there is no loss-of-pairs from in-plane motion as the strength of the noise does not vary significantly as the size of the second window increases. Inplane loss of pairs is avoi.ded if the second interrogation window is sufficiently large to contain all particle images that were in the first window. The penalties for using a large second window include longer computation time and reduced pixel resolution in the first window. Out-of-plane loss of pairs is avoided by using a sufficiently thick second light sheet. The penalty associated with this procedure is the increased energy requirement for the second sheet. Detection bias caused by low seeding density is not present because the lower more efficient seeding levels of the procedure ensure 95% valid detection probability.

An extended range of velocity fields has been utilized in this analysis and in a Monte Carlo simulation of the cross-correlation method to reveal the critical dimensionless parameters of the mean cross-correlation function and its fluctuations. These parameters are an extension of thosc uscd for multiple-pulse systems, namely (NJ )1, Do, IX2 - Xll/d!, I,1XI/d!, d2 /d!, M l,1u 1M/do dr/d!, Iwl,1t/ ,1zo I and ,1Z02/ ,1Z01'

To optimize the system performance, the following broad criteria, similar to those used in auto-correlation analysis of double- and multiple-pulse systems, are recommended. For single-exposure images, choose


(b) MIL1uIAt/d1 < 0.03; (c) MIL1ulL1t/dr < 1; (d) 1.0 < Do < 1.2.

23

For double-exposure images, the particle image density must be higher as the noise peaks are doubled so we must choose (a) (N I) 1 > 10 and F 0 = F 1= 1. The remaining criteria are identical to those above, (b)-(d).

In these cases, proper selection of the light sheet thickness, the interrogation spot sizes and separation can ensure that there is no loss-of-pairs due to either in-plane or out-of-plane motion with pulse separation selected to provide sufficient spatial resolution by maintaining adequate image displacements. While these criteria are a good general combination, more precise values can be used for a particular experimental flow field by reference to the appropriate figures.

To achieve optimal operation, some a priori knowledge of the flow field must be used to estimate the bounds of in-plane and out-of-plane motion and the extent of variation in the velocity field caused by velocity gradients. For flow fields which do not have appreciable velocity gradients, choose the seeding density C, d 1 and ~ZOI to ensure adequate spatial resolution and At to maintain adequate image displacements and then choose d2 and L1zo2 to eliminate loss-of-pairs and to obtain a satisfactory valid detection probability.

In the presence of significant velocity gradients, conditions (b) and (c) above place a physical limit on the extent of the velocity variation that can be tolerated in designing an optimal group of parameters. In this case both d1 and L1t must be chosen to satisfy these constraints with the seeding density C to be determined to enable (N1)1 > 7 for single-exposure analysis or (N1)1 > 10 for double-exposure analysis. The remaining parameters can then be chosen to ensure that there is no loss-of-pairs.

Appendix

To develop mathematical expressions for the components of R(s) in equation (7), it is useful to reformulate the expression for the transmissivity of a distortion free nearly paraxial photographic recording of a single exposure flow field in equation (1) as follows:

(A-i)

where

g(x, t) = L: t5(x - Xj(t)) (A-2) i

indicates the locations of the particles in the flow at time t. A similar expression can be


used for equation (3). It is convenient to decompose y(x, t) into mean and fluctuating parts

g(x, t) = C(x, t) + llg(x, t), (A-3)

where C(x) is the mean number of particles per unit volume and is assumed to be time independent and <llg) = O.

Substituting (A-I) and (A-3) and similar expressions into earlier expressions for II (X) and 12(X) in equations (2) and (4) respectively, shows that R consists of three components as shown in equation (7). They are

Rds) = f dXI l1 (X - X1)IdX - X 2 + s) f dxlo1CTo(X - Mx)

x f dx' I~2C'To(X - Mx' + s), (A-4)

RD(S) = f dXII1(X-X 1)IdX - X 2 +s) f f dxdx'To(X-Mx)To(X-Mx' +s)

x IOII~2llg(x, t)llg(x', t + M), . (A-5)

RF(S) = f dXII1(X-XJldX-X2 +s) f f dxdx'to(X-Mx)TO(X-Mx' +s)

xlolI~2{Cllg(x', t + M) + C'llg(x, tn. (A-6)

The conditional average for each component of R can be determined by using the equation for <llylly'lu) from Adrian [1], namely;

<llg(x, t)llg(x', t+ M)lu) = C(x, t)b(x' -x-u(x, t)llt). (A-7)

Acknowledgment

This material is based on work supported by the National Science Foundation under Grant No. ATM 89-20605.

References

1. Adrian, R. J., Statistical properties of particle image velocimetry measurements in turbulent flow. Laser Anemometry ill Fluid Mechanics, Vol. III. LADOAN Institute Superior Tecnico, Lisbon, Portugal (1988) pp. 115-129.

2. Adrian, R. J. and Zoltani, C, Measurement of particulate motion using a high resolution solid-state camera and high speed electro-optic double framing. Abstract ICALEO (1990).

ANALYSIS OF PlV IMAGES 25

3. Arroyo, M. P., Yonte, T., Quintanilla, M. and Savir6n, 1. M., Particle image velocimetry in RayleighBenard convection: Photographs with high number of exposures. Optics and Lasers in Engineering 9 (1988) 295- 316.

4. Cenedese, A. and Paglialunga, A., Digital direct analysis of a multi-exposed photograph in PlY. Experiments in Fluids 8 (1990) 273-280.

5. Goss, L. P., Post, M. E., Trump, D. D. and Sarka, B., Two color particle velocimetry. Proc. ICALEO, LIA 68 (1989) pp. 101 ~ Ill.

6. Keane, R. D. and Adrian, R. J., Optimization of particle image velocimeters. Part I: Double-pulsed systems. Measurement Science and Technology 1 (1990) 1202-1215.

7. Keane, R. D. and Adrian, R. J., Optimization of particle image velocimeters. Part II: Multiple-pulsed systems. Measurement Science and Technology 2 (1991) 963-974.

8. Keane, R. D., Adrian, R. J. and Ford, D. K., Single exposure double frame particle image velocimeters. Proc. ICALEOn (1990) 91-110.

9. Kimura,!. and Takamori, T., Image processing of flow around a circular cylinder by using correlation techniques. In: Veret, C. (ed.), Flow Visualization IV Washington, D.C.: Hemisphere Publishing Corp (1986) pp. 221- 226. 10. Lee, M. M., Hanratty, T. J. and Adrian, R. J., An axial viewing photographic technique to study turbulence characteristics of particles. Int. J. Multiphase Flow 15 (1989) 787-802.

11. Lourenco, L. M. and Krothapalli, A., The role of photographic parameters in laser speckle or particle image displacement velocimetry. Experiments in Fluids 5 (1987) 29-32.

12. Meynart, R., Simpkins, P. G. and Dudderar, T. D., Speckle measurements of convection in a liquid cooled from above. J. Fluid Mech 182 (1987) 235-254.

13. Prasad, A. K., Adrian, R. J., Landreth, C. C. and Offutt, P. W., Effect of resolution on the speed and accuracy of particle image velocimetry interrogation. Experiments in Fluids 13 (1992) 105-116.

14. Utami, T., Blackwelder, R. F. and Ueno, T., A cross-correlation technique for velocity field extraction from particulate visualization. Experiments in Fluids 10 (1991) 213-223.

15. Willert, C. E. and Gharib, M., Digital particle image velocimetry. Experiments in Fluids 10 (1991) 181 193.

16. Yao, C. S. and Adrian, R. 1., Orthogonal compression and l-D analysis technique for measurement of particle displacements in pulsed laser velocimetry. Applied Optics 23 (1984) 1687-1689.

Decay of rotating turbulence: some particle tracking experiments

STUART B. DALZIEL Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB39EW, U.K.

Key words: rotating turbulence. particle tracking. experiments, inertial waves

Abstract. Recent development of measurement techniques based on particle image velocimetry (PlY) are enabling more detailed measurements to be made over extended regions of a flow than have been previously possible. These techniques are of particular value for turbulent flows where the structures present within such flows are incompletely understood and are not readily accessible to traditional measurement techniques. Unfortunately the considerable processing time and specialised equipment required with most PlY techniques limits their applicability when ensemble statistics are required for an evolving turbulent flow. This paper reports on the development and application of an efficient, fully automated particle tracking system. The system was developed as part of a study of the decay of turbulence in a rotating environment. Ensemble descriptions of the temporally evolving flow were required over an extended measurement domain. For each set of parameters particles were tracked with a sampling frequency of 12.5Hz over 60 seconds for 25 realisations. Typically 350 particles were identified and tracked at each time step. Processing speeds in the region ten to fifteen sample images per minute were achieved using a PC/AT compatible computer. The results of the experiments were found to be in broad agreement with previous investigations. However it was found that the method of generating the initial turbulent flow had a profound affect on the subsequent evolution due to the forcing of a strong, large scale systematic flow.

1. Introduction

Turbulence is widely considered one of the most difficult problems in fluid dynamics. Except in special cases, theoretical advancements are hampered by an inability to find exact solutions to the Navier-Stokes equations; approximations to these equations using closure models has met with only limited success. The range of scales involved in a turbulent flow means that only relatively low Reynolds number flows are accessible numerically to the present generation of computers. As a result experimental work continues to playa major role in developments, the scope being limited by our ability to measure and understand what is happening, rather than the use of approximate fluid dynamics. The quantity of primary interest in such experiments is the fluid velocity as a function of both space and time.

Over recent years the velocity measurement techniques available have advanced significantly. Starting from flow visualisation and hand-eye digitization, hot wire/film anenometry and then laser doppler anenometry were able to yield accurate point measurements of the flow. Fuelled by modern image processing technology there is renewed interest in flow visualisation as it now offers accurate quantitative information about a two or three dimensional region of the flow rather than just a single point. A wide range of image processing techniques have been developed to measure

27

F. T. M. Nieuwstadt (ed.), Flow Visualization and Image Analysis, 27-54. © 1993 Kluwer Academic Publishers.

28 S. B. DALZIEL

the fluid velocity. The key feature in common with most these techniques is the seeding of the flow with particles: it is the motion of these particles which is actually measured. A thorough review of particle image velocimetry (PlV) techniques may be found in Adrian [IJ and elsewhere in this volume.

In this paper we describe a simple, efficient method for measuring velocity by tracking a large number of individual particles. This method was developed as part of a study of the temporal decay of rotating turbulence, but has subsequently been used to measure a variety of other flows (turbulence near a stress-free surface, turbulent plumes and jets, two dimensional turbulence in a stratified environment, RayleighTaylor instability, baroclinic instability and baroclinic vortex interaction). The particle tracking forms a part of a wide ranging image processing package, DigImage, developed for analysing fluid flows. In this paper we illustrate the particle tracking (PT) as it was applied to the rotating turbulence problem.

Considering its importance to a range of fields (e.g. meteorology, oceanography and turbo machinery), turbulence in rotating systems has been the subject of relatively few experimental studies. Part of this is due to experimental difficulties in generating and measuring such turbulence, and part due to the as yet incomplete understanding of the classical nonrotating problem. As a consequence there are very few data sets available for comparison with theoretical and numerical results. The results which arc in the literature fall into two broad categories: inhomogeneous turbulence (typically produced by an oscillating grid - we shall comment on these briefly in the final section), and decaying homogeneous turbulence (typically produced by a single pass of a grid or flow past a grid). The work reported in this paper falls into the second category: the temporal decay of grid generated turbulence with a zero mean flow.

The majority of early work on classical turbulence has been undertaken in wind tunnels looking at the evolution ofturbulence generated by a uniform flow past a grid. It is generally accepted that ensemble statistics may be replaced by their temporal equivalents in this set up. The space-time transformation and Taylor's frozen field hypothesis suggest point measurements are equivalent to a one dimensional transect in the direction of the mean flow. Such grid generated turbulence may be considered homogeneous and isotropic to a reasonable approximation. Unfortunately it is not practical to rotate a sufficiently large wind tunnel at adequate rotation rates to study the effect of rotation, particularly if we are interested in the later stages of decay. Instead we must consider an ensemble of experiments in which the turbulence exhibits a temporal evolution. The requirement for ensemble statistics means that a large number of experiments must be processed, and so the processing time was a fundamental design consideration during the development of the particle tracking.

Ibbetson and Tritton [11J were the first to make detailed measurements of the temporal decay of rotating turbulence. The turbulent flow was generated in air by pulling two grids apart vertically within a shallow rotating annulus, and measurements were made by sweeping a hot wire probe around the annulus. Their primary concern was to determine whether rotation produced an increase or decrease in the

ROTATING TURBULENCE 29

decay rate: they found it to increase, but concluded this was due to inertial wave dissipation in the viscous bondary layers on the tank top and bottom.

The most recent work is by Jacquin et al. [12] who considered a tube rotating rapidly about its axis. Air was blown down this tube at up to 20 ms -1. The air passed through a suitable diffuser to bring it into solid body rotation and a grid to generate the turbulence. The decay downstream of the grid was then observed using hot wire probes. The main conclusion that rotation slows the decay is consistent with present thinking. As with Ibbetson and Tritton, Jacquin et al. found a strong anisotropy develop in the length scales parallel with and perpendicular to the rotation axis. In contrast very little change in anisotropy in the corresponding velocity fluctuations was observed.

In section 2 we detail the experimental set up, and in section 3 outline the particle tracking technique employed. We present classical and rotating results in sections 4 and 5 (respectively), before concluding in section 6.

2. Experimental apparatus

Figure 1 sketches the tank in which all the experiments were performed. The tank measures 1.2 m long (x direction), 0.25 m wide (y direction) and 0.6 m deep (z direction). The tank is positioned on a precision rotating table capable of rotation rates of up to 2n radians per second about the vertical z axis. The basic experiment

c:b I

Fig. 1. Sketch of experimental setup showing tank, traversing mechanism and grid.

30 S. B. DALZIEL

consists of a grid being towed down the length of the tank filled with water. The grid is machined from a single sheet of aluminium to give a square mesh consisting of 6.35 mm square bars at M = 31.75 mm centres. The grid is towed along the length of the tank at up to U = 0.24 ms - 1 by means of a traversing carriage located on the top of the tank. The grid Reynolds number (ReM = U M/v, where v is the kinematic viscosity of water) is constant at ReM ~ 7500 over most of the length of the tank. The carriage and other associated equipment are controlled by means of a small microcomputer mounted on the rotating table, which in turn communicates with a terminal in the laboratory via a set of slip rings.

The design and construction of the tank and traversing mechanism included the ability to traverse a conventional hot film probe for velocity measurement. However the vibration in the traversing mechanism proved too great for reliable turbulence measurements. The development of the particle tracking system described in this paper grew out of the need to measure the flow field and the inability to utilise hot film anenometry (the project did not have access to an LDA system at the time).

The optical arrangement used in the particle tracking experiments is sketched in figure 2. The need to keep the effective diameter of the table relatively small (less than 1 m) forced all imaging to be indirected through a front silvered mirror. Reference points for image registration during the analysis were provided by fluorescent strip lights, covered in aluminium foil and painted matt black. The individual points were

Slide ~,.

Video -=) camera

Fig. 2. Sketch of optical arrangement for experiments. The three mirrors are all front-silvered.


produced by piercing the foil against the glass tube with a sharp needle. This technique provides diffuse reference points (so the angle at which they are viewed is not critical) with an easily adjusted size. The points were superimposed as a row down each side of the image through the use of front silvered mirrors. The fluorescent lights were positioned so that the reference points lay in the in-focus plane of the camera.

The flow was illuminated using a sheet of light approximately 7 mm thick «()y)

provided by a 250 W quartz halogen slide projector mounted above the tank. The light rays were approximately parallel to the rotation (z) axis with the sheet cutting along the length of the tank (x) at the centre line (y = Yo). The flow was imaged using a medium cost single CCD Super VHS camera (625 lines, 50 Hz field rate), fitted with a zoom lens (fl.6, focal length 10.5 mm to 126 mm), feeding through slip rings to a Super VHS video tape recorder. While the camera produced a Y/C colour signal, only the luminance (Y) component was passed through the slip rings to be recorded. This arrangement provided approximately 400 lines horizontal (intensity) resolution. Electronic shutter speeds of 1/50 sand 1/120 s were employed. At the higher shutter speed it was necessary to utilise the camera's automatic gain circuitry. The minimum useful illumination is quoted as 7 lux at f1.4 in the camera documentation: the camera was typically operating close to this limit. The signal noise due to the recording medium was generally less than that due to the camera at the relatively low light levels. The signal to noise ratio on playback remained adequate, allowing successful particle tracking even when the automatic gain was necessary. The equipment utilised to track the particles after the experiment was completed will be described in conjunction with the tracking process in the next section.

The particles used were produced by crushing and seiving pliolite (a resin used in the manufacture of white paint) to obtain sizes in the range 150 pm to 250 pm. The very small particles and dust remaining after the seiving were removed by a decanting process after suspension in fresh water. Photographic wetting agent was used to prevent the particles adhering to the free surface, and around 3 wt% salt used to adjust the water density so that the particles were approximately neutrally buoyant. While the particles were of irregular geometry, they were sufficiently small and close to spherical for their precise geometry to be of little importance.

3. Particle tracking

In this section we describe the particle tracking approach to PIV used in this study. The underlying philosophy is one of simplicity and computational efficiency. Efficiency is of particular concern with the need for ensemble statistics in this evolving turbulent flow.

The particle tracking process may be divided into a number of distinct tasks: experimentation, image acquisition and enhancement, particle location, particle pairing and subsequent analysis. Generally, due to the volume of data involved, it is desirable to combine some of these tasks into a single phase. Here we shall consider

32 s. B. DALZIEL

the tracking phase which consists of the acquisition, location and pairing tasks. The end product is six bytes of data per particle per time step, describing the particle positions, characteristics and relationships.

3.1. Image acquisition

Images are acquired from Super VHS video tape recorder (VTR) with a resolution of 512 x 512 pixels at 256 grey levels using a Data Translation DT-2862 arithmetic frame grabber. Due to the nature of video tape technology, it is not possible to acquire suitable high quality images from a "paused" VTR. A combination of the VTR's internal frame strobe, markers on the audio channel and interrupt driven routines on the host computer allowed efficient, reliable control of tape transport and acquisition of the exact frame required while in "play" mode.

During the tracking process, images are acquired as a sequence of four frames (not necessarily adjacent) from the playing VTR. The tape is then repositioned as a background task while the subsequent processing is undertaken. If desired, a background image is subtracted from the incoming signal to enable the subsequent application of uniform thresholds. If the particles are moving more than a small fraction of their diameter between each video field (1/50 s - each frame consists of two interlaced fields), it is necessary to remove half the information contained on the interlaced frame to ensure that a given particle appears at only one location on a given sample image. This also provides the possibility of sampling at the 50 Hz field rate, provided the small vertical displacement between the two fields is not important (the video camera may be modified so that it always produces only the odd or the even lines).

The fluorescent reference points are located and used to generate a mapping from the current pixel coordinate system to a reference pixel coordinate system using a least squares routine. In the majority of cases only a simple translation is required. The root mean square (rms) error in this mapping is calculated from the residuals and used as a check on the quality of the image. If the rms error exceeds a preset limit, then a further attempt will be made to acquire the image in the hope of improving its quality. The reference point mapping is used in conjunction with a pixel to world transformation to obtain the true world coordinates from the current pixel system.

3.2. Particle location

To locate a particle the image is first searched for a point satisfying a spatially uniform threshold condition (the optional subtraction of the background image during acquisition means that this threshold may be spatially varying relative to the raw image). The region of the image satisfying the threshold and connected to this first pixel is fathomed, and statistics are collected describing this region or blob. The shape, size and intensity characteristics of the blob are then checked against the allowable limits to determine if the blob represents a valid particle. The location of the particle is


determined from the volume (mass) centroid of the region satisfying the threshold and the position transformed to the world coordinate system. This strategy produces good estimates of the relative position of a given particle in a sequence of images, provided the particles are sufficiently close to spherical that any rotation of the particle is of no concern. As an estimate of their absolute position, the volume centroid is less accurate, but the small errors in this are much less important. More complex and computationally expensive models such as those assuming a Gaussian intensity profile (e.g. Perkins and Hunt [14]) are not appropriate for the current method of illumination and are not expected to yield significantly better results.

Particles less than one pixel in size may only be located with pixel accuracy. However, for particles occupying more than one pixel in each direction the volume centroid gives subpixel accuracy, the error being related to the area A occupied by the particle through

(J ~ a((J.i)A- 1(2 x ~, , (1)

-where (J x is some measure of the error in estimating the position of the particle, (Jii is a measure of the relative intensity variations within a pixel (which may include both the unresolved portion of the intensity signal and the noise introduced by camera, recording medium and the digitizing process), and a is some constant. Experience shows that particles extending at least three pixels in each direction may typically be located to an accuracy of better than 0.2 pixels (typically 0.1 pixels) relative to the same particle at some other position.

In practice we search for particles in an image twice. The second pass utilises less stringent threshold requirements than the first and is used to pick up blobs which we are less certain about. As will be explained in section 3.4, this second set of particles is handled somewhat more cautiously.

3.3. Particle pairing

A wide variety of methods have been used to pair the particle images between two successive image samples (see Adrian [1]). Here we develop an efficient pairing algorithm based on the transportation algorithm [9], a graph theory technique frequently used in operations research for determining optimal associations between two sets. With careful coding the computational time for this method increases only slightly more rapidly than the number of particles (in many cases the matching process is faster than the particle location).

The basic idea is to minimise some objective function' which is linear in the associations it includes between the two sets. Here one set is the particle images P = {P[i] for i = 1, M} at the old time, tIn] (say), and the other the particle images Q = {%] for j = 1, N} at the new time, t[n+ 1]. We define a set of variables aij recording the set of particle pairings: if aij = 1, then particle image P[i] at tIn] is considered to be the same particle as the image qU] at t[n+ 1]. If P[i] and %1 are considered as different

34 S. B. DALZIEL

physical particles, then rl. ij = O. We express the objective function, as the linear combination

(2)

where cij is the cost for the association rl.ij to be included in the pairings. The subscript i is summed from 0 to M, where M is the number of particle images at tIn]' The value i = 0 is reserved for all particles outside the set of particles P found at tIn) (i.e. all particles outside the viewing region plus those obscured within the viewing region). Similarly, subscriptj is summed from 0 to N, where N is the number of particle images at tIn + 1] and j = 0 represents all particles outside the set of particles Q found at tIn + 1]'

We shall discuss how to assign the costs cij in the next subsection. Before that we shall consider in more depth the structure of the problem and how to solve it.

The constraints on rl.ij arise from a physical particle only being able to be in one location at a given time. Thus, for any i = K =f 0, there must be one and only onej in [0, NJ for which rl.Kj = 1. Similarly, for any j = L =f 0, there must be one and only one i in [0, MJ for which rl.iL = 1. An association of the form rl. ij = 1 for i, j =f 0 represents a pairing between images in the two sets of particles, while rl.Oj = 1 indicates that qUI was not in the set P. Similarly rl.iD = 1 indicates that Pm is no longer in the set Q of physical particles we have located at t[n+ 1]' Note that there may be many values of i and j giving rl.Oj = 1 or rt.iO = 1 as there can be many particles entering or leaving our viewing region and hence associated with the special index O. In general rl.oo will not be zero as some of the particles not in the set P will also not be in the set Q. However the value of rt.oo is of no concern and so we assign it a zero cost.

The transportation problem differs from our present particle pairing problem in that the former does not have the possibility of associations outside the sets P and Q (i.e. i = 1, M and j = 1, N). Nevertheless the structure of our present problem is very similar to the transportation problem. Both may be written as a linear program of the form BOI = ~ with, = min e 01, where all the coefficients of the B matrix are either 0 or 1 and are arranged in such a way that B is totally unimodular. For our present problem the right hand side ~ is integer, so that the only solutions for 01 = rt.ij are also integer. The differences lie in that the present problem includes a set of constraints for the associations with i = 0 or j = 0 (particles external to P or Q, respectively). While these constraints follow the simple block identity structure of the transportation problem they are inequality constraints and so require the introduction of artificial variables (these artificial variables do not have the same block identity structure). With ~ = 1 all the variables, except rt.oo, are constrained to be either zero or one. The variable rt.oo may be any nonnegative integer, depending on the total number of particles in the flow. However, as we have already stated, the value of rt.00 is of no consequence.

Solution of the particle pairing problem can be tackled in the same manner as the graph theory approach to the transportation problem. We consider rl.ij and cij as two


dimensional tables and define the relational index 1(j) as the value of i, for givenj ? 1, for which (Xij = 1. Similarly we define J(i) as the value of j, for given i ? 1, for which (Xij = 1. Note that /(J(i» = i and J(1U» = j, provided the corresponding association is between particles in the two sets P and Q. In the transportation problem, this identity means that only one of /U) and J(i) are needed. However, for the particle pairing problem, 1(0) and J(O) can not be defined uniquely, hence the need for both relational indices.

The solution process starts by assigning some arbitrary set of associations which satisfy the physical constraints on particles. Careful choice of this initial assignment of 1U) and J(i) will lead to the optimal solution being attained much more rapidly. Suppose some association akl is currently not in the set of associations. If we were to allow the a kl pairing to be made, then for the constraints on the P[k] and q[l] particle images to remain satisfied, the associations (XI(l)1 and akJ(k) must leave the set of pairings. Further, to maintain the constraints on the PrIll)] and q[J(k)] particle images, we must introduce the association aI(I)J(k)' Now the change in the overall cost (reduced cost) associated with this change over is

(3)

If rCkl is negative, then entry of (Xkl into the set of pairings is favourable. The reduced cost is calculated for all akl = 0 with k ? 1 and I ? 1, although if it has

already been evaluated for rCI(l)J(k) it need not be evaluated again for rCk/' Note that associations to the i = 0 andj = 0 sets may still be made through the aI(I)J(k) variable in such a way that does not violate the inequality constraints on these variables. The k, I pair which yields the most negative value of rCkl will be allowed to enter the set of pairings along with (XI(l)J(k)' The aI(I)1 and akJ(k) associations must leave the set of pairings. These changes may be reflected by simultaneously updating the relational indices

/(1):= k, J(k):= I, /(J(k»:= J(1(/», J(1(f)):= 1(J(k». (4)

This process is repeated until no association has a negative value of rc k1 • The linear nature of ( ensures that this solution has the minimum value for the objective function and corresponds to the optimal set of associations. (In principle this optimal set as defined by cij may not be unique, but in practice this situation rarely arises.)

3.4. Costing associations

The only algorithmic restriction on the assignment of costs cij is that they may not include information on the state of (Xij for the current time step. If information about (Xij were used in cij then the objective function would be nonlinear and a more sophisticated algorithm (e.g. simulated annealing) would be required to determine the optimal solution. Each cost may include information about the positions of P[i] and

36 S. B. DALZIEL

q[j)' the velocity history (normally only of P[i)' but may include other particles at t[n)),

their geometric properties (e.g. shape), intensity, colour, fluid dynamics or a wide variety of other possible factors. However, experience has shown that a relatively simple costing strategy is all that is required in the majority of cases. In this subsection we outline the strategy we have employed for the current set of experiments. While we have utilised more sophisticated strategies in other flows (e.g. flows containing dye as a tracer), they are beyond the scope of this paper.

The basic method of costing an association r1. ij is a modification to looking for the nearest neighbour. We define the basic cost of the association as

(5)

where xli) and x[j) are the locations of P[i) and q[j) (respectively) projected onto the Y = Yo plane, uri) is some estimate of the velocity (in the Yo plane) of P[i) at t[n), Y is some weighting function (typically unity), c5t = t[n+ 1) - t[n) is the time step, and m is some positive value (typically m = 2). If y = ° and m = 1, then bij forms the basis for a nearest neighbour solution. For y = 1 and m = 1 we are applying a minimum acceleration criterion, while y = 1 and m = 2 may be viewed as minimising the rate of work on the fluid elements containing the particles.

To overcome difficulties which arise when considering associations with i = ° or j = 0, the possibility of multiple particles casting a single image, or with other anomalous behaviour, we utilise a number of functions or modifiers in conjunction with the basic cost. The cost of an association r1.ij is written as

(6)

where the modifier functions '1i' ej' f0j and Lj are typically step functions acting as price premiums or discounts. The purpose of these functions is explained below.

The function C(·) controls the maximum cost, Crnax (say), an association is allowed to have. If the expression inside C(.) yields a result larger than CrnOX' then C(.) will set the resultant value of cij to infinity. The pairing algorithm outlined in the previous section may be modified to ensure that associations with infinite costs are never in the solution. The associations r1. iO and r1.0j are cos ted either on the cost of associating with a particle lying just outside the physical boundaries of the image, or at the maximum cost CrnOX' whichever is smaller. Thus the maximum cost effectively places a limit on how far a particle may be from its predicted position; this is equivalent to a bound on the apparent acceleration or rate of work.

The use of infinite costs in place of cij > Crnax greatly improves computational efficiency as we need not evaluate costs for associations between well separated particles, nor worry about such associations in the pairing algorithm. The number of finite values of cij will be greater than or equal to Max(M, N) but much less than (M + 1)(N + 1).


One of the difficulties with basic cost function bij is the need to have an estimate for the velocity uri]" If a pairing was made for P[i] at the previous time step, then we may use this to estimate uri]' However, if we have no history for P[i] then we must make a reasonable guess at uri] but make an allowance for the guess being wrong. This guess could be based on a knowledge of the flow or the behaviour of other particles in the neighbourhood of P[i]' For thin light sheets or flows where there are no appreciable velocity gradients parallel to the viewing direction within the illuminated region, the local mean velocity is a good predictor of u[i]' For thick light sheets in three dimensional flows we may be able to do no better than predict u[i] as some constant. Regardless of how we predict uri] where it is not known from the particle history, we shall increase the maximum distance between the predicted position of P[i] and any paired image q[jl by reducing the cost of any such association by multiplying by some factor 1]i ,;:; 1 (we may wish to prevent matches on large, small or elliptical particles when Uti] is not known by setting 1]i = 00 for such particles). Typically this discount factor will be derived from some estimate of the maximum error in the estimated value of uri]' If 1]i is set too large, then only particles conforming very closely to the predicted uri] will be able to enter the tracking process. If too small, then initial matchings may be made over unaceptable distances. However, provided Cmax is sufficiently small, any erroneous pairings introduced by a value of 1]i which is too small will not persist past thc frame pair at which they entered the problem: uri] at the next step will not predict the particle's position with sufficient accuracy. For particles Pri] which were matched at the previous time step we know uri] so offer no discount and set 1]i = 1.

We utilise a number of ways of detecting and using particles which appear at almost the same location at t[n + 1]' The first is based on the shape of a particle: if a blob appears very elliptical, then it may represent two particles close together. We assign to the elliptical blob one ordinary particle with the modifier Cj = 1, and a possible second particle with Cj set to some constant greater than one. Associations with the ordinary particle proceed as normal. Associations with the second particle are costed higher by the factor Cj. If Cj is sufficiently large, and Cmax sufficiently small, then an association with this second particle will only be made if we predict two particles should be there on the basis of the information at t[n]' Note that particles with no velocity history are prohibited from matching with such particles.

The modifier Pfi; operates in much the same manner as [,) except that it is based on the overall area of the particle rather than the shape. Particles smaller than a preset limit have Pfij = 1, while those larger are treated as two, the second with a price premium of 9) ~ 1.

As noted in subsection 3.2, we locate particles using two different thresholds. Particles satisfying the more stringent threshold criteria are assigned a modifier value of Tj = 1, while those satisfying only the less stringent threshold suffer a premium set by T j ~ 1, restricting the size of the error between their predicted and actual locations

at t[n+ 1]'

Extensive use of this costing and pairing strategy in a variety of different flows suggests only a very weak dependence on the 8j , Pfij and OJ modifiers. It is for this

38 s. B. DALZIEL

reason, combined with simplicity, that we have adopted step rather than continuous functions. Only 1]i and Cmax have a marked effect on the tracking process.

3.5. Implementation

The tracking algorithm we have outlined in the previous four subsections has been implimented on a PC/AT compatible micro computer using a Data Translation DT-2862 arithmetic frame grabber. All software has been written by the author in a combination of assembler and Fortran in such a manner as to allow fully automated tracking. To improve computational efficiency the association costs cij are evaluated only once and stored after rescaling to a single byte integer value. The cost matrix cij requires (M + 1)(N + 1) bytes of memory limiting this simple implementation to 511 particles in each sample (cij is 256 Kbytes). An alternative implementation of the algorithm making use of the fact that many cij are infinite allows up to 4095 particles to be tracked by the PC/AT. Practical considerations limit the number of particles to around 2000. (When the experimental work reported in this paper was undertaken only 511 particle could be tracked.)

The speed of operation depends on the number of particles and quality of the images. When tracking a relatively small number of particles, the execution speed is limited to around 25 frame pairs per minute by the need to acquire images in groups of four samples from a playing video tape. When tracking more than around 200 particles the computations for the location and pairing of particles becomes important. With approximately 400 particles the processing rate varies from six (20 MHz 80386) to fifteen (25 MHz 80486) frame pairs per minute depending on the speed of the computer.

For each time step the pairing algorithm is executed twice. The first pass operates on P[i] and qU] at t[n] and t[n + 1]' respectively. Any particles at t[n] which are not matched are stored away for later use. Particles at t[n + 1] which were not matched are then compared with particles a t[n~ 1] which were also left unmatched, to determine if any further pairings may be made. This second pairing operation increases the number of matches made by between 1% and 10%. The number of matches made at any time step depends primarily on the velocity component normal to the light sheet and the thickness of the light sheet.

3.6. Velocities

There are a number of different methods of estimating the velocities from the positions of a particle along a path. The simplest and arguably most effective of these fit a surface to the positions of a given particle in x, t space. In practice we fit a straight line to the x, t and z, t data sets over some time interval sc5t during which the change in velocity is relatively small. The slope of the fitted lines may then be used to estimate the velocity. In the limit of s = 1 this degenerates to the simple finite difference approach. The fitting process should utilise all the information in sc5t, suggesting either


a least squares fit or a Chebeshev polynomial. Both these strategies are in some sense optimal and in practice yield very similar results. The results presented in this paper employ the least squares approach.

With the least squares fit the error in the estimated velocity (Ju is given by the standard error in the slope of the line which may be expressed as

{ 12 }1/2 (J

(Ju = (s + 2)(s + 1)s ,5; . (7)

With the values used in this paper ((Jx < 0.2, s = 5,6t = 0.08s) we have a velocity error (J u < 0.24 mms -1. Typically the error is less than half this value.

We should note that the particle tracking process and the method of calculating velocities over paths with s> 1 leads to a sampling bias in favour of regions of the flow in which the velocity normal to the light sheet is relatively small. This arises through the need for particles to remain in the light sheet over s + 1 samples. The value s = 5 has been chosen so that relatively few particles will ever cross the 7 mm light sheet in less than sl5t = 0.4 s. In the present experiments this bias is only significant in the very early stages of the decay process and will not affect the discussion in the following sections.

4. Nonrotating turbulence

We shall consider first the nonrotating (f = 0) limit as the behaviour of this well known flow will show any major problems with our experimental apparatus or measurement method. There are a wide range of methods of characterising such a flow. For the present we will confine our attention to relatively simple velocity statistics.

We define the spatial-ensemble mean velocity for the illuminated region as

use(t[n]) = I ~ U[i][r](t[n])/I M[r](t[n])' r , r

(8)

where U[i][r](t[n]) is the velocity for particle i in realisation r at time t[n] in the light sheet. There are M[r](t[n]) particles for which the velocity is known in realisation r at time t[n].

The summation is over all particles located in each realisation (approximately 350 particles) and all realisations (typically R = 25 realisations) at the specified time. As our tank represents a closed system the flow must have a zero mean velocity. Therefore we might expect (8), which is a sample of the global mean velocity, to evaluate to zero. However, as we can see from Fig. 3, Use does not vanish as expected. Two features are readily apparent in the horizontal component of use: an oscillation with a period of approximately 1.25s (Ut/M:::::; 9.5), and an exponentially decaying mean value. The first of these has the correct frequency for deep water waves in the

40 S. B. DALZIEL

0.0 50.0 100.0 150.0 200.0 250.0 300.0 150.0 400.0 20.0 +-__ ----I ___ -'-___ '--__ -'-___ .l..-__ --L. ___ -'--__ --L.-.,

1."i,O

10.0

'i.(l

-15.0

Ut

M

t(s)

o.us

ii, iN 0.06

U

0.04

0.02

.().06

.().<I!

-20.0 +----r--,----r--,----r--..-----,---~-_,_--r--_.---i (l,() S.O IOJ) 15.0 20.0 25.0 10.0 35.0 40.0 45,0 50.0 55.0 60.0

Fif:. 3. Evolution of the mean velocity within the light sheet for f = O. The solid line is for the horizontal component, and the dashed line for the vertical component.

tank. These waves are initiated by the passage of the grid. The mechanics of driving the grid prevented a rigid lid from being used. While slats were fitted between the bars of the grid to reduce the effect of the free surface, a significant wave component still exists.

The origin of the decaying flow in Fig. 3 is also due to the passage of the grid. Careful observations show that the passage of the grid sets up a significant large scale circulation within the tank. This motion is predominantly in the direction of the grid near the centre of the tank, with return flow near the boundaries. Increasing the solidity of the grid near the walls and making a double pass with the grid reduced the size of this circulation, but it did not prove possible to eliminate it. The embarrassing existence of the systematic flow is not unique to the apparatus used in these experiments (Maxworthy, personal communication) although much longer tanks do not seem to suffer this problem (e.g. Britter et al. [4J). However, most studies do not address the issue, partly due to the difficulty with measuring and allowing for such a flow, especially using traditional techniques. The majority of experiments on homogeneous turbulence have been in wind tunnels where the problem is much less severe with a much large ratio of mesh length to tunnel width (giving a much better separation of scales). A similar phenomenon is known to exist in oscillating grid experiments with the production of a larger scale systematic flow (e.g. McDougall [13J). Tn both cases the scale of the systematic flow is set by the overall tank geometry and not the mesh size.


Due to the existence of spatial structure in the surface waves and in the systematic circulation, statistics should be evaluated relative to a spatially varying ensemble mean, De' say. In practice the mean is evaluated for each time step by distributing the particle velocities onto a regular grid (using a local weighted least squares fit) and averaging the grid over all realisations. Figure 4a plots the evolution of the turbulent fluctuations relative to this mean. If the initial turbulence field is approximately homogeneous and isotropic then during the initial stages of decay, starting some time t > to after the passage of the grid, and extending until Ut/M is greater than around 200, the turbulence intensities are then found to follow the power law

u2 = A [U(t - to)]-P U 2 M '

(9)

where A and f3 are constants and to is the virtual origin at which the fluctuations would be infinite. Batchelor's [2J initial period decay law gives f3 = 1 for the type of grid used in these experiments. More extensive data sets suggest a value of f3 between 1.2 and 1.4 (e.g. Comte-Bellot and Corrsin [5J). As is frequently the situation with power laws, the exponent f3 for the optimal fit is sensitive to location of the virtual origin to. Typically the virtual origin is found to be located at Uto/M between 1 and 6, with the initial period, during which the power law applies, starting some ten to twenty grid time scales (M/U) later.

Figure 4b reproduces the fluctuation data in a log-log format. Between approximately 10 and 75 grid time scales after the passage of the grid the energy decays following a power law with f3 ~ 1. The size of the ensemble is not sufficiently large to pin down f3 more precisely as the value of to is also unknown. The initial much flatter curve is due to a combination of the behaviour immediately behind the grid and the conditional sampling provided by the particle tracking (u < (jy/{sbt». The bias due to the conditional sampling is significant only in the very early stages of the decay. The other notable feature of this plot is the marked decrease in the decay rate at times larger than Ut/M ~ 75 (t ~ lOs). The decay does not follow a power law after this point and cannot be attributed to the normally increased decay during the final period. Rather it would appear as though the largest turbulent scales are starting to interact with the large scale systematic flow set up by the initial passage of the grid. Moreover, as this circulation is on the same scale as the dimensions of the tank, the overall experimental geometry will be playing a significant role. A power-law decay appears to establish again after Ut/M ~ 250 once the scales of the systematic flow are fully incorporated into the decay process. The decay rate after this point will be influenced by the geometry of the tank but is again with f3 ~ 1.

Clearly the large scale circulation imposed by the initial passage of the grid imposes severe experimental difficulties. A variety of methods were tried to minimise its generation, the results presented in this section representing the best alternative. As will be seen in the next section, rotation compounds the difficulties with the generation of inertial waves.

42 S. B. DALZIEL

0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 20.0 +-----''----....&..----"'------'------'------'-----'------'---,

lH.O

11),0

14,n

12(' \

\ , lOJl

H.O

(dl

4.0

2.0

\ \ \ \ \ \ \ \ \

\ ....

Ut

M

u',w'

U

\~ ~------~--~------------~'.

' .... -'-------------------~.::-~-==::::::==:--------' --------:::::

t(s)

0.08

0.D7

0.06

0.05

0.04

0.03

0.02

0.01

(U1 0 -1.(-, ---5r.0---I'O.-0 --""15'.0---2TO.0---2"S-.0---30r.0---3'S.-0 --4"O'.0---4S!":.0---:s:r0.:-0 --5J5r:.0--:6(]t.o 0.0

(a)

20.0 40.0 60.0 80.0100.0 200.0 400.0 600.0 0.8 1.0 2.0 4.0 6.0 8.0 10.0 20.0 -,..::;:...:.L:....-__ ..::L.. __ ---':.-......:.._-'--...L-__ ---' ___ -'-_-'---J..--I. ___ -'-__ --'-_-'-i

10.0

9.0

8.0

7.0

6.0

5.0

4.0

3.0

2.0

Ut

M

t(s)

0.08

0.07

0.06

0.05

0.04

0.03

0.02

om 0.009

0.008

0.007

0.006

0.005

1.0 .L._-~---._-.___r~---._--__r-__r-r_._--__r---r_-r_--.-; 40.0 60.0 80.0100.0 0.12 2.0 4.0 20.0 6.0 8.0 10.0 0.2 0.4 0.6 0.8 LO

(b)

Fig. 4. Evolution of the turbulent velocity fluctuations about the spatially varying ensemble mean flow for J = O. Plot (a) has a linear scale while (b) a logarithmic scale showing the power law behaviour of the decay (to = O.4s). Solid lines indicate the along-tank horizontal component, while dashed lines denote the vertical component. The initial period decay law with f3 = 1 is shown as a dot-dash line in (b).


5. The effect of rotation

In the interior of the tank, rotation affects the motion through the Coriolis force acting primarily on large scales and low velocities (i.e. small Rossby number). The systematic flow found for nonrotating turbulence has both these features and will hence be affected by the rotation from a very early stage. Figure 5 shows the evolution of the Use mean velocity at a rotation rate of Q = f /2 = 0.5 rad. s -1 (for the sake of brevity we shall present results only for this rotation rate). The relatively short period (~ 1.25 s) free surface waves (now Kelvin waves) are still present. In addition, we are able to see two (or more) relatively large amplitude inertial wave modes in place of the simple mean flow found in the previous section.

Plane inertial waves in a rotating system may be shown to have the velocities obeying

u = ±(k2A.3 -k3A.2)cos(K(k1X + k3Z)+kJ!t)-A.1 sin(K(k1x + k3Z)+kJ!t),

w = ±(k1 A.2 - k2A.l)COS(K(klX + k3Z) =+= kJ!t) - ..1.3 sin(K(k1x + k3Z) =+= kJ!t),

(10)

in the plane of the light sheet (e.g. Greenspan [8]). The magnitude of the wave number is given by K and the orientation by the unit vector k. The amplitude vector A, may be

0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 4(X}.O

20.0

Ut 0.08

u,w -(mms-') M

15.0

U,W (HI6

U 10.0

004

5.0 0.02

~------ " " ~ " " " ." , '" ' 0.0 U.O / ..... ~--' - ...... , '" '-, ./ , '" .... ,-_ ...... , ,_./ " -5.0 -0.02

-0.04 10.0

-0,(16 -15.0

t(5) -O.OI'!

-20.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0

Fig. 5. Evolution of the mean velocity within the light sheet for f = 1 rad/s. The solid line is for the horizontal component and the dashed line for the vertical.

44 S. B. DALZIEL

any vector perpendicular to k. Subscripts indicate the components in the three axis directions. The frequency OJ of these plane waves is related to the angle between the rotation axis and the unit wave number vector by

OJ = kd (11 )

The solution given by (10) satisfies both the linear and nonlinear intertial wave equations. Note, however, that superposition of modes is not possible in the latter case.

While the inertial waves in this flow are likely to be nonliner, we may make some progress by assuming a degree of superposition is possible. As noted above, Fig. 5 shows evidence of two inertial wave modes. Experiments at different rotation rates have shown the relative amplitude of these modes appears to depend on the rotation rate, presumably the result of variations in the ratio of the time to traverse the grid and the rotation period. By considering the vertical mean velocity we can see a mode with OJ ~ 0.4f, implying the wave number vector is inclined at an angle of approximately 66° to the rotation axis. In addition to this OJ ~ O.4f mode, the horizontal mean velocity shows a response at OJ = f The frequency and absence of vertical motion at this frequency are consistent with the wave number vector parallel to the rotation axis.

As with the nonrotating flow in the previous section, we may determine the spatiotemporal structure of the mean flow and utilise it when evaluating the statistics of the turbulence. Figure 6a plots the evolution of the fluctuations about the varying ensemble mean flow. Figure 6b plots the same data on logarithmic axes from which we see the existence of a power law giving a somewhat lower decay rate than observed for f = O. While the value of f3 is not known for either case, overlay in the two plots allows a direct comparison to be made if we assume to is the same for both cases. The decay process does not show the abrupt change in behaviour found for f = 0, suggesting that interaction between the inertial waves is generating turbulence at relatively large scales preventing an initial scale separation between the inertial waves and the turbulence (we had such a scale separation in the nonrotating flow in the previous section). Note that the energy contained in the inertial wave field is significantly larger than the systematic flow in the f = 0 limit.

A series of four plots showing the spatial structure and temporal evolution of the ensemble mean velocity field is given in Fig. 7. This series has been chosen to show the spatial structure of the inertial waves and the relatively rapid changes it undergoes. Note that the standard error in this spatially varying ensemble mean is proportional to U~/Rl/2, where u~ are the fluctuations about ue and R is the number of realisations (25 for the present experiments). While this represents a relatively small degree of uncertainty, statistics for a much larger ensemble would be desirable.

The degree of anisotropy between the horizontal and vertical components is illustrated by the structure function

K = (u~ - w~)/(u~ + w~), (12)


0.0 50.0 100.0 1500 200.0 210.0 JOO.O 310.0 400.0 20.0 +-----'-----'----......... -----'-----'----......... ----'------'---,

18.0

16.0

14.0

12.0

IO,n

S.o

6.0

4.0

2.0

(J.()

10.0

9.0

8.0

7.0

6.0

S.O

1.0

2.0

Ut 0.08

M U',W'

(mms") u', W' 0.07

u 0.06

0.05

0.04

om

0.D2

------~~=-:-:::-::::-::--::::-_-_-_ ..... _-_-_-_-_-_-_-_-_-_-_-_ ... _-_-_-_-_-_~_-_-_-_-_-_-1. 0.01

t(s)

0.0 <0 +----,---'lOr.0---ISr.0---20r.O---2S'.0---30'.0---3I'.0---40'.0---41!":.0~--::10!":.0~--::55!":.0~-:60t:.0 0.0

(a)

0.8 1.0 2.0 4.0 6.0 8.0 10.0 20.0 40.0 60.0 80.0100.0 200.0

t(s)

u',w

U

0.08

0.D7

0.06

0.05

0.04

O.DJ

0.02

0.01

0.009

0.008

0.007

0.006

0.005

1.0~'---r---~--'-'--r---~---'--r--r~---'---~r--r--r~ 0.12 0.2 0.4 0.6 0.8 1.0 2.0 4.0 6.0 8.0 10.0 20.0 40.0 60.0 80.0100.0

(b)

Fig. 6. Evolution of the turbulent velocity fluctuations about the spatially varying ensemble mean flow for f = I rud/s. Plot (a) has a linear scale while (b) a logarithmic scale showing the power law behaviour of the decay (to = O.4s). Solid lines indicate the along-tank horizontal component, while dashed lines denote the vertical component. The initial period decay law with fJ = I is shown as a dot-dash line in (b).

~ ~

0 ~

~ g

~ 8

~ ~

8 ~

8 o

0 0

0 0

0 b

oo

b

oo

b

~ $

~

ttl!!tttllllll

~ttt!ttttlll'

tttlt!ltlll

-ttttttttill

g-it

tttt!tlll

tttt!!tlil

~~tttttt!II

t ttl

t t

, I

I

t f t

t t t

, I

I

s, t t

t 1

f 1

1 I

I

~ I~~ttftll

g 8- o o

rttll/I

tifltl!

tttl!f!

tltllt"

tttl

tt!!!.

t ttl

t ,

, \

\ !

, \ ,

,

tttl"'\

\!\\",

ittl""\\\\"""

1tttllll'

o 0

IN

g ~

6 ~

~ ~

~ ~

~ ~

~ ~

~ ~

8 S

o b

<=>

l,tlt1

ttlttttll

t g-

1~,

1/lt

fttttttttt1

t

~"

Illffttttttttt!

I ~~,'

IIIf

1t1

1tt

1tt

1t

t o

Ilt!

t11

11

11

1t1

1!!

t

, I

I I

! t

1 1 tI

t t

t f

1 t

t f

t ~

1llltttftttttttlt!

f 1

IIIIItfttftfttllll

"]

I I

I tit

t t

1 1

1 1

t ttl

I I

I t

<;;'

-1

11

/ltlffft1

flttll!

t ~

III/tlftfttffttttl!

I g

j I

! I

/ ttl t

t t

t f

ttl I

tt

I I

If/fllttttttftl!!!

I 81

t

f f

t :

t j

t t

t t

t tt

l f

t t

t t

o tlflttttttttttttttl

t tt!ttttttttttttt'l

l

j

~1Iftltttfttttttti\1\

t Itt!!lt/ttttlli1

\\

I i ~

I I 1

ft ttl I

/ ti

t I

\ I

i 1

, \

I I

III

I! tttft/ffttt!!1

~

fI' ~ ~ ~


260.0

240.0

220.0

2(Xl.O

180.0

160.0

140.0

120.0

1((1.0

sO.n

60.0

40.0

20.0

00

-20.0

40.0 " 151J.0 .I()o.O ·50.0 0.0 5d.0 100.0 Isb.o 200.0

(e)

,- ,- ,- ,- ..-260.0

240.0

220.0

200.0 -- .......

180.0 ..... -- V"

..... ..... -- ....-160.0 ..... ..... ....... ....-140.0 /' /' ....... ....... ....- ..-

120.0 /' /' /' -- -- -- --,- ".- /' ..... -- ....... ..-100.0 ,- ".- ".- ".- ..... r'

80.0 " " ,- ,- '" '" 60.0 .- .- .- " " ..- ..-

I .- .- .-40.0

20.0

0.0

-20.0

-40.0 .150.0 .Ido.o .50.0 0.0 56.0 100.0 Isb.o 200.0

(d)

Fig. 7. Series of plots showing the evolution ofthe spatially varying ensemble mean motion for I = 1 rad/s. The passage of a wave crest through the series is clearly visible. The velocity scale is indicated on the first .'igure. The times corresponding to these plots are (a) 4s(Ut/M = 29), (b) 6.24s(Ut/M = 46), (c) 8.48s(Ut/M = 62), (d) lO.72s(Ut/M = 79). The arrows at the top and right-hand extremes of the plots are the averages for the corresponding column or row.

48 S. B. DALZIEL

which is plotted in Fig. 8 for both the f = 0 rad/s and f = 1 rad/s cases. In the absence of rotation we see a persistent 10% anisotropy, a figure consistent with previous investigators for classical turbulence. For f = 1 rad/s contamination from the systematic flow yields a smaller degree of anisotropy at early stages. As the decay rate of vertical fluctuations is greater than for horizontal fluctuations, there is a gradual increase in the anisotropy.

The dominant effect of rotation on the flow is in the growth of the associated length scales. We define the two-point correlation functions

L L (u;(x)uj(x + Ax) - u;uj) Rij(Ax, t = tJ = I I '

uiu j

(13)

where Ax is the separation (in the y = Yo plane) between the points at which the two velocity components (projected onto the y = Yo plane) are evaluated. Subscripts are used here to represent different velocity components rather than different times. The summation is over all particle pairs with the appropriate separation, falling in the specified time interval (tc ± AtJ, and over all realisations. Note that the finite thickness of the light sheet (()y) means that the correlation functions defined by Rij will in general be slightly smaller (numerically) than the true correlation functions for velocities lying in the y = Yo plane. This will not be significant, however, for IAxl »()y.

Figure 9 shows the Ruu = Rll and Rww = R33 two-point correlation functions

0.0

0.5

0.4

0.3

0.2

0.1

0.0

-0.1

-0.2

-0.3

-0.4

-0.5 0.0

50.0

u' - w'

u' + w'

5.0

100.0 150.0

10.0 15.0 20.0 25.0

200.0 250.0 300.0

30.0 35.0 40.0 45.0

350.0

Ut

M

t(5)

50.0

400.0

55.0 60.0

Fig. 8. Evolution of the structure function K = (iie - we)/(iie + we) for the classical (solid line) and rotating (dashed line) flows.


(a)

-7 7 ct

'" ~ .~

~ ~ ()

-

(b)

Fig. 9. Two-point correlation functions for the f = 1 rad/s ensemble. (a) Ruu = Rll and (b) Rww = R33 at t = 3s (Ut/M = 22); (c) Ruu = Rll and (d) Rww = R33 at t = 20s (Ut/M = 147).

50 S. B. DALZIEL

(c)

(d)

Fig. 9 (Continued)


evaluated at two different times. The first pair of plots is evaluated relatively early at t = 3s (Ut/M = 22) before the effects of rotation are dominant. The behaviour as a function of spatial separation follows the normal form of correlations between velocities parallel to the separation remaining positive, while those perpendicular pass through zero. The Ruv = R 13 correlation function (not plotted) is approximately zero for all separations, as would be expected. The integral length scales evaluated from these correlations are Luux = 0.69 M = 22mm for horizontal velocities and separations, and Lwwz = 0.50 M = 15 mm for vertical velocities and separations. The inequality of these two length scales reflects the growing anisotropy introduced by rotation in addition to the weak anisotropy inherent in grid turbulence and contamination from the systematic flow. For isotropic turbulence, the length scales with the velocity and separation perpendicular should be half the value of that when the velocity and separation are parallel. Here we find they slightly exceed this relationship (Luuz = 12mm and Lwwx = 10mm), an affect probably due to the anisotropy introduced by rotation and the systematic flow.

The correlation functions at t = 20s(Ut/M = 147) show clearly the enhanced vertical correlation of the velocities, and so will yield larger values for the vertical integral length scale. However, the systematic flow set up by the initial passage of the grid contaminates these figures to a significant degree, with the correlation functions increasing with vertical separation for larger separations. This makes evaluation of the integral length scales from these statistics very difficult. Further discussion on these length scales and how they are affected by rotation and the systematic flow are beyond the scope of this paper.

One of the strengths of particle tracking is the ability to obtain Lagrangian statistics much more efficiently than the manual digitization. Figure 10 plots the Lagrangian autocorrelation functions for particles in the light sheet at t = 5s(Ut/M = 37). Due to the relatively thin light sheet we are considering there is a strong bias at longer times to particles with small velocities normal to the sheet. The number of particles used in this calculation is also plotted in the figure, and can be seen to decline approximately exponentially, as would be expected for a random distribution of cross-sheet velocities. Caution should be used when interpreting this plot as the turbulence is decaying with time; this is also a feature of comparable Lagrangian statistics in wind tunnel experiments (e.g. Snyder and Lumley [15]). The present results, however, are also contaminated by the systematic flow. The fluctuations in the trace apparent for separations greater than around 2 seconds are due to the free surface Kelvin waves: our Eulerian method of correcting the velocities for the systematic flow is not appropriate for Lagrangian autocorrelations (further discussion is beyond the scope of this paper). Note also that there are too few particles in the sample for separations exceeding 4 seconds.

6. Conclusions

In this paper we have outlined a simple, efficient PIV method of automatically tracking individual particles in a fluid flow. The method has been implemented on a

52

I:: 0.5 o .;::

'" e o u o :; '" c 0.0

.~ Cl C

'" C, '" ...J

-0.5

0.0 5.0 10.0 15.0 20.0

-.----.. -::.":::":-:,:-... - ....... - - - -.-

25.0

Ufl.t

M

S. B. DALZIEL

30.0 35.0

fI.t (5)

.1.0+-------r-------.-------,r------,--------1 0.0 1.0 2.0 3.0 4.0 5.0

Fig. 10. Autocorrelation functions with f = 1 radjs for particles tracked from t = 5s (UtjM = 37). The u - u component is plotted as a solid line, w - w as dashes, u - was dot-dash, and w - u as dot-dot-dash. The relative number of particles belonging to the original set still illuminated by the light sheet is also plotted as a dotted line. Note that at large time separations the number of particles becomes too low for the autocorrelations to be meaningful.

PC/AT compatible micro computer utilising a medium cost frame grabber and Super VHS video tape recorder. While the algorithm described in this paper has been implemented to allow up to 4095 particles to be tracked simultaneously, practical considerations impose a limit of approximately 2000 particles. The experiments described in this paper utilised an early version of the particle tracking system following typically 350 particles. The high throughput (up to 25 frame pairs per minute in some situations) makes this method ideal when ensemble statistics of a temporally varying flow are required. Both Eulerian and Lagrangian views of the flow may be obtained from the particle paths.

The particle tracking technique has been applied to a wide range of flows and is illustrated here as part of a study of the decay of rotating turbulence. In addition to being a convenient method for measuring the velocities in this zero mean flow experiment, the ability to gather information over a two dimensional region has proved invaluable in analysing the systematic flow contaminating the experiment.

Severe limitations in the experimental set up have been discovered. The initial passage of the grid is found to set up a large scale systematic flow. In the absence of rotation this flow is predominantly in the direction of the grid near the centre of the tank, with the return flow near the boundaries. Multiple passes of the grid and increased solidity near the walls reduced this effect but did not prove possible to


eliminate it. When the system is rotating, the systematic flow breaks up into a large scale inertial wave field in the order of an inertial period. The wave field appears to consist of the superposition of two modes, one with the wave number vector parallel to the rotation axis, and the other with it inclined at approximately 66°. This wave field has a high degree of coherency throughout an ensemble of 25 realisations. The spatially and temporally varying structure of this wave field is established from the ensemble flow, and used as the basis of a correction to establish the turbulent component of the flow. The separation in length scales did not prove sufficient for this technique to be entirely effective.

The generation of such a strong systematic flow during the passage of the grid is of deep concern in the rotating system. Earlier investigators have not reported such difficulties, primarily as a result of such information being less accessible to traditional measurement techniques. The problem may be expected to be less severe with the symmetric grid arrangement of Ibbetson and Tritton [llJ as the velocity component of the large scale flow introduced by the passage by the grid will be parallel to the rotation axis over most of the tank. However, at the initial mid-height position of the two grids and at the top and bottom of their annulus we would expect a significant radial component which may be strongly affected by the Coriolis force. The experiments of Jacquin et al. [12J will be less affected due to the mean flow parallel to the rotation axis and the absence of boundaries perpendicular to that axis. Moreover, as our present difficulties are the result of initial transients, the statistically steady flow in their tube does not have the same form of forcing function.

The oscillating grid experiments of Bretherton and Turner [3J, Hopfinger et al. [lOJ, Dickenson and Long [6J and Fluery et al. [7J may also be adversely influenced by the existence of a large scale systematic flow. Experiments by McDougall [13J show that in the absence of rotation such oscillating grids generate a systematic flow with an amplitude comparable to the fluctuations. This flow will interact with the Corio lis force in regions of the tank remote from the grid. This feature is being investigated further by Drayton (personal communication) using the present particle tracking technique. The advection of smaller scale turbulence by this flow may well lead to the formation of the intense vortices and rapid spatial change from three dimensional to nearly two dimensional motion characteristic of such experiments.

Despite the experimental limitations, rotation has been found to decrease slightly the rate of decay of turbulent fluctuations. In addition to a small increase in the anisotropy as the decay progresses, rotation leads to a marked increase in the integral length scales. These findings are in broad agreement with those of other authors.

References

1. Adrian, R. J., Particle-image techniques for experimental fluid mechanics. Annu. Rev. Fluid Mech. 23 (1991) 261-304.

2. Batchelor, G. K.: The Theory of Homogeneous Turbulence. Cambridge University Press (1953).

54 S. B. DALZIEL

3. Bretherton. F. P. and Turner, J. S., On the mixing of angular momentum in a stirred rotating fluid. J. Fluid Mech. 32 (1968) 449-464.

4. Britter, R. E., Hunt, J. C. R., Marsh, G. L. and Snyder, W. H., The effects of stable stratification on turbulent diffusion and the decay of grid turbulence. J. Fluid Mech. 127 (1983) 27-44.

5. Comte-Bellot, G. and Corrsin, S., The use of a contraction to improve the isotropy of grid-generated turbulence. J. Fluid Mech. 25 (1966) 657-682.

6. Dickinson, S. C. and Long, R. R., Oscillating-grid turbulence including effects of rotation. J. Fluid Mech. 126 (1983) 315-333.

7. Fluery, M., Mory, M., Hopfinger, E. J. and Auchere, D., Effects of rotation on turbulent mixing across a density interface. J. Fluid Mech. 223 (1990) 165-191.

8. Greenspan, H. P., The Theory of Rotating Fluids. Cambridge University Press (1968). 9. Hichcock, F. L., The distribution of a product from several sources to numerous localities. J. Math.

Phys. 20 (1941) 224. 10. Hopfinger, E. J., Browand, F. K. and Gagne, Y., Turbulence and waves in a rotating tank. J. Fluid

Mech. 125 (1982) 505-534. 11. Ibbetson, A. and Tritton, D. J., Experiments on turbulence in a rotating fluid. J. Fluid Mech. 68 (1975)

639-672. 12. Jacquin, L., Leuchter, 0., Cambon, C. and Mathieu, J., Homogeneous turbulence in the presence of

rotation. J. Fluid Mech. 220 (1990) 1-52. 13. McDougall, T. J., Measurements of turbulence in a zero-mean-shear mixed layer. J. Fluid Mech. 94

(1979) 409-431. 14. Perkins, R. J. and Hunt, J. C. R., Particle tracking in turbulent flows. In Fernholz, H.-H. and

Fielder, H. E. (eds), Advances in Turbulence, Vol. 2. Springer-Verlag (1979) pp. 286-291. 15. Snyder, W. H. and Lumley, 1. L., Some measurements of particle velocity autocorrelation functions in a

turbulent flow. J. Fluid Mech. 48 (1971) 41-71.

Measurement of product concentration of two parallel reactive jets using digital image processing

H. STAPOUNTZISt, 1. WESTERWEEU, 1. M. BESSEM2 ,

A. WESTENDORp2 & F.T.M. NIEUWSTADT2

1 University of Thessaloniki, Dept. of Mechanical Engineering, Box 443, Thessaloniki 54006, Greece 2 Delft University of Technology, Faculty of Mechanical Engineering and Marine Technology, Laboratory of Aero- and Hydrodynamics, 2628 AL De(ft, The Netherlands

Key words: jet flows, mixing, concentration fluctuations, LIF techniques, digital image processing

Abstract. The chemically sensitive LIF technique [9J is employed to study the mixing of two reactive axisymmetric jets, one of which carries fluorescein, in an ambient quiescent fluid. The degree of mixing depends on the jet spacing and the axial position downstream of the jets and power laws are found to hold for some concentration characteristics. Unlike the far velocity field of dual plane jets, self preservation laws are not found to hold in general for the concentration field.

1. Introduction

The interaction and mixing of separate jets is a fundamental problem of turbulent shear flow and it is used in a wide variety of engineering applications such as burners and combustion chambers, chemical reactors, propulsion, thrust augmenting ejectors for V/STOL aircraft and fluidics. Both the near or far fields of the jets may be of interest as in the recirculation regions of confined jets and the far field of acoustic noise.

There is little information on dual-jet velocity and pressure fields available in literature and hardly any work on the concentration field of two mixing jets. The conclusions drawn from the study of the interaction of two parallel plane jets [1-6J can be summarized as follows (see Fig. 1): Initially, in region (A), extending 15 to 20 jet diameters, d, from the nozzles and for nozzle distances less than about 30 d, the two jets converge towards the centerline between the nozzles. Reversed flow and two counter rotating vortices are observed inside this region. In the merging region, (B), further downstream a stagnation point exists, whose position scales almost linearly with the transverse distance between the jets, 1. Finally in the combined flow region, (C), the flow characteristics are resembling those of a single jet. Self preservation for the mean velocity is observed in the regions (A) and (C), and the effect of the Re number is significant. The region (B) is highly anisotropic, with the centerplane

longitudinal velocity fluctuation, u2 , reaching a minimum, the transverse, v2 normal to

the jet centerplane reaching a maximum and w2 showing little change. The intensities

there are ordered in value as v2 > u2 > w2 • It is anticipated that the commencing of mixing and reaction (for chemically reactive jets) should occur in this regime. It is not

55

F T. M. Nieuwstadt led.), Flow Visualization and Image Analysis, 55--69. © 1993 Kluwer Academic Publishers.

56 H. ST APOUNTZIS ET AL.

-------

x

1_ ~~;:---~"" ,.9~~_ ~~ _______ _

Common I Converging I Merging I Combined end wall f-- region A -+- region 8 --1 region C

Fig. 1. Mean velocity profiles for plane dual jet flow [6].

known what the posItIon and extent of the merging region would be in dual axisymmetric jets. However, data for axisymmetric dual coflowing reactive jets [7] indicate that it should lie much further downstream from the jet nozzles compared to the dual plane jet position.

The lack of experimental data is partly due to the difficulty in describing mixing in turbulent flow and especially in the measurement of instantaneous concentrations of rapidly reacting species [8].

A useful technique which enables one to distinguish fluid that has been molecularly mixed from that which has been merely stirred, is that of chemically sensitive Laser Induced Fluorescence [9, 10]. This method exploits the pH-sensitive characteristics of a fluorescent dye excited by a laser light beam (or plane sheet) in conjunction with a nearly isothermal (for low concentrations) reaction between a base and an acid into which liquids (in both or in either of them) the dye has been dissolved. With the advent in the development offast data acquisition and image processing systems, this method can be used to measure the time dependent overall character of entrainment and mixing and the detection of coherent structures, information which cannot be obtained from the statistically averaged concentration field. On the other hand, one aspect of jet mixing which can be directly obtained by this technique is the axial distance, M, required to molecularly mix the jets to at least a given mass ratio. As an extension, the technique may be used to determine the locations of the upper and lower flammability limits, when a flame mixture is being studied.

The purpose of this paper is to apply the chemically sensitive LIF non-intrusive technique together with digital image processing in order to study the mixing of two parallel axisymmetric jets.

TWO PARALLEL REACfIVE JETS 57

2. Experimental set-up

The measurements described were carried out in a tapered vertical water tank 104m high with upper rectangular cross section 0.21 x 0.21 m and lower rectangular cross section 0.25 x 0.32 m, see Fig. 2. The tank had glass windows on all four side walls and the floor. The jets issued vertically downwards into the tank via two parallel stainless steel tubes of internal diameter d = 5 mm and outside diameter D = 6 mm. The distance between the centers of the tubes, l, was varied between I = 1.2 d to 8.2 d. The

mirror

reflected lIght-sheet

d

Base

1 Acid + Fluorescein

1

y

O.2lm

LED referencelIght

watertank

14m

t--------- O.32m ---..

Fig. 2. Experimental setup.


upper value of I was dictated by the size of the tank to avoid interference from the side walls and bottom of the tank. The jet Re number based on the mean exit speed U j

(approximately 1.9 m/s), the diameter d and the kinematic viscosity of water was Re :::::; 9000. The reactants used were Nitric Acid and Sodium Hydroxide. Different tests were made with initial acid or base concentrations between 0.1 Nand 1 N. Even at the high concentrations the density differences from that of water were small and the jets could be considered as momentum driven [11].

The laser light sheet was produced by a rotating (1000 rpm) polygonal mirror exposed to a 0.8 W Laser Argon ion beam (488 nm) via a fiber optic probe. The sheet was in the plane defined by the centerlines of the jets (x, y plane), its thickness was approximately 1 mm and its vertical extent about 0.65 m. In this present work measurements were taken only at that plane.

A CCD video camera recorded the fluorescence on the x-y plane in a square array of 512 * 512 pixels. If the jets were not interfering, the minimum and maximum Kolmogorov microscales under the present conditions would be about 25 11m and 210 11m respectively [12], meaning that not all of the smallest scales of turbulence might be resolved. Similarly, the temporal resolution for the smallest time scales should be at least 1 ms as opposed to the 40 ms frame rate of the CCD camera. Therefore, no detailed investigation of the structure of the concentration field was sought in these experiments. Rather, attention was focused on the mean and rms concentration fluctuations obtained from the intensity of fluorescence.

3. LIF technique

One of the jet fluids consisted of an aqueous acid solution of HN03 homogeneously mixed with a small amount of the pH-sensitive fluorescent dye, fluorescein. The other jet fluid was an aqueous solution of NaOH. The concentrations of the acid and base tried in the experiments ranged from 0.1 N to 1 N and the concentrations of the dye from 10 - 6 to 5 x 10 - 6 moles/It of acid solution. The fluid carrying the fluorescein is only visible (i.e. it fluoresces) when it reacts with the base or when it is diluted with the ambient water until the local pH of the solution crosses a certain threshold value.

Preliminary tests were performed in a small cubic glass container (100 x 100 x 100mm internal dimensions), placed inside the tank, and in the working tank itself. The purpose of these tests was to check: (a) the absorption and scattering of the laser light sheet, (b) the effect of temperature on fluorescence, (c) the homogeneity of the laser light sheet over the area of interest and (d) the dependence of fluorescence intensity on the base-acid concentrations. The procedure was to mix thoroughly acid + fluorescein and base solutions, wait until large scale motions have died out and then use the video camera and the image processing system, in situ, to acquire the data. The small container was used to save on chemicals, when used at high concentrations (1 N). The most severe case for absorption and scattering was to use 1 N concentrations of acid and base and the maximum fluorescein concentration. The


base-acid reaction is exothermal and for the aforementioned concentrations the overall temperature rise was less than 7.5°C, after reaction and mixing were completed. Reacted solutions of 16DC and 36DC were tested (ilT= 20D e), the latter solution being artificially heated to this temperature level.

Figure 3 shows the variation of the intensity of fluorescence on a horizontal cut between the two parallel vertical walls of the container. A mild exponential decay of fluorescence with distance is observed and an overall drop of about 20% due to the temperature rise of ilT= 20°e. Tests with more dilute solutions in the working tank indicated a fluorescence decay of almost linear type. A vertical cut in the tank (i.e. in the streamwise, x, direction) midway between the tubes, showed good uniformity of fluorescence over the area of interest (0 to 80 d) with some attenuation on both sides due to the longer optical path of the light beams.

The effect of scattering at 488 nm was tested by placing a band pass optical filter in front of the video camera. The sensed wavelength was thus centered at 514 nm (the wavelength of fluorescence). The shapes of the fluorescence-distance curves were similar with and without the filter, a sign that elastic scattering was small. However, the filter attenuated the overall fluorescence intensity by 60%, rendering the measurement of low levels of fluorescence inaccurate. Care was taken to allow for the release of the small air bubbles present in the water tank and the solutions in order to keep the light scattering to a minimum.

The main experimental results presented here were not corrected, at this stage, for the above mentioned effects, because at relatively high concentrations the extent of the reaction zone was small (compared to the size of the test container) and thus the absorption was small, while far downstream, where the reaction zone is wider the reactant concentrations were very low. In support of this, was the recovery, using the

Q)

::J

o

250.00

200.00

> 150.00

Q)

> Q)

100.00 » Q) I....

C>

50.00

0.00 I I I

0.00 40.00

deg. C

I iii I I i I I

80.00 120.00 Transverse distance Y (mm)

Fig. 3. Variation of fluorescence intensity across the calibration container (normal to the jet flow).


present method, of the concentration decay characteristics of a single, non-reacting jet of water and fluorescein issuing into the water tank.

Depending on the strength and nature of the acid-base solutions, the pH range over which the dye undergoes fluorescence transition, can be crossed in a very narrow range of mixture ratio, ([9], strong base-acid solutions, pHthreshold ~ 4.5) or in a wider range ([8], mild base-acid solutions, pHthreshold ~ 7.0). In the latter case the relationship between fluorescence intensity, I J' and the acid volume fraction is not linear because of the gradual change of I J' and the acid volume fraction is not linear because of the gradual change of I J' For strong reactants, I J suddenly rises from almost zero, to its maximum value I Jrnax at the threshold mixture ratio of the acid-base and thereafter it changes linearly with that ratio (so that the dye is merely diluted by the base). This is shown in Fig. 4 which presents data obtained in the small glass container. If Cp is defined as the concentration of the molecularly (dye-bearing) fluid whose local pH is above the fluorescence threshold then,

(1)

The threshold mixture ratio, also called stoichiometric ratio, defined as the mass ratio of the base fluid to the acid fluid can be determined by gradually adding some known base solution to some known acid solution and marking the relative acid-base concentrations when fluorescence suddenly starts. Since the fluorescence transition across the threshold is reversible and occurs on a very short time scale, the dye

o o c:i co

--.J Wo >c::: wo --.J"<t

>W cr: C)

0.40 0.45 0.50 0.55 0.60 0.65

VB/(VB+VA) Fig. 4. Fluorescence intensity against base volume fraction. Initial concentration of reactants is IN.

1WO PARALLEL REACTIVE JETS 61

fluorescence determines directly if the instantaneous local extent of molecular mixing between the jets has crossed this threshold.

The present experimental conditions are however different and more complicated than those of previous workers, in the sense that fluorescence could also be excited by the mixing of the acid jet fluid with the ambient water in the tank, a process which also raises the pH of the mixture. For this reason, strong acid-base solutions were selected for most of the runs and experiments with acid-dye solutions in different ambient fluid environments were planned in order to ascertain the overall effect of ambient fluid mixing. Equation (1) was therefore assumed to be approximately valid, at least in the central area between the jets.

4. Image analysis technique

For the image processing the PC-SEMPER software was used, on a HP-Vectra RSj20 PC-AT equipped with a DT2851 frame grabber and a DT2858 auxiliary frame processor. The image processing software was extended with additional commands for image alignment and image averaging.

Runs at different transverse jet spacings ljd, were recorded in the non-interlaced mode on a video tape (duration approximately 12 s) and were processed later on a workstation. Each frame was grabbed manually, but the subsequent processing was automated. Only halfframes were processed, i.e. those containing the "odd" or "even" lines of the field. The resulting image consisted of 256 rows parallel to the stream wise, x, direction and 512 columns in the transverse, y, direction, i.e. a total of 256 x 512 pixels of grey level intensity 0 to 255. This scale sets also the best accuracy in measuring the relative intensity of fluorescence (product concentration). An LED light positioned at the lower right hand side of the tank during the experiments, served as a reference point for the alignment and superposition of the individual frames for statistical averaging. This was accomplished by computing the cross correlation function in a small interrogation area round the reference light for every two consecutive frames. Knowing the relative displacement vector of these frames, of the order of a few pixels (which might have been caused by vibration of the tubes, camera movement etc.), it was easy to align the individual images and perform statistical operations on them to obtain the mean and rms of the concentration fluctuations. With the existing system up to 50 frames were analyzed due to storage limitations. Work is now in progress to enable sequential grabbing of up to 200 frames and subsequent processing on an HP 9000 computer with a much faster software.

5. Experimental results and discussion

Figure 5 shows an example of image grey level representation for the mean and rms product concentration averaged over 25 frames for acid-base solutions of IN. Higher

62 H. STAPOUNTZIS ET AL.

256 E 512

Mean product concentration

128 a: w CD

y E :J Z

:3 Cl a:

256

COLDUI'IN NUI'IBER 256 512

Rms product concentration

Fig. 5. Grey level representation of the mean and rms concentration fluctuations for two parallel reactive jets. Rows are parallel to jet axes, columns are normal. There are 256 rows and 512 columns. lid = 2.2,25 frames.

mean product concentration (fluorescence intensity) is observed in the central region between the jets and higher fluctuations outside this central region. This is also what one would expect for a dual plane jet, where both the mean and the fluctuating velocities peak off-centerline till the far downstream self preserving region is reached [3, 5, 6]. Of course the instantaneous composition of mixed fluid cannot be represented by the mean. Figure 6 contains plots ofthe concentration averaged over 1, 10 and 30 frames and shows that there could be large regions across the combined jet flow where the "instantaneous" composition of mixed fluid may be uniform, that is regions of rather well mixed flow.

The distribution of product is not symmetric with respect to the dual jet centerline. This is more clear in Fig. 7, where averaged profiles of mean concentration, c, are plotted versus the transverse coordinate for various streamwise locations. The profiles are sharper on the acid side because of the sharp fluorescence transition in a still acidic environment. The gradual dilution of the product on the base side is responsible for

TWO P/,RALLEL REACTIVE JETS 63

Fig.6(a).

Fig.6(b).

Fig. 6( c).

Fig. 6. Effect of number of averages on image statistics. Mean concentration, lid = 1.2. (a) 1 frame; (b) 10 frames; (c) 30 frames.


160.00

.1 I/d=3.2 CURVE COLOUMN A 1 E

..j B 64 Q) C 128 ::J 120.00 D 192 0 E 256 > F 320

G 384-

Q) x 80.00

0..

c 0 Q) 40.00 :2

IU 0.00 I •

50.00 90.00 130.00 170.00

Pixel number, y direction

Fig. 7. Development of mean product concentration profiles in the streamwise, x, direction. lid = 3.2. For column position refer to Fig. 5.

Plate 1. Laser induced fluorescence in the mixing region of two parallel reactive jets. Acid with fluorescein is carried on the right hand side stream, base on the left hand side. lid = 1.2.

TWO PARALLEL REACTIVE JETS 65

the "skewed" shape of C in Fig. 7, see also Plate 1. The shift in the maxima of C is also attributed to this behavior.

The development of the mean maximum product concentration, Cmax , with downstream distance x/d, is shown in Fig. 8. Significant mixing and reaction signalled by fluorescence "turn-on" occurs after certain distance, x., downstream of the nozzles, depending on their spacing, lid. In a similar configuration of coflowingjets [7], with Uj/U 0 = 54 (U 0 is the ambient coflow speed), mixing appears to start much later, but at large jet spacings this difference diminishes. In the same figure, (i.e. Fig. 9) appears

100

v ~ JlO

II o

X

E u

10.00 20.00 30.00 40.00 50.00

AXIAL DISTANCE x/d

Fig. 8. Maximum product concentration in the stream wise direction .

•

• o

• o o

o

00000 xc~cJd present experiments "66 tH, xs / d present experiments ••••• xs/d axisym. jets in coflow U/Uo=54 00000 xMP/d dual plane jets meeting point

lid 10

Fig. 9. Streamwise position on centerline of jets where mean concentration is max., Xc =c ,reaction is initiated, X" or jets first meet, XMP . m"


the position, X MP ' where the two plane jets meet [6], and which it was thought that it could bring some relevance to the mixing. For lid> 4 the plane jets meet at a streamwise location much shorter than the one it takes for axisymmetric jets to mix. This could be partly due to the greater dilution of the (three dimensional) axisymmetric jets.

Figure 9 shows that the mean concentration reaches a maximum at streamwise location, XC=Cmax, which is also a function of the jet spacing lid. A corresponding quantity for two parallel nonreactive plumes from line sources A, B is the "interference

coefficient" r = cAcB/(cArrnscBrrns) [15, 16]. In that case as well, it is found that the amount of turbulent mixing midway between the plumes depends on the spacing of the plumes. In Fig. 9, XC=Cmax is plotted against lid and it is found that it grows according to a power law, approximately as (ild)0.45. If the two jets were not interfering with each other, then using the known results for the concentration field of single jets [14], one finds that XC=Cmax should grow linearly with lid. Therefore, although it is claimed that in the far field the combined jet behaves as a single jet from an appropriate virtual origin [1], there could still be characteristics of the concentrations on which the initial conditions have a persistent influence. There is not sufficient length of working section in the present work to estimate the downstream position where fluorescence becomes very weak.

The effect of jet spacing on the maximum concentration is shown in Fig. 10. In general, the maximum concentration decreases with lid since the two jets entrain more ambient fluid as ljd increases. Data from a dual plane jet [3], pertaining to the decay of mean maximum velocity, U max are included in Fig. 10. This maximum velocity is seen to be related to the jet spacing by an approximately (l1d)-o.5 power law. Urnax

L[) L[) N )

o E

UO.l

66666 Cmax/255 present experiments -- Umax/Uj duol plane jets L3J

I/d 10

Fig. 10. Maximum concentration and maximum velocity on the center plane of dual jets. Concentration is non-dimensionalized by max. pixel intensity and velocity by the mean exit velocity.


occurs prior to the combined self preserving region of the two jets (region "c" in the introduction) and appears to control the development of this region. In turn, U max

depends on XM~!2, where XMP is the distance from the nozzles to the meeting point of the two jets. Again, it is observed that some characteristics of the concentration field of the axisymmetric dual jets are not self preserving.

Figure 11 shows the variation of the maximum product concentration with xld at spacing lid = 3.2. It is found, though not shown systematically here (except for the plot of Fig. 5b), that for small lid there are two maxima off the center-line of the jets. For lid above approximately 5.2 the maxima move closer to the center-line form with a tendency to disappear. For every lid, an absolute maximum is found and the position where this maximum occurs, xcrms=crmsmax, is plotted against lid in Fig. 12 .

~ • :::le:;' • ---.J'> • « • > ---.J W x Q

i §

u C)

'" 0_00 10_00 20_00 30.00 40.00 50.00

X/d RUN3

Fig. 11. Maximum rms product concentration, lid = 3.2 in the streamwise direction.

o l!l

0 u<t

'-..

i

I §

u II

~

I 0

X

gl

(1/ d)Ol

\

I/d Fir;. 12. Streamwise position where rms concentration is maximum.


This quantity, as opposed to the corresponding one for the mean concentration, increases very slowly with lid. The ratio crmsmax/Cmax is roughly equal to 0.3 for the range of values lid considered in this experiment. In single jets this ratio is in the range of 0.25 [10,11].

6. Conclusions

The degree of mixing between two axisymmetric parallel jets, monitored down to the molecular scale by means of the chemically sensitive LIF technique, was found to depend on the axial distance from the nozzles and their spacing. The closer the jets, the higher the maximum mean product concentration and the earlier it appears along the jet center-line. This position is related with a power law to the jet spacing. The RMS product concentration shows a similar behavior but the maxima occur off the jet center-line and their axial position, although following again a power law, is not as strongly dependent on the jet spacing. The velocity field of interfering dual plane jets is known to be self preserving at large axial distances, but there is evidence that this is not always the case for the concentration field of dual axisymmetric jets.

The LIF technique is proved to be a useful and powerful technique but care should be exercised when dealing with more than two mixing species.

Acknowledgments

The authors would like to thank Mr C. Gerritsen and Mr W. Kracht for their assistance in setting up this experiment.

References

1. Miller, D.R. and Comings, E.W., Force-momentum fields in a dual-jet flow. Journ. Fl. Mech. 7 (1960) 237~256.

2. Tanaka, E., The interference of two dimensional parallel jets. Trans. Bull. of the JSME 13 (1970) 272~ 280.

3. Tanaka, E.: The interference of two dimensional parallel jets. Trans. Bull. of the JSM E 17 (1974) 920~ 927.

4. Krothapali, A., Baganoff, D. and Karamcheti, K., Development and structure of a rectangular jet in a multiple jet configuration. AIAA Journ. 18 (1980) 945~950.

5. Elbanna, R. Gahin, S. and Rashed, M.l.l.: Investigation of two plane parallel jets. Al AA Journ. 21 (1983) 986-991.

6. Lin, Y.F. and Sheu, M.1.: Investigation of two plane parallel unventilated jets. Exp. in Fluids 10 (1990) 17~22.

7. Stapountzis, R., Tzavellas, P. and Moros, T.: Effects of turbulence on the mixing and chemical reaction for cross flow and coflowingjets. In: Johansson, A.V. and Alfredsson, P.R. (eds), Advances in Turbulence 3, Springer Verlag (1991) pp. 300~311.

8. Komori, S. Kanzaki, T. and Murakami, Y.: Simultaneous measurements of instantaneous concen-

TWO PARALLEL REACTIVE JETS 69

trations of two reacting species in turbulent flow with a rapid reaction. Phys. Fluids A 3(4) (1991) 507-510.

9. Koochesfahani, M.M. and Dimotakis, P.E.: Mixing and chemical reactions in a turbulent liquid mixing layer. Journ. Fluid Meeh. vol. 170 (1986) 83-112.

10. Dahm, WJ.A. and Dimotakis, P.E., Measurements of Entrainment and mixing in turbulent jets. AI AA Journ. 25 (1987) 1216-1223.

11. Papanicolaou, P.N. and List, EJ.: Investigations of round vertical turbulent buoyant jets. J ourn. Fluid Meeh. 195 (1988) 341-391.

12. Miller, P.L. and Dimotakis, P.E.: Reynolds number dependence of scalar fluctuations in a high Schmidt number turbulent jet. AIAA Phys. Fluids A 3(5) (1991) 1156--1163.

13. Shy, S.S. and Breidenthal, R.E.: Turbulent Stratified interfaces. Phys. Fillids A E(5) (1991) 1278-1285. 14. Blevins, R.D.: Applied Fluid Dynamics Ilandbook. Van Nostrand Reinhold (1984) p. 243. 15. Stapountzis, H.: Covariance and mixing of temperature fluctuations from line sources in grid

turbulence. In: Hirata, M. and Kasagi, N (eds), 2nd Int. Symp. on Transport phenomena, Tokyo Hemisphere (1988).

16. War haft, Z.: The interference of thermal fields from line sources in grid turbulence. J ourn. Fluid M echo 144 (1984) 363-387.

Image analysis of oil film interferometry-a method of measuring wall shear stress distributions

HENRI A. SILLER 1, RICHARD J. PERKINS2 & GERD JANKE1

I Hermann Fottinger Institut fur Thermo- und Fluiddynamik, Muller-Breslaustrasse 8, DW-IOOO Berlin 12, Germany 2 Department of Applied Mathematics and Theoretical Physics, University of Cambridge. Cambridge. U.K.

Key words: interferometry, shear stress measurement, image processing, fringe patterns

Abstract. A method is described of measuring a wall shear stress distribution that varies in the direction of the flow. Variations in the height of a very thin oil 1iIm moving under the boundary layer generate interference fringes, which are recorded and digitised using image processing equipment.

The evolution of the film surface in space and time can be reconstructed from the interference fringe patterns and used to calculate the shear stress field. This reconstruction is achieved by comparing the picture data with images that were calculated for prescribed heights that are adjusted iteratively, until the calculated intensities match the data.

The method is applied to a flow approaching a step, and the results are compared with pulsed-wire measurements.

1. Introduction

The first theoretical analysis of the oil film technique for surface stream line visualisation was developed by Squire [7]. Tanner and Blows [8] introduced oil film interferometry as a method of measuring wall shear stress. The method was then developed further [9,5,4], but apart from Tanner's original work the later methods only made point measurements. Using image processing technology, the oil film interferometry technique can be extended to measure shear stress at many locations simultaneously.

Oil film interferometry is based on observing the changing height of a thin oil film moving under a boundary layer: it will spread out, and its surface height will change with time, influenced primarily by the local shear stress, the pressure gradient and gravity.

The height of the oil film is typicall y of the order of micrometres and is very difficul t to measure directly, but changes in height can be visualised by interferometry. On a reflecting surface that is illuminated by monochromatic light, interference fringes are visible in the oil film, and lines of constant light intensity are lines of constant height. By observing the changes in the fringe pattern, it is possible to obtain information on the local film height and its derivatives.

The advantages of oil film interferometry are that it is a direct method that does not have to be calibrated against a reference device, and that it is not dependent on the

71


72 H. A. SILLER ET AL.

flow properties - unlike, for example, a Preston tube, where the law of the wall has to be satisfied. Of course, the temperature dependence of the oil's viscosity needs to be known.

This paper describes a new way of acquiring and processing data from oil film interferometric experiments that allows the measurement of shear stress distributions that vary in the direction of the mean flow. During the experiment, the changing fringe pattern along a line parallel to the flow is recorded in an x-t-diagram. When the data is processed, the unknown height distribution in the x-t-diagram is approximated by trying to find a known height distribution that produces a matching fringe pattern. The shear stress distribution can be calculated from these heights.

The technique is illustrated with two sets of experimental data from a flow approaching a step, and the results are compared with measurements from a pulsed wire probe.

2. The motion of a thin oil film under a boundary layer

In order to obtain the relationship between wall shear stress and oil film thickness, we make the following assumptions:

- The presence of a very thin oil film does not change the conditions in the boundary layer.

- The shear stress is constant across the air-oil interface [7]. This is valid provided the slopes of the oil film are small.

- The curvature of the oil film is small, so surface tension effects can be neglected and the pressure is continuous across the oil-air interface.

In a two-dimensional flow on a horizontal surface, the oil film changes its shape under the influence of the local shear stress T, the external pressure gradient op/ox, and gravity g [6].

(2.1)

where '1 stands for the viscosity and p for the density of the oil. In order to obtain the shear stress, this equation can be integrated, starting from the

edge of the oil film where h = 0, if h(x, t), its derivatives with respect to x and t, and the pressure gradient are known:

2'1 IX ah 2 op 2 oh T(X) = - - -dx + -h - - - hpg-.

h2 0 ot 3 ox 3 ax (2.2)

The first term on the right-hand side dominates over the pressure gradient term and the gravity term, except in the vicinity of a separation point. We neglect these terms

WALL SHEAR STRESS DISTRIBUTION

y

60

flow ---< .....

50 40 30 x[mm]

x: upstream distance from the step y: coordinate parallel to the step

20

Fig. J. Video picture of interference fringes in the oil film.

73

10

because in areas where they become large, the slopes of the oil film will be too steep to produce a visible fringe pattern.

3. Interference and fringe patterns in a thin oil film

In a thin oil film on a reflecting surface that is illuminated by a diffuse background of monochromatic light, interference between rays of light reflected from the upper and the lower surfaces of the film produces a pattern of fringes (Fig. 1).

The fringes are lines of equal height and the relation between the reflected light intensity I and the film height hex) is

I - 1. ex cos (7 nh cos 0 ), (3.1)

where I. is an offset intensity, A the wavelength, n the refractive index of the oil, and 0 the angle of the light rays in the oil film. This angle is determined by the refractive index of the oil n and the angle of the camera 0' following Snell's law:

sinO' n = sinO.

The difference in height between two neighbouring minima of intensity (dark fringes) is

A 1 ,1.h =---

2n cos 0 (3.2)


and is 0.213 11m in these experiments. Note that (3.1) maps a multitude of heights onto the same intensity. It is therefore not possible to invert the height to intensity mapping directly.

4. The method

To calculate the shear stress distribution -r(x) that generated a given pattern of interference fringes, it is necessary to know the evolution of the height of the oil film in x and t. Unfortunately,

- a pattern of interference fringes only contains information on height differences, not on the heights themselves;

- it is impossible to determine whether the change in height between two fringes is positive or negative.

One way of analysing the interference pattern is with a line-following algorithm, since lines of equal intensity correspond to contours of equal height. It was expected that this approach would be sensitive to small scale disturbances in the fringe pattern and that it would only use a small amount of the data in the picture. Instead, we have developed a surface fitting method because

- surface fitting algorithms can exploit physical limitations on the surface curvature; - surface fitting algorithms integrate over larger areas and are therefore much less

sensitive to noise than line following algorithms which differentiate to find minima and maxima of the intensity.

experiment

~96 Vl - 0

90 80 70 60 50 L.O x[ mml

experiment 2

96 Vl

-- 0

60 50 L.O 30 20 10 x[mml

Fig. 2. x-I-diagrams from experiments 1 and 2.

WALL SHEAR STRESS DISTRIBUTION

OSOdium lamp A= 589 nm

7 transparent paper

z step

--/-- -,

\ black paint

experiment (2): x-pos. of leading edge = 65 mm

experiment (I) : x-pos. of leading edge = 95 mm

Fig. 3. Experimental setup.

75

This approach has the advantage of being very general and possibly useful for the extension of the method towards the measurement of two-dimensional shear stress distributions.

The method was developed using a numerical model to simulate the evolution of oil film heights under a prescribed shear stress distribution. The computed height distribution was then used to synthesise appropriate images. (Full details of this are given in [6].) This approach was important in the development of techniques for enhancing real images, and it enabled us to investigate the sensitivity of the method to noise in the image data.

The method consists of:

- acquiring picture data from an oil film experiment (Fig. 3); -arranging the data in an x-t-diagram (Fig. 2); - preparing that data using image processing techniques;

calculating an approximation for the surface heights; - calculating the shear stress from the height data.

4.1. The experimental procedure

The technique is illustrated using measurements made in a separating flow in front of a small step in a wall jet facility (Fig. 3). The data presented in this paper were acquired


in two experiments. In experiment 1, the leading edge of the oil film was located 95 mm upstream of the position of the step, and in experiment 2 this distance was reduced to 65 mm. The mean flow velocity at the nozzle was approximately 14 mls for both experiments. The plate was made of glass, its underside was painted black. The light-source was a sodium lamp behind a paper screen. This setup provides an extended light-source, thus our fringes are localised [1]. Light reflected off the back surface only adds an offset to the light intensity but does not alter the fringe pattern. The step was 31.2 mm high, 80 mm long in the stream wise direction, 370 mm wide, and it could be removed from the plate. A video camera was fixed overhead and connected to an image processing unit.

The later approximation of the oil film heights required that the heights of the oil film were known at one point in time. For this reason, the experiments were conducted as follows:

- the video camera was positioned, the camera angle ()' and the size of the area seen by the camera were determined;

- the oil was applied in a line perpendicular to the direction of the flow; - the wind tunnel was turned on and reached its final velocity after approximately 35

seconds; - under the influence of the flow, the oil spread out into a thin film, forming a ramp

with constant slope (this shape is typical of flows with spatially constant shear stress distributions, the heights can be determined from (3.2) by counting fringes);

- a line in the streamwise direction where the oil film was relatively free of disturbances from dust particles was selected and data acquisition began (the choice ofthe line does not have great effect on the result because disturbances from dust are only local and do not convect);

- the step was placed onto the plate to introduce a shear stress distribution that varied in the streamwise direction.

4.2. Representation of data in x-t-diagrams

For a two-dimensional flow, all the information from the observation of the interference fringes can be expressed in a very compact form, by arranging the data in an x-t-diagram. These diagrams contain the information on the deformation of the interference fringes with time in a narrow region of the oil film along the x-axis and are constructed by

- extracting one line from the video picture parallel to the flow at each time step, typically every 1.5 seconds;

- placing all these lines next to each other in the x-t-diagram.

Figure 1 shows a typical video image and Fig. 2 the two x-t-diagrams (containing 512 x 64 pixels each) from experiments 1 and 2. Figure 1 shows irregularities in the fringe pattern about 50 mm upstream from the step. They are caused by dust on the plate and have only local effects, because they do not convect.

WALL SHEAR STRESS DISTRIBUTION 77

4.3. Image processing

The aim of the image analysis is to determine the most probable height distribution h(x, t) which would generate the observed fringe pattern J(x, t). Fringe patterns can be calculated for arbitrary heights, camera angles, and refractive indices using (3.1). By iteratively changing the heights, calculating the corresponding intensities for the given camera angle and oil, and comparing them with the given data, the heights can be made to converge towards the height distribution that most probably underlies the intensities in the experimental data.

Before the observed fringe pattern can be compared with patterns calculated from (3.1) using an assumed height distribution, it is necessary to improve the image quality, to remove noise and large scale distortions, and increase the dynamic range of the image.

First, the variations in background intensity are removed by determining the mean intensity variation in the x-direction of the image and subtracting it.

Then, to increase the dynamic range of the image, the intensity histogram is modified using a direct histogram specification, as described in [3]. The model intensity histogram is calculated from the intensity histogram that would be generated by an oil film with constant slope (which is the same as that generated by a random set of heights). Tests with intensity distributions calculated from (3.1) with arbitrary heights showed that the histograms of even complicated fringe patterns always resembled this histogram.

Finally, the picture was smoothed along all the rows with a moving average over three pixels.

4.4. Finding the surface height from the intensity data

In order to calculate the shear stress distribution it is necessary to compute the oil film heights throughout the x-t-plane. It is assumed that the oil film thickness increases linearly with x along the line t = 0 and that the height at the leading edge of the oil film remains zero. The problem is then to determine a surface which is consistent with these boundary conditions and with the observed fringe pattern.

4.4.1. A model for the surface The surface height is modelled by dividing the picture into rectangular patches (Fig. 4), where the height in each patch is defined by bilinear interpolation between its four corner points. The accuracy of this model depends on the relation between the surface slopes and the size of the patches:

- regions with low surface curvature are well modelled by larger patches; - regions with high curvature can only be fitted by smaller patches.


Fig. 4. Calculation of the surface height by surface patches.

4.4.2. Approximating the height over one surface patch For each surface patch the heights at three of the corners are known from the initial conditions or previous calculations. To determine the height at the fourth corner, an initial height is assumed, the corresponding fringe pattern is calculated using (3.1), and the sum of the squared differences between the calculated and observed intensities is computed. The height at the fourth corner is varied iteratively, so as to minimise the sum of the squared differences.

4.4.3.Approximating the height over a larger area To approximate the surface height distribution over an x-t-diagram contammg 512 x 64 pixels, the area is divided into 8 boxes of 64 x 64 pixels. Knowing the heights along the axes of the area, the heights in the upper right hand corner of each box can be calculated by stepping from box to box from left to right and using the result from one box as input value for the upper left hand corner's height of the next box.

If the height approximation algorithm fails to find the height over a box of 64 x 64 pixels, it will split this box into four sub-boxes and try to minimize over these consecutively, starting with the box at the lower left. This algorithm is applied recursively, the minimum box-size being 4 x 4 pixels.

4.4.4. Assigning the heights along the axes The initial height distribution along the x-axis is obtained by taking a Fourier transform along the line in the picture that represents the initial condition. The dominant frequency is a measure of the number of fringes along that line and the phase angle a measure of the height at the first pixel. From the data of the Fourier analysis, we can construct a model height distribution which gives a fringe pattern that closely resembles the original.

4.5. Calculating the shear stress

To calculate the shear stress according to equation (2.2), it is not necessary to know the height over the whole picture. In theory, data from two consecutive lines parallel

WALL SHEAR STRESS DISTRIBUTION 79

to the x-axis would be sufficient to calculate oh/at for each point. In practice, however, it may be necessary to average over several lines to obtain a good result.

The height approximation over the 64 lines of the picture yields the heights on a rectangular grid with a spacing of 4 pixels. To obtain a smooth surface and the derivatives and integrals in an analytical form, we use a cubic b-spline approximation of the surface.

5. Results

The diagram in Fig. 5 shows the shear stress values calculated from the x-t-diagrams in Fig. 2 in comparison with pulsed wire measurements. The shear stress cannot be calculated for the whole length of the x-t-diagrams, because the height approximation breaks down on the far right hand side, where the fringe pattern becomes too irregular. It does so because in this region of the pictures the oil film is higher, due to the Initial height distribution when the film was formed. Small relative changes in the film height have more dramatic effects on the fringe pattern here than in regions where the film is very thin. The data from experiment 2 were obtained in the area around the separation line, where the shear stress and its gradient are very low. A wave develops and travels upstream towards the separation line. The algorithm fails on the steep slopes of the wave.

Near the upstream edge of the oil film, (2.2) produces large errors because both h and oh/ot approach zero. Apart from these effects, the results from both experiments are consistent and they agree with measurements taken with a wall pulsed wire probe (for more information on this technique see e.g. [2J).

0.4 I I I I

flow ----expo I 1-- --exp.2 0.3

f'"-, )( pulsed wire I X , " I ,

0.2 , ~, I

I x, I 'x.

x x

;:;-~ 0.1 z

~ 0 x

-V ""

-0.1

I I I I X ~ step

//.. ,,//..."~,,//~ 60 40 20 o -0.2

100 80 x !mml

Fig. 5. Resulting shear stress distributions for experiments 1 and 2 compared with pulsed wire data.


6. Conclusions

This paper describes a method of measuring wall shear stress distributions that vary in the direction of the main flow. It is based on interferometric visualisation of changes in height of a thin oil film and the reconstruction of the surface height in the x-t-plane by a surface approximation method.

Applied to a separating flow in front of a step, the surface approximation produces good results as long as the fringe pattern does not become too complicated and the shear stress results agree with those of wall pulsed wire measurements.

Since the method provides an entirely general way of determining a surface from a fringe pattern, given a few boundary conditions, it should be possible to extend it to the measurement of surface shear stress distributions in a plane.

Acknowledgements

We gratefully acknowledge financial support from the British Council and the Deutscher Akademischer Austauschdienst. R. Perkins was funded by the Wolfson Foundation. H. Siller was funded by the Technische UniversiHit Berlin, Corpus Christi College Cambridge, and under the Erasmus scheme.

References I. Born, M. and Wolf, E., Principles of Optics, 6th edn. Oxford: Pergamon Press (1980) p. 291. 2. Castro, J.P., Dianat M. and Bradbury, LJ.S.: The measurement of fluctuating skin friction with a pulsed

wire wall probe. In: Turbulent Shear Flows 5 New York, Berlin, Heidelberg: Springer (1987). 3. Gonzales, R.C. and Wintz, P.: Digital Image Processing. Reading Massachusetts: Addison-Wesley

(1987) pp. 152-158. 4. Kim, K.-S. and Settles, G. S.: Skin-friction measurements by laser interferometry. AGARDograph 315

(1988), Chapter 3. 5. Monson, DJ. and Higuchi, H.: Skin friction measurements by a dual-laser-beam interferometer

technique. AIAA Journal 19 (1981) 739-744. 6. Siller, H.A.: An optical method of measuring wall shear stress using oil film interferometry and image

processing techniques. A dissertation submitted to the University of Cambridge for the Certificate of Postgraduate Study (1991).

7. Squire, L.c.: The motion of a thin oil sheet under the steady boundary layer on a body. Journal of Fluid Mechanics 11 (1961) 161-179.

8. Tanner, L.H. and Blows, L.G.: A study of the motion of oil films on surfaces in air flow, with application to the measurement of skin friction. J. Phys. E: Sci. Instrum. 9 (1976) 194-202.

9. Tanner, L.H., A skin friction meter using the viscosity balance principle, suitable for use with flat or curved metal surfaces. J. Phys. E: Sci. Instrum. 10 (1977) 278-283.

Title: Recent Advances in LSV, PIV and PTV Author: L. Lourenco

Paper presented at the EuroMech Colloquium 279 Delft 2-5, July, 1991

1. Introduction ............................................................................................................. 3 2. Application of PlY to the measurement of an unsteady flow .................................... 5 3. Yortex interactions in the transition region of a rectangular jet ................................ 8 4. Digital or "On-line" PlY .......................................................................................... 9 6. Conclusions ............................................................................................................. 13 References ................................................................................................................... 14

Abstract

Since the pioneering experiments of Dudderar and Simpkins (1977), Grousson and Mallick(l977) and Barker and Fourney (1977), Particle Imaging Yelocimetry has evolved and is rapidly becoming an essential tool for the measurement of instantaneous two dimensional velocity fields. This rapid development is carried out by a growing number of fluid mechanics experimentalists which recognize the unique capabilities of the technique to measure velocity fields in both space and time, and supported by the even faster development and availability of microcomputer hardware and software. In this paper some of the most significant and recent developments are described and illustrated using examples of the application of the technique to map various flow fields. These include both optical systems for image acquisition as well as digital, "on-line" methods for integrated image acquisition and processing.

81


82 L. LOURENCO

1. Introduction

Whole field velocimetry techniques such as Laser Speckle (LSV) Particle Image (PIV) and Particle Tracking Velocimetry (PTV) are different implementations of the same basic principles measuring instantaneous fluid velocities by recording the position of images by small tracers suspended in the fluid at successive time instants. The common underlying assumption is that these tracers follow closely, and with minimal lag, the fluid motion. This assumption holds true for a wide variety of flows of interest provided that the tracers are small enough and/or that their density approaches that of the fluid.

Besides their common goal LSV and PlY, on one hand, and PTV on the other hand do not share the same historical development and practice. An important difference is that in LSV and PlY, which have been established as two operating modes of the same method (Meynart and Lourenco, 1984, Adrian, 1984 and Lourenco and Krothapalli, 1987) the concentration of tracers is rather high and the measurement of a "local" fluid velocity results from an average over many tracers contained in a interrogation cell. Usually the cell is regularly spaced and its size determines the spatial resolution. This is in contrast with PTV where the velocity is determined at random locations using the images produced by a single tracer.

Particle Tracking Velocimetry has evolved as a means of extracting quantitative information from conventional flow visualization data, such as streak photography or multiple exposure photography (Elkins et ai, 1977), (Kobayashi et ai, 1983), (Utami and Ueno, 1984) and (Gharib et aI, 1985). While these methods excel as a means for fast and easy mapping of the flow basic structures, they fail in providing an accurate velocity field map with high spatial density. Typically, the evaluation of the particulate displacements directly from their images, such as evaluating streak length or spacing between successive images of the same tracer, requires processing large amounts of data and rather sophisticated software. These techniques tend to be quite laborious and usually not very accurate, as the measurement errors become large when the mean distance between tracers is of the same order of magnitude of the streak length, or the spacing between successive particle images decreases. As a consequence the tracer density is kept low, resulting in velocity measurements with poor spatial resolution. The problem of having sparse velocity information at random locations has been addressed with the introduction of rather sophisticated interpolating schemes, but the validity of this approach is questionable considering that is most cases the spacing between data points is larger than the flow scales, and the velocity may be interpolated at best with first order accuracy.

LSV and PlY offer a valid alternative to the more conventional PTV methods because of their ability to provide higher temporal and spatial resolution of the instantaneous flow field. These two techniques are based on essentially the same principles and method of operation with exception to the amount of seeding employed. Meynart and Lourenco (1984) argued that best results are obtained when the particle concentration is kept as high as possible as to ensure a maximum number of particle images in an interrogation cell but not so high to overlap and generate speckle.

RECENT ADVANCES IN LSV, PlV AND PTV 83

Generation of speckle patterns is not desirable in most applications due to dependence of the speckle formation with the fluid motion. Other reasons to limit the concentration are related to the degradation of the light sheet propagation and possible interference with the mechanics of the fluid motion.

In this article the application of the PIV technique to the study of practical flow fields is used as a means to illustrate both the techniques ability to map spatially and temporally developing flows, as well as the most recent features incorporated in the technique.

2. Application of PIV to the measurement of an unsteady flow Pitching NACA airfoil.

The unsteady flow past a NACA 0012 airfoil in pitching up motion is experimentally investigated in a water towing tank using the Particle Image Velocimetry technique ( C. Shih, L. Lourenco, A. Krothapalli and L. van Dommelen, 1991). The Reynolds number, based on the airfoils chord and the free stream velocity, is 5000. The airfoil pitching motion is from 0 to 30 degrees angle of attack at a dimensionless pitching rate ofO.13l.

The water towing tank facility is 180 cm long and 43x55 cm in cross sectional area. The towing carriage is driven by a D.C. motor and has a speed which varies from 0.3 to 30 em/sec. A motor speed control system controls the towing speed via a digitalto-analog converter. To ensure a smooth traverse of the towing carriage, an acceleration ramp at the beginning of the travel and a deceleration ramp at the end of travel are implemented on the controller. The NACA 0012 airfoil has a chord length of 6 cm and an aspect ratio of 6.67. A recording camera with a scanning mirror system are mounted on a reinforced, vibration free platform that extends from the airfoil carriage. The scanning mirror is used as a means of introducing a velocity bias in order to resolve flow reversals and stagnant flow regions. The airfoil pitching motion is provided by a stepping motor with a programmable controller, which is pre-programmed with the test profile and activated by the host computer. The airfoil changes linearly from 0 to 30 degrees after the airfoil is towed more than one chord length. A DEC Vaxstation II computer monitors all motions (figure I).

A dual pulsed laser system, consisting of two frequency doubled Spectra-Physics DCR 11 Nd-Yag pulsed laser systems is used to provide the double illumination pulses. A system of prisms and polarizing cube beam combiners make the light beams emitted from the two lasers collinear (figure 2). Time separation between the laser light pulses can be varied from a fraction of one microsecond to a few seconds by adjusting a pulse generator used as a trigger. A cylindrical lens is used to form the combined beam into a sheet that illuminates the mid-span section of the airfoil. Metallic coated particles (TSI model 10087), with an average diameter of 11 microns, are used as the flow tracers. A phase-triggered 35 mm camera (Nikon F-3) with a 105mm Macro lens is used in the recording of the flow field. Synchronization between components is accomplished using a Tektronix modular electronics system. This system also provides the phase-reference between the motion of the airfoil and the PlV photographic timing sequence.

Once the photographic film is processed contact prints are made. It has been shown by Meynart and Lourenco (1984) and Pickering and Halliwell (1984) that when

84 L.LOURENCO

using the Y (lung's fringe approach to retrieve the displacement data this process increases the SNR and fringe visibility, These prints are used to obtain the velocity data by point wise scanning following a Cartesian grid. The step size of the scanning operation was chosen to be 0.5 mm in both directions; this corresponds to a step size of 1.25 mm in the physical plane. The interrogation beam diameter corresponds to an equivalent physical probe size of 0.625 mm. which is approximately half of the mesh size. The probe laser beam size can be varied, using a telescopic lens arrangement, to accommodate other

spatial resolutions. In the present application the above mentioned resolution was found to be sufficient to resolve the relatively large spatial flow scales. The Young's fringes produced by the laser beam were digitally processed to produce the velocity at the interrogated point on the negative, and the vorticity was found by numerical differentiation.

As mentioned earlier, the airfoil is started with a ramp type motion at zero angle of attack and is allowed to move one chord length, after which the flow field is fully established with minimal effects of the starting process. Next the airfoil is impulsively given its pitching motion and allowed to pitch to 30 degrees angle of attack. During the pitch up motion the airfoil travels approximately four chord lengths. Once the pitch up motion stops the airfoil is allowed to travel for an additional four chord lengths before it comes to a complete stop. In the present experiments the airfoil pitches about its quarter chord point.

Figures 3 and 4 show the time development of the flow around the pitching airfoil using the PlY technique. Figure 3 presents the velocity field at different times as uniformly scaled velocity vectors. The corresponding vorticity fields are presented as isovorticity contour plots in figure 4.

The essential component of the technique that makes it possible to measure complex flow fields, such as the one presented above is the velocity bias method. Due to limited range in the fluid velocity, about 20 cm/sec, the velocity bias method was successfully implemented in this experiment using a scanning mirror. However. the scanning mirror approach suffers from some restrictions. Firstly, it requires the motion of mechanical components and precise synchronization with the camera shutter and the laser illumination pulses. This makes the use of the mirror very unpractical, if not impossible, in cases when asynchronous recording of flow events is necessary, e.g. turbulent events, and when the required bias velocity magnitude exceeds the mirror's mechanical capabilities, Therefore the implementation of a passive velocity bias method which does not require the movement of mechanical components is very desirable.

A new method introduced by Landreth and Adrian (1988) uses the polarization of light scattered from seeding particles by switching the polarization of the illumination beam and placing a birefringent crystal, e.g. calcite, in front of the photographic lens to displace the particle image between illumination pulses. In the arrangement proposed by Landreth and Adrian a Pockels cell is used to rotate by 90 degrees the polarization of the laser output providing the illumination pulses. The method is quite effective provided that the most of the light scattered by the tracers retains its original state of polarization. This happens to be the case for the particles and recording optics geometries used in most

RECENT ADVANCES IN LSV, PIV AND PTV 85

current PlY applications: these generally include micron size particles ( less than 5 microns ). e. g. smoke or polystyrene, which do not absorb an important fraction of the illumination laser light, and moderate viewing angles of the recording optics ( less than 30 degrees). There are. however. some inconveniences when using this method. The magnitude and orientation of the shift changes in the film plane, due to the fact that the light is collected from a finite view angle. Also, the shift amount for a given camera magnification cannot be changed unless the crystal thickness is varied or more than one

crystal is used in series. In order to resolve these problems another shifting method using the light polarization effect is proposed by Lourenco. The birefringent crystal is replaced by a polarizing beam splitter cube and two isolators 0/4 wave plate) arrangement (figure 5 a.b,c). The light paths for the light scattered by a particle having both the sand p polarization states are also illustrated. The principle of operation of the apparatus is as follows. Consider a stationary particle that is first illuminated by p-polarized light. The scattered beam travels through the cube undisturbed in its path and changes its state of polarization from p to s when it propagates through the 1/4 wave undergoes a total reflection and reverses it sense. Now having its state of polarization rotated by 90 degrees, the light beam is totally reflected at the cube's interface and directed towards the camera objective. In contrast, if the light beam is s-polarized then it undergoes first a total reflection at the interface, followed by a change in its state of polarization while interacting with the 1/4 wave plate (figure 5b). It then proceeds through the cube and into the camera lens. Providing that the light paths have similar distances and that the 1/4 wave plates are parallel to the cube's surfaces, the image of the scattering seed will appear in the film plane at the same location. A uniform displacement (shift) can be generated by slightly tilting one of the 1/4 wave plates with respect to the cuber's surface. The amount of displacement is proportional to the angle of tilt (figure 5c).

This method is currently being used to examine both high and low speed flows. An example of its application while studying the transition region of a rectangular jet is given below.

3. Vortex interactiuns in the transition region of a rectangular jet

The transition region of a rectangular jet of aspect ratio 4 at a Reynolds number. based on the e4uivalent diameter. of 4500 was investigated. A low speed air supply system is used to flow air in a settling chamber 25 cm in length and 7.5 cm in diameter. A honeycomb and a series of screens at the inlet of the settling chamber were used to reduce the flow disturbances. The cross-section area of the nozzle contraction changes graduaJly from a circular cross-section, 7.5 cm in diameter, to a rectangular nozzle. The long (L) and short (W) dimensions of the nozzle are 3 em and 0.75 em respectively and streamwise contours of the contraction are fifth order polynomials. The contoured nozzle has a contraction ratio of 1 <).6: lover a length of 1.6 times the inlet diameter. In order to obtain appropriate jet seeding, smoke particles in the micron range are produced. The smoke and the ambient air are mixed in a large mixing and settling chamber. The air smoke mixture is then supplied to the settling chamber of the jet using a small axial fan. A second smoke generator of the same type is used to seed the outside air surrounding the jet.

86 L. LOURENCO

The laser sheet illuminates the central plane of the jet through the small dimension of the nozzle. Two laser pulses with a duration of 10 nsec and separation of 50 microseconds are used for the double exposure recording. The time separation is chosen such that the recorded particle displacements are in between 50 - 250 microns and thus easily resolved when using the Young's fringe approach. Furthermore these displacements appear to be three to four times less than the smallest flow scale we needed to resolve. The particle image doublets are recorded using the 3S mm camera with the 105 mm Macro lens. The camera is installed in an assembly containing the polarizing cube and 1/4 wave plates to obtain a shift during the recording. Photographs of the flow field were taken at random times. After development the photographs were analyzed using the Young·s fringe method, and the velocity vectors were obtained on a Cartesian grid (60xgO). Using a central difference scheme, the instantaneous vorticity

was obtained, with an estimated accuracy of 100 sec-I. This absolute error was evaluated using the error estimate of 1.S % full scale ( 4.S m/sec ) on the velocity vector.

Typical double exposure photographs of the jet, for two different times, are shown in figure 6. The pictures display the flow field from the nozzle exit to a downstream location of 8 widths. From a number (about 500) of similar photographs the near field (X/w<20, where X is the downstream coordinate) of the jet can be characterized by three distinctive regions. The first region in which the initial shear layer is unstable and rolls up into discrete vortices; an interaction region in which two or more vortices interact with each other, and a region in which vortices break-up into threedimensional motion.

Figure 7 shows a typical instantaneous velocity field corresponding to one of the double exposure photographs in figure 6. The results are shown after removal of the velocity bias. Using a central finite difference scheme the instantaneous vorticity and strain rates are calculated and shown in figure 7. Also included are the instantaneous Reynolds stresses. The results represent, with great fidelity, the aforementioned regions of the flow field.

4. Digital or "On-line" PIV

As demonstrated by the previous examples PIV is a superior tool for the diagnostics of flows that evolve both in space and time, as it is the only method capable of providing the researcher with flow field strain and vorticity information. However, the current bottleneck and difficulty in the utilization of the technique still remains the photographic step. This difficulty stems from the fact that before the double exposure frames can be analyzed there is one or two wet photographic processes involved, depending on whether or not contact prints of the original negatives are required to improve the data quality. This does not account for the fact that for proper and optimal operational use of the technique, quite a few photographic parameters (e.g. time between exposures, velocity bias amount) and seeding rates must be adjusted by trial and error. It is, therefore, desirable to develop an alternative approach that facilitates this set-up stage by making the image acquisition step less laborious and time consuming.

Several investigators, having recognized the above mentioned shortcomings, have proposed various methods of solution. Okada et al (1990) proposed the replacement of the photographic film by a liquid crystal television (LC-TV). The double exposed


photographs are recorded using a video camera and stored in a frame buffer for later display on the LC-TY. The velocity information is then retrieved using a point by point analysis method based on the Young's fringes approach. The current limitation if this approach is the limited resolution of about 320x220 pixels of the LC-TY. An obvious variation of this method is the digital processing of the double images stored in memory. In 1999 Cho first formulated the theory associated with the digital counterpart of the Young's fringe analysis method. In his approach the double exposed digital image frames were obtained by digitizing single exposed video images and then adding successive frames. This method has also been implemented by Westerwell and Nieuwstadt (11)90) who used such a technique to study the flow behind a circular cylinder. The major shortcoming of these methods is that they can only be applied to the measurement of slow moving flows and lack the resolution and accuracy of their analog counterparts. The limitation in velocity range is due to the fact that the image acquisition is tied to the timing of the video cameras (25 or 30 Hz); an additional timing problem is that not all the pixel elements are acquired at the same time, due to the video image scanning. Another implementation of the same idea was put in practice by Willert and Gharib (191) 1). The merit of their approach is that instead of adding together the individual frames to generate a double exposure image for processing using autocorrelation algorithms, the cross-correlation of successive images is carried out to obtain the displacement. As such, because the frame's phase information is not lost, the technique provides displacements without directional ambiguity and with no need to implement any other method like the velocity bias method.

The emphasis on the current pry research at the Fluid Mechanics Research Laboratory at the Florida State University is mainly in the are of so-called "on-line" systems. These systems use non-standard video cameras together with high capacity frame digitizers and buffers for the acquisition and storage multiple exposed frames. Present capability allows the digitization of images in various levels of spatial resolution and area coverage. Available systems are the standard resolution (512x512 pixels), the high resolution (1 280x 1024 pixels) and the very high resolution (2048x2048 pixels). Processing of the digital frames is carried out using auto- and cross-correlation algorithms. Results are very encouraging and seen to indicate that at the present time some applications can be fully met using these easier to use systems. A brief presentation of one of such systems based on the standard resolution format (512x512 pixel) CID camera follows.

5.1 Hardware configuration

The multiple exposed frames are acquired by a cm video camera with a 512x512 pixel sensor. The square pixels are 15 microns on each side. Unlike commonly available RS-170 format commercial cameras which "scan" the images at a rate of 30 Hz this camera has the capability of grabbing extremely fast (strobed) scenes due to the fact that all the pixels are illuminated and charged at the same time. The image read-out is performed in a sequential. non-interlaced fashion, at rate of about 10 MHz, after which a new image can be acquired. The timing for image acquisition and read-out corresponds, approximately, to a maximum image refresh rate of 30 Hz.

The non-interlaced cm camera output can be viewed directly by inputting this

88 L. LOURENCO

analog signal into a monitor equipped to handle the non-standard progressive scan format, or as shown in figure 8, as the input for a frame grabber/buffer combination board resident on a PC-AT type microcomputer. The frame grabber which allows for the input and digitization of non-standard video formats, digitizes the image retaining its 512x512 resolution, square aspect ratio, and with 8 bits resolution of gray scale. The frame grabber board also has the capability to convert the digital frame information on one of its memory buffers into an analog video signal (RS-170 format) which is displayed in a conventional RGB video monitor. The system synchronization is achieved by means of an especially developed interface which ensures that the camera stays in the acquisition mode during the time the laser illumination pulses are fired, and in the readout mode during the image acquisition phase by the frame grabber. With this arrangement a maximum number of 30 frames per second can be digitized and stored. The present on-board memory of the frame buffer allows for the storage of a maximum of 16 images. However, images can be transferred to the PC-AT hard disk for permanent storage and/or later processing. Another method available to achieve long record lengths of flow events is to record on a high quality super VHS type tape the analog video output of the frame grabber, for later playback and re-digitization. The inconvenience of this approach is that some image degradation will occur.

Using this arrangement a variety of laser sources can be used without restriction. In the application described herein a CW laser with a Bragg cell as the shutter is used.

The processing of the multiple exposed frames is performed using the PC-AT microcomputer and an array processor in order to achieve computational efficiency and speed.

5.2 Processing of the digital frames and typical results

Using the hardware described in the previous section a whole field velocity measurement of the flow past a circular cylinder towed in a water tank has been carried nut. The cylinder is 25.4 mm in diameter and is towed with a velocity of 2.5 cm/sec corresponding to a flow Reynolds number of approximately 600. The flow is seeded with II micron in diameter metallic coated particles and the F# number for the recording is 16. Consequently the particle image size is mostly dominated by diffraction effects and is evaluated to be around 20 microns, i.e. just slightly above the pixel size. The magnification computed as the size of the sensor divided by the size of the scene being recorded is equal to .07. In order to resolve the ambiguity of the velocity vector a scanning mirror method was implemented in the recording of the frames.

Figure 9 is a typical example of a velocity field obtained after processing of the triply exposed frames, and removal of the velocity bias. This velocity map corresponds to the flow after the cylinder has been impulsively started from rest and moved 3 diameters. Analysis of this figure reveals that with this technique it is possible to map the velocity field with reasonable detail and accuracy.


Point by point processing of the multiple exposed frames is accomplished using an autocorrelation algorithm. The frames are divided into interrogation cells 16, or multiples of 16, pixels in both the vertical and horizontal axis. The aspect ratio of these regions need not be square and may be changed to accommodate flows with a dominant velocity component. Much like in the Young's fringe analysis method where the particle displacements are only a fraction of the analyzing beam diameter, for a successful measurement the linear dimensions of the region being analyzed with the autocorrelation (or crosscorrelation) algorithm need to be at least three times larger than the local displacement. The choice of the optimum analysis area as well as its impact on the overall accuracy of the technique will be discussed on a forthcoming publication. The digital autocorrelation in two dimensions of the particle image doublets is perfonned using a FIT algorithm. The algorithm is implemented using the array processor and allows for the entire processing of a full frame image containing about 1000 vectors in just 4 minutes. Figure 10 is autocorrelation map from one of such interrogation cell. The off-center peak location of this autocorrelation array corresponds to the average displacement of the tracer images contained in the interrogation cell. An interpolation algorithm is implemented in order to achieve sub-pixel resolution for the evaluation of the peak location. Discrimination between a valid measurement and spurious data (e.g.,cylinder wall, drop-out regions without seeding) is performed by comparing the amplitudes of the three most important peaks in the correlation array, including the zero order peak.

The a<.:curacy of the technique was evaluated first by means of a simple experiment. A uniform flow was generated towing the carriage of the towing tank with constant velocity. This flow was recorded using the "On-line" PIV arrangement, while changing the time between exposures and the number of exposures according to the s<.:hedule presented in table 1. Also included in the table are the results of the average displacement measured in pixels and the corresponding rms. The average displacement and rms were determined averaging the results of about 1000 interrogations per frame. A measure of the techniques' accuracy is obtained comparing the variation of carriage velocity measured as the ratio of pixel displacement divided by the time between exposures. This measurements are within 1 % to 1.5% of each other. Another measure of the technique'S accuracy is given by the scatter of the measurement in terms of its rms. The scatter does not exceed .1 pixels. The reasons for this surprisingly high accuracy will also be discussed in a forthcoming publication.

90 L. LOURENCO

6. Cunclusions

Whole field measurement techniques, in particular PlV, have demonstrated the ability to record the instantaneous velocity field with accuracy comparable to that of LDV while retaining its non-invasive nature. The development of special recording techniques, such as the velocity bias make it possible to measure complex flows and within wide ranges in velocity. The most noteworthy development is. in the author's opinion, the introduction of video techniques as means of not only eliminating the wet photographic processing stage. but also to provide the operator with an almost real-time capability of data access. Due to the introduction of higher resolution cameras, e.g. 204gx2048 pixel format. these systems are undergoing rapid development. and at the present stage, the video. "on-line" approach can replace in quite a few applications the 3S mm format camera without loss of accuracy and spatial resolution.

Adrian, RJ. Applied Optics. vol 23. 1984

Barker. D. B. and Fourney. M. E. Applied Optics. vol 1, 1977

Cho, Y. C. Applied Optics, vol 28, 1989

Dudderar, T. D. and Simpkins, P. G. Nature, 270,1977

Elkins, R.E. et al Rev. Sci. lnstr., vol 4g, 1977

Gharib. M et al AIAA paper 85-0172. 1985

Grousson. R. and Mallick. S. Applied Optics. vol. 16. 1977

Kobayashi, T. et al

References

Proc. Int. Symp. Flow Vis., Ann Harbor. Michigan, 1983

Landreth, C. C. and Adrian R.J. Applied Optics, vol 27, 1988

Lourenco. L. and Krothapalli Exp. in Fluids, vol 5, 1987

RECENT ADVANCES IN LSV, PlY AND PTV

Meynart, R. and Lourenco. L. VKI Lecture Series. 1994

Okada. E. et al Proc. App. Laser Vel. Fluid Mech .• Lisbon. Ponugal, 1990

Pickering, C 1. and Halliwell. N. A. Applied Optics, vol 23. 1984

Shih. C. L. Lourenco, van Dommelen. L. and Krothapalli. A. to appear in AIAA Journal 1991

Utami. T. and Ueno. T. Exp. in Fluids. vol 2. 1984

Westerwell.1. and Nieuwstadt. F.T. Proc. App. Laser Vel. Fluid Mech" Lisbon. Ponugal. 1990

Willert. C E. and Gharib, M. Exp. in Fluids. vol 10, 1991

Yao. C S. and Adrian R. 1. Applied Optics. vol 23. 1984

91

92 L. LOURENCO

Table 1 : Time between exposures vs displacement

Time (msec) Number of Average Rms displ(pixels) exposures displ(pixels)

10 2 3.21 .089 3 3.36 .078 4 3.24 .075

15 2 5.02 .095 3 5.06 .074 4 5.07 .064

20 2 6.99 .120 3 7.01 .061 4 7.05 .079

30 2 10.77 .139 3 10.87 .133 4 10.76 .170

35 2 12.32 .146 3 12.56 .127 4 12.29 .137

Figure 1: Schematic a:Tangement [or the dual pulse laser system

o nvu 51.H," 11

1:::1 !10:1 r~~~~~;Oliet

Figure :;: Schematic of tne airfoil's motion comrol system


- ....::::

(a) t lI../C=2.56. a= 19.2 del'. (d) I lI../C=5.13, a= 30.0 deg.

(b) t U /C=3.84, a= 28.8 del'. (e) I lI../C=5.99. ex= 30.0 deg.

(c) U /C=4. 70. ex= 30.0 dec. (I) t U /C=8.56. a= 30.0 del'.

Figure 3: Velocity map of fiow past !\ACA 0012

94

(a) tC_/C = 2.56

(0) i.C_/C = 3.64

(c) t~..,/.: =:: ';.70

c

o

C:> -

o .. :.{[::'," '-:c>;~;; :<9;.'<~NN/:

(d) tU_/C = 513

(e) tU_/c = 5.99

(f) u:~/c = 856

Figure 4: Vorticity map of flow past KACA 0012

L LOURENCO

(

RECENT ADVANCES IN LSV, PIV AND PTV

~rShCcl

fr- ci r---: --r-..... I ~s

..¥' Anli Rc.nccuon Surface ,----..,---'=--1

.......... Totally Rcne~ivt Surlacc

Pamcle

Lase: Sheet

~s

C======,,=Ajnu ReOectiof: Suri::u:c

......... ToaJly Reflecli,'" Su,,"acc

==========~z~==~====

/1

,~ i j I '

:j-I I ~

i ~s:: She::

1

/'----; / -------u

Lens ~ Anli Rtf/cellor; SI!;i.lc:

,----,--=-'1

h:1a~!: Plan:

lm!l.;: Plan:

Figure 5: Velocity Bias: beam splitter/isolators arrangement a) p-polarized light path b) s-polarized light path c) opcra.r.ion for velocity bias

95

t P - Polarized T

• S - Polarized

96 L. LOURENCO

A

B -, ...

r' I>

Figure 6: Illst.alltancOllS double exposurc photographs of the ccntral pI aile cOllt<tillillg thc small dimcnsion of the nozzle

RECENT ADVANCES IN LSV, PIV AND PTV

30

0.0 u

:.C

L: 1

t..( [

0.0 1.0 2.0 ),0 -4.0 ~.O 6.0 7.0 6.C 2.' L 2.0 I (d)

, :: ! , , , '0~~Od2.,

(. ~2.0

";::: -0 .... ~

-1.2 r -2.0 ~

i -2''\,.0

2.0

~ .2

~~~~l.2 ~ (§5'" ~~O.'

o ~~~:z ';,-0., . ,'-"-.-., 0 '--" 0 ~

~: 'I-\':;,'

00, ~~-20 ~O ~.

. . . • f 1.0 l.C ~.O ".0 5.0 6.C 7.0 t..O-J..·

x/W

(I)

~ -~ ~ ~ w ~

- ',.:: I i

-2.C :--C

- ',.2 ;.. c

-2.0 ~

~ ~. ... ~C ~ i.0 2.0 :1.0 -.0

x/W '.0 E.e 7.0

Figure 7: Instantaneous quantities corresponding to the photograph in figure 6a.

a)Two-dimensional velocity field, b ):\ormalized velocity field, c) ~;;, d) (:, e) ud ' f) ilo~:'

97

98 L. LOURENCO

~oooo

RECENT ADY ANCES IN LSY, PlY AND PTY

Figure 9: Velocity field for fiow past impulsively started circular cylinder

D.C. Peak

/

Data Peak

Figure 10: Two-dimensional autocorrelation map, 32x32 pixel analyzing window.

99

DEVELOPMENT OF PARTICLE IMAGE VELOCIMETRY:

A NEW COMPUTATION METHOD WITH DIRECTIONAL RESOLUTION

P. GUIBERT, Q.C. DUAN, M. MURAT, J. JULLIEN

Universite Pierre et Marie CURIE (Paris VI) - CNRS URA 879 Laboratoire de Thermodynamique Appliquee aux Machines et Thermique

2, place de la Gare de Ceinture - 78210 Saint Cyr L'Ecole - FRANCE

ABSTRACT

This paper describes a new processing, interdistance histogram,

to resolve PIV (Particle Image Velocimetry) photographs. The aim is to

determine the module and the direction of the flow velocity. The

module is obtained by measuring the displacement of the particles in a

predetermined time lap. The directional ambiguity is eliminated by

using an asymmetrical laser light exposure sequence. The method is

based on the search of interval histogram on a band with a fixed

width. This band is moved virtually with a fixed steepness 9 over the

whole image. The new processing is compared with the well known

spacial autocorrelation by application to computed images. Similar

results were obtained and histogram computation takes about the same

time as the autocorrelation calculation which uses an array processor.

In addition, the histogram method enables us to eliminate the

directional ambiguity.

KEYWORDS

PIV - Histogram - Velocity measurement - Laser applications -

Image processing.

101


102

INTRODUCTION

The application of the technique

P. GUIBERT ET AL.

PIV (Particle Image

velocimetryl involves two steps. First, an image representing the flow

field is grabbed with particular operating method [1,2,3,4]' Then, the

signal contained in this image is extracted and treated either by a

computation analysis or by an optical method [5,6].

The flow is seeded with tiny particles to be visualized. A thin

sheet of light is necessary to have the image taken. Its orientation

should be such that it contains the dominant flow direction. The light

source should be sufficiently powerful and may be coherent,

monochromatic or not [7].

In order to determine the velocity vector, it is necessary to

know not only the particle positions but also the moment when the snap

is done. A time reference is therefore used. It can be obtained by

either a dual, a multiple laser shot or by a special camera.

Theoretically, when the visualization field is lightened twice, the

photograph shows the superpositon of the first position of the

scattered light of particles and the translation of this pattern. The

distance of a particle image pair !J.X is a fonction of the time

interval between the two exposures !J.t, the magnitude of velocity V and

the camera magnification M. This fonction can be represented by:

!J.X = VM!J.t

The image grabbed is then treated. The difficulty of the image

processing is to realize the pairing of different exposures. Different

processings are suggested to resolve the particle image transparency

according to the image particle density: LSV ,PIV and PTV [1,2,5]'

I CREATION OF AN ARTIFICIAL IMAGE

To evaluate or write any treatment, we need to have a wide

range of PIV images and the creation of artificial images is a good

DEVELOPMENT OF PARTICLE IMAGE VELOCIMETRY 103

means. The computed image is an interrogation spot or a mesh of the

whole image in which the velocity is assumed constant. The analysis of

the measurement line calls to mind the introduction of many

parameters. They come from: tracer particles, illumination, recording

optics, fluid characteristics and exposure parameters, ....

A summary is given on the following section. We try to estimate

the effects of differents parameters on the real image and to choose

the significant ones for the creation of the artificial image.

A set of parameters is choosen according to the above diagram.

These parameters are:

d: mean displacement of particle during Tl3 time

Dim: dimension of the numerized mesh (pixels)

De: boundary angle deviation (degree)

Del: boundary angle deviation of the ith particle (degree)

Fm: boundary module fluctuation (%)

Fml: boundary module fluctuation of the ith particle (%)

Nl: number of isolated points (monopoint)

Nz: number of bipoints

N3: number of tripoints

Np: number of light point (Np = 3N3 + 2Nz + NIl

Ngthr: threshold grey level (PIV binarized image)

Sthr: threshold area of light points (PIV binarized image)

S/N: signal to noise ratio ([3N3 + Nzl/NIl

TIz: time interval between the first and the second exposures

TI3: time interval between the first and the third exposures

TI3 = Rt x TIz (2:s Rt :s 4)

e: inclinaison of the flow field in the interrogation spot

INFLUENCE ON REAL IMAGE

pattern (LSV, P I V, PTV)

[ Number 0 flight po ints Np

AR T I F I C I A L IMAGE PARAMETERS

104 P. GUIBERT ET AL.

EXPERIMENTAL CONDITIONS

Particle charac t er i st i cs

Recording lens aperture

Aberration and magnifica.

Light source power

Sensitivity INFLUENCE ON REAL IMAGE

Dust (big size) Diffraction particle sizel

Grey level Sorting out right particles Sthr

Threshold level Ngthr

ARTIFICIAL IMAGE PARAMETERS

INFLUENCE ON REAL IMAGE

the studied flow field

Grid size and accuracy Dim


INFLUENCE ON REAL I MAGE

Numerical biaisis

Ca I culation error I I . S i gnaliNoise rat i 0 S/N



Flow characteristics:

three dimensional unsteady INFLUENCE ON REAL IMAGE t urbulen t cycl ical

Pairing of particles

Isolated particles

Fluctuation in the grid

Velocity range Signal/Noise ratio S/N

Module fluctuation Fm

Angle deviation De

Particle displacement d




Exposure parameters:

duration interval between exposures INFLUENCE ON

REAL IMAGE number of exposures

I Light point luminosity

Particle displacement

Preproccessing Dim

eho ice of displacement d

bi or tripoint image


In addition to the numerous parameters, in order to eliminate

the directional ambiguity, we use an asymmetric coding by three

exposures of the flow. The time interval between the first exposure

and the second one is greater than between the second one and the last

one. According to the bibliography [5,8,91, for an area about 1mm

diameter numerized on a 128x128 pixels dimension (DIm), the number of

light points Np is estimated to be about 50 to 100. Each set of light

points can be an isolated point (monopoint), a bipoint or a tripoint.

An isolated point and a bipoint are the images of the particle seen

during one or two of the three exposures. A tripoint is the image of

the particle seen during all the three exposures.

The displacement of each set of light points (bipoint or

tripoint) is modulated by Fm, Fml, De and Del. Del and Fml are

computed with a random constant distribution. The fluctuations of

displacements are determined to be less than the boundary fluctuation:

- Displacement particle i = d (1 ± Fmi)

- Angle particle i = €I ± Dei

The light point of each particle is supposed to be square and

is selected as two square pixels. The reasons for this choice can be

justified by the fact that we must:

- increase the usable square area of the computed Fourier

transform.

- have the same diffraction pattern of the aperture of the

particle to increase Young's fringes contrast.


sort out inadequate particle (big dust or packing of

particles). It is done by a criterion based on the particle area Sthr.

What is more, the application of the new processing works on a

binary image and is summed up by the particle gravity center.

II TREATMENT

The aim is to determine the module and the orientation of the

particle velocity and to eliminate the directional ambiguity. To see

whether a processing is good, one has to look over its reliability,

its accuracy, its application field, the time it requires to resolve a

mesh, the hardware you need to execute the processing. At the moment

many computation programs are applied, we can keep in mind especially

the most common methods. These methods analyse a small portion of the

mUltiple-exposure photograph. They use the fact that it produces a

locally periodic random image, and bring about the use of spectral

analysis either by the analysis of Young's fringes or by determining

the two dimensional correlations. The result is the visualization of

maxima at the coordinates corresponding to the average displacement of

the tracer particles. The major drawback of these methods is that the

computation of the autocorrelation function is very slow if you don't

have special hardware. We suggest a new computation method with

directional resolution.

The first step is the preparation of the original image. The

grey level image is transformed with an adequate threshold into a

binary image Ngthr. Then the center of gravity and the area of each

light point are computed. The whole development is based on these two

results. The area S determines whether a particle is valid according

to the dynamic condition (S < Sthr), so we can sort out the right

light points from the original array.


The second step calculates interval histogram on a band with a

fixed width b. This band is moved virtually with a fixed steepness 9

over the whole image, 9 varying in the 0-180 range (Figure 1). The

array histogram is the aparition number of a distance. The distance

unity is the pixel. When the mean flow angle is reached, three peaks

with the distance interval corresponding respectively to d12, d23, d13

appear on the histogram. This result can be represented as shown on

the figure 2. The abscissa axis is length scale d varying from 1

to Dim. Dim represents the dimension of the analysed mesh. The

ordinate axis represents the angle 9 varying from 0 to 1800• When the

quality of the image is not very good (image taken with very important

flow fluctuation, for instance), a standard deviation criterion

enables us to find the right angle. This criterion is necessary since

the three peaks are no more visible.

Virtual representation

o • o o •

o •

r------- .. 10 .01 1..-------.1

o .1st laser shot

• .2nd laser shot

() .3ed laser shot

Figure 1 - Principle of the interdistance histogram.

Artificial image made with various

combination of mono, bi, tripoints.


The figure 2 gives an example of histogram representation

obtained with an three exposures artificial image. The band width is

kept at 3 pixels. The angle scanning step is equal to 1 degree.

Apparition number

from 1 to 360 d

from 1 to Dim (pixels)

Figure 2 - Image representation of the histogram

The last step consists in determining the right direction of

the velocity vector [10,11). Two masks of dissymetrical pattern are

used. The dissymetrical pattern is defined by the previous calculation

of the distance and the time ratio (Figure 3). They are moved

virtually at the flow orientation over the whole image and the

computer calculates the frequency of each pattern. One can notice that

the mask pattern dimension gives the accuracy of the solution.

(1) (2) (3)

• •• •• •

Figure 3 - Masks with dissymetrical patterns

DEVELOPMENT OF PARTICLE IMAGE VELOCIMETRY

FORMULATION:

If Gx, Gy are coordonates of the gravity center of the light

point and b the band width:

with

and

III APPLICATION

Np Np

Histo (d,a) = ~ ~ O(o:).Ob(/3) 1=1 j> 1

o = 1

0= 0

(I Gx(j,a)-Gx(i,a) I-d) (I Gy(j,a)-Gy(i,a) I)

when o

0: '" 0

1

o

when 0 < /3 :s b

f3 > b

109

The principle of creating artificial images enables us to

evaluate a few characteristics of the processing by varying parameters

quoted previously.

As the direction of velocity vectors is known, applications

were realized with a two-exposure image. The purpose of this section

is to find the limit of each processing (autocorrelation and

histogram) and their accuracy. The angle parameter is kept constant

because the processings are invariant by rotation. The most

significant parameters are the fluctuation parameters (module

fluctuation and angle deviation) and the signal to noise ratio which

is the number of bipoint over the number of monopoint.

The first set of application is a study of the variation of the

signal to noise ratio. Two parameters are studied: relative error on

the distance and the angle deviation. The relative error on the

distance is the difference of distance found by the calculation

against the mean displacement introduced in the creation of the image.

The angle deviation is the difference between the introduced angle and

the calculated angle. The mean distance d is kept constant during all


these tests and is equal to 30 pixels. The S/N ratio takes value in

the range of 0.04 to 12. The whole number of light point is chosen to

be 50. These tests are made with three different module fluctuations

(Figure 4).

S/N=O.2~

130% No module fluctuation 100 0

~ .d 0 60 Vi"

<D 100% '" 0 o"i '" C 60 0, .'!l

60% +: <D

'" :g,. '5 '" c + 40 C

0 60% "": .2 ..

20 OJ g 0 '>

<D 40% 'i' . ~ ....... ~&-.

<D

• '0

'" A <D > : Ol ~ 20% : C -.; 10% .. -;;i;"' ........

<{

a: 0% III .. '"

0,01 0,1 1 10 100

SIN ratio (nb bipoints/nb monopoints)

130%

S/N=O.5~

... l Module fluctuation: 10% r 100

~ 0 60 Vi" <D 0 c ~ '5 C 0

g " <D >

~ a:

100% " ~ 60 0 ""

0> 80%

<D 0

* ,., :g,. ... 40 0 C

60% .. ~ ... 0

!--1 20 '>

40% ....,. + ~,k~ll--.t.

<D

'" '0 <D

20% Ol C

10% + ~~'" .. <{

+ 'I! ~ 0%

0,01 0,1 1 10 ,00


S/N- 2 -...........

130% ...... i Module fluctuation: 16% I <D o c lOO'JI ~ · '5 c o g <D

j ~

80%

· 20% 10% 0% ·

U,01

80

' . .:- i

0 0 40

I ... , 2Q

"'~Il-f-I -~ ++0 + ULUiSIi U :0

0.1 1 10 100


+ Histo. (dist) CJ Correl.(dist) liE Histo (angle) ... CorreL(angle)

Figure 4 - Variation in signaL to noise ratio

DEVELOPMENT OF PARTICLE IMAGE VELOCIMETRY III

The main result we obtain is that both proccessings supply the

right solution above a boundary value of the S/N ratio. The boundary

value is very low. In this case, there are 7 bipoints with 36

monopoints in the image. The two proccessings have similar

performances. Another illustration can be done by putting the

evolution of the Signal to Noise ratio with the module fluctuation

(Figure 5).


Figure 5 - Module fluctuation and signal to noise ratio.


A second set of test is carried out. The studied parameter is

the influence of the particle displacement (Figure 6 a,b,c).

., 0 c .s ., '6 c 0

g ., ., > . ., <II

£

50%

45%

40%

35%

30%

25%

20%

15%

10%

5%

0%

10

10

9

I 8 0, 7 ., :g.

1

10

......

I No module fluctuation I -I No angle deviation

. Maximum relative error: 10% -- " .. ~ ~ .. .. ~ • 20 30 40 50 eo

Distance (pixels)

... Histogram D Autocorrelation

I No module fluctuation I l No angle deviation

maximum error: 2 degrees

A ~ .! - ,

.. E3

20 30 40 50 60

Distance (pixels)

... Histogram D Autocorrelation

Figure 6a - Variation in distance with no fluctuation


90%

~ CI> 70% 0 c ~ '" '5 50% c 0

g 30% CI> CI>

~ 10%

a; ex:

10

110

., 90 II)

~ Ol II)

70 :g. c 0 50 ., .,

0> Ol

30 "0 CI> rn c 10 q;

10

l~odUle fluctuation: 5% I Angle deviation : 5 degrees

validity lim\ 0 ... ... 0

~

20

~ -...

0

0

Ii '" liil ... lii

30 40 50 60

Distance (pixels)

A Histogram o Autocorrelation

I Module Huctuation : 5% • ~ Angle deviation : 5 degrees I

validity limit o

... 0

... ... ... 1\ iii t a ... ~ 20 30 40 50 60

Distance (pixels)

... Histogram a Autocorrelation

Figure 6b - Variation in distance with increassing fluctuation

113


Validity limit ~aluu~toco~IT~el~a~ti~on~;-h~i~st~p~grn~m~ ______ ~~~~~~~~l <7\,I"'"T . • \. l Module fluctuation: 10% I

~ 70%+-__________ \_\.+-1 .... D3-·~\.':_----=:=----_An--~~:~d""'--.au~:~: 8_d-J.egrees C , ~

~ ~t-----------_L_"_'''_''-_''~'~~~: __ n-__________ ~ c 0 o g ~+-------------------+-~~~~--------~ CD CD o

~ 10%dr-------<1a-------~--+-----....., ---'"'-...... .--'f

£ 10 20 30 40 50 60

Distance (pixels)

~ Histogram o Autocorrelation

Validity limit

110 autocorrelation, histo~rnm l Module nUCluation : 10% I ., Angle deviation: 8 degrees

CD 90

'i ~ 0

g' 70 i :g, c 0 0 50 ''ij 0 .s:

-~ ~ CD 30

.... _- ~ '0

CD ...

0 rn 0 0 ... C 10 .i. t ... a A '"' ~

10 20 30 40 50 60

Distance (pixels)

... Histogram o Autocorrelation

Figure 6c - Variation in distance with increassing fluctuation

We have chosen a S/N ratio not too high but beyond the last

boundary value (S/N 0.5). The accuracy is displayed if any

fluctuation is introduced. The relative error on the distance is

smaller than 10 7. and the maximun angle deviation is not greater than

2 degrees. When the module fluctuation and the angle deviation are

increased, the boundary distance reaches 35 pixels. One can notice a

better performance with the histogram computation. It is due to the

band width value. A value larger than two disminishes the effect of

angle deviation. This parameter operates as if a projection were made

in the direction of flow calculation of every light point positionned

in the band.


VI CONCLUSION

115

The histogram treatment is a new way to resolve PIV pattern.

Its principle is simple. Working on only an array of the gravity

center of each light point simplifies development of the calculation

and the time it requires to determine the expected informations is

very short without any special hardware. In addition, finding the

direction of the velocity vector is now possible with the use of a

dissymetric pattern. The application of the two masks is easy because

the gravity center of each light point is already known.

We would like to point out that the fundamental part of this

technique is its experimental part and the treatment doesn't raise a

real problem when the image has a good quality.


REFERENCES

1. Adamczyk, A.A., Rimai, L.: 2-dimentional particle tracking velocimetry (PTV): Technique and image processing algorithms. Expts in Fluids. 6 (1988) pp 373-380.

2. Lourenco, L.: Advances in tluid mechanics measurements. Lecture notes in engineering. M. Gad-el-Hak (editor) 45 (1989) pp 127-199.

3. Arefi, S., Joulin, M., Prabel, F., Schon, J.P., Lowitz, G., Courbon, M., Zeboudj, R.: Apports des techniques de traitement d'image a la visualisation d'ecoulements. Rev. Gen. Therm. Fr. W 320-321 (1988).

4. Settles, G.S.: Modern developments in flow visualization. AIAA Journal. Vol 24 W8 (986) pp 1313-1323.

5. Adrian, R.J., Fansler, T.D., Reuss, D.L.: Instantaneous planar scale vorticity and strain rate velocimetry. SAE 890616.

French, D.T., Landreth, C.C, measurements of velocity and large in an engine using particle image

6. Grant, I., Smith, G.H.: Speckle velocimetry applied to wake flows. Int. Cont. on Holography Applications. S.P.I.E. Vol 673 (1986) pp 358-365.

7. Merzkirch, W.: Flow visualization. Academic Press Inc. 1974.

8. Lourenco, L., Krothapalli, A.: The role of photographic parameters in laser speckle or particle image displacement velocimetry. Expts in Fluids, 5 (1987) pp 29-32.

9. Marko, K.A., Li, P., Rimai, L., Ma, T., Davies, M.: Flow field imaging for quantitative cycle resolved velocity measurements in a model engine. SAE paper 860022 (1986).

10. Adrian, R. J., Landreth, C. C.: Electrooptical image shifting for particle image velocimetry. Applied optics. Vol 27 N°20 (1988).

11. Landreth, C.C, Adrian, R.J, Yao, C.S.: Double pulsed particle image velocimeter with directional resolution for complex flows. Expts in Fluids, 6 (1988) pp 119-128.

ALGORITHMS FOR AUTOMATIC MEASUREMENT OF

SIZE AND VELOCITY OF SPRAY DROPLETS

FROM HOLOGRAPHIC RECONSTRUCTIONS

Anselmo Chavez and Franz Mayinger

Lehrstuhl A fiir Thermodynamik, Technische Universitiit Miinchen

Arcisstr. 21, 8000 Miinchen 2, W. Germany

ABSTRACT

Software procedures to evaluate single and double pulsed holograms of

sprays with special emphasis on automated focussing and droplet identifi-

cation techniques are described in detail. They base on techniques of the

digital image processing and are implemented on a personal computer which

makes the work suitable for applications in small optical laboratories. Fi-

nally, new results of the evaluation of a large series of holograms of sub cooled

spray droplets injected into saturated vapour are presented as an example of

application.

Keywords: droplet sizing, evaluation of pulsed holograms, spray character-

ization, image processing of holographic reconstructions.

117


118

INTRODUCTION

A. CHAVEZ AND F. MA YINGER

In the last 10 years many efforts have been directed toward evaluation of

pulsed laser holograms of particle fields by applying digital image processing

techniques. An insight into this problem was presented by Haussmann &

Lauterborn (1980). The evaluation of holographic reconstructions of particle

fields consists of scanning the three dimensional image with a videocamera,

in identifying well-focussed particles within the depth of field of the imaging

optics (e.g. microscope objectives), and in measuring and classifying the

selected particles with respect to the depth coordinate. With this information

it is possible to reconstruct the history of the particles (i.e. their trajectories

and time of residence in the control volume) and, depending upon their

changes in shape, size, number and concentration, to deduce information on

heat transfer or other transport quantities associated with these changes.

The main problem in the evaluation of holograms of particles consists of

selecting and classifying well-focussed particles while the videocamera scans

the three-dimensional holographic image. From the conglomerate of particles

imaged on the camera sensor at a given value of the depth coordinate, only

those few which satisfy a given selection criterion are allowed to remain in

scene for later processing. Ligthart & Groen (1982) discussed the possibilities

of a series of filtering algorithms which can be applied as a criterion to select

the particles. Some of these algorithms are used today by the autofocus

pocket cameras. Recently, Schaffer & Umhauer (1987) presented a semi

automatic method for evaluation of double pulsed in-line holograms. In this

SIZE AND VELOCITY OF SPRAY DROPLETS 119

method, the operator searches for well-focussed particles while a computer

controlled video camera stepwise scans the holographic image. Finally, with

help of image processing techniques the selected particles are measured and

classified. Now, the tendency is to eliminate more and more of the interactive

participation of the operator and in this manner to avoid the permanent

repetition of human decisions along the evaluation process.

This article presents two computer-aided procedures for automatic eval

uation of pulsed off-axis holograms (single and double pulsed holograms re

spectively) of spray droplets. The first one is used to recognize sharply fo

cussed droplets from single pulsed holographic reconstructions. It is based on

an idea by Haussmann (1979), who applied gradient operators as a sharpness

criterion in the focussing and mesurement of bubble sizes. For the present

study, the resolution of the Haussmann's method was enhanced by two orders

of magnitude making it suitable for droplet size measurement. The second

procedure is applied to recognize the two successive positions of droplets from

double pulsed holographic reconstructions and to measure the corresponding

velocities. Both procedures were implemented on a personal computer. By

using these procedures, the operator is released from the situation of making

decisions interactively during the evaluation process. This allows for a more

efficient application of drop focussing and classifying criteria, resulting in a

substantial increase in the accuracy of the measurements and in an effective

reduction of the time dedicated to the evaluation.

120 A. CHAVEZ AND F. MA YINGER

SCOPE OF THE WORK

The aim of this work is to describe an approach for automated evalu

ation of pulsed laser holograms of sprays. The holograms to be evaluated

are of "off-axis" type as can be obtained from the holographic arrangement

presented in Fig.1. They contain information about size, position and veloc-

ity of the spray droplets and about the form of the spray cone. The spray

is produced by injecting sub cooled liquid of the refrigerant Rl13 (Trifluoro

trichloroethane) through a 60° simplex (hollow cone type) pressure nozzle of

0.6 mm in bore diameter, into an environment formed by its own saturated

vapour. A schematic description of the spray flow is presented in Fig.5.

" ---- ----- ---- --------------------~-( Energy 1 Joule '1:

i A. 694 nm : pulse duration 30 ns i ; pulse interval! 1 - 800 fl~)

PULSED

RUBY LASER

AL divergent lens AS imaging system H holoplate MS ground glass S mirror SL convergent lens ST beam splitter

EXPERIM. CHAMBER ...--_____ .. 0 bj ect

rot,,,,ronce beam

HOLOCAMERA

Fig.l. Holographic Arrangement

Special emphasis is given to the discussion of our technique developed

to scan the holographic image, the criterion adopted to select well-focussed

droplets and, in the case of double pulsed holograms, to identify those spot

pairs in the scanned image corresponding to droplets imaged at two successive

positions (double exposure of the same holographic plate). The results of the

evaluation of a series of 160 pulsed holograms of the Rl13 spray at different

injection mass flow rates (0.8, 1.37,2.0,2.72,3.86 g/s) and vapour pressures

(0.1 - 1.0 MPa) are included as an example of application.


OPTICAL SET-UP AND EXPERIMENTAL CHAMBER

Holographic Technique

The pulsed laser holography represents one of the more suitable non

invasive measurement methods for the study of transport phenomena (e.g.

heat and mass transfer) in dispersed transparent flows. It provides one or

more three dimensional scenes of the volume of interest taken at a very short

exposure time ('" 30 ns). The recorded holograms can be reconstructed with

help of a continuous laser beam and analysed off-line at any time. Fig.1 shows

the optical set-up for the recording of off-axis holograms. Using a separated

reference beam, the holographic reconstructions can be observed directly or

with help of a microscope as in a photograph. The resulting reconstructed

images are very clear for particle sizes d > 10 A where d and A are the drop

diameter and the wavelength of the laser light used to record the hologram,

respectively. Position and velocity of the particles of the dispersed phase

can also be obtained from the reconstructions. The principal features of the

method and some advanced adaptations are explained in detail by Trollinger

(1975) and Chavez & Mayinger (1988).

Experimental Chamber

Figure 2 shows a scheme of the experimental chamber. It consists of a

thermally insulated cylindrical autoclave of 206 mm interior diameter and

650 mm in height, designed for pressures up to 2 MPa. Two quartz glass

windows (</> 100 mm) installed in the cylindrical wall provide the optical

access. The liquid is injected from the top of the vessel through the hollow


cone nozzle. The nozzle, concentric with respect to the cylindrical wall, can

be moved axially to permit the observation of any section of the spray. The

lower third of the vessel is filled with liquid refrigerant Rl13 which is heated

by an electrical resistance (1.2 kW) installed on the lower plenum, to produce

the saturated vapour environment in the injection space of the autoclave. A

funnel is placed between the boiling liquid and the spray to collect the spray

droplets and leads them to the outlet.

~ I'IiESSUIIIZEIl uaulO

flESEnvom

,

t

Fluid: Refrigerant R113 ,

TEST ClIl\.MIlEn

f ,...-------, liquid Inlel

liEATEn; -

I SI •• m l,.p

Olops Outlel

~-= IHv9

5 <m< 70 kglh l=20't 2 < P < 1.B MPa

~ ___ ~ _____ ~ ________ .L-_______________ ~ ______ __

'------------._------

Fig.2. Experimental Facility

Measurements of temperature and pressure at different points of interest

in the facility were carried out with conventional thermocouples and pressure

sensors monitored by a personal computer.

SIZE AND VELOCITY OF SPRAY DROPLETS

IMAGE PROCESSING OF

HOLOGRAPHIC RECONSTRUCTIONS

123

One of the principal problems appearing in the application of pulsed laser

holography consists in handling the large amount of information contained in

the holograms. Theoretically, holographic materials are able to store the in

formation on position, texture and brightness of more than 106 particles per

square millimeter. In our case, the droplet concentration is very much lower

(a few drops per cubic millimeter of reconstructed space). Nevertheless, one

single hologram can contain information about position, size, and velocity of

many thousands of droplets. Comprehensive studies of the characteristics of

the droplets and their interactions with the gaseous environment necessarily

require the help of computer-aided particle counting and measuring methods.

Fortunately, the rapid development of the computer technology in the last

years, as well as the miniaturization and mass production of computer com

ponents has made many applications of the digital analysis and processing

of images possible. Many tasks of the pattern recognition, handling of im

age data, and computer graphics, which were earlier reserved only for large

computing centers and TV companies, can be today also carried out in small

optical laboratories with help of the personal computer.

Image Processing System

The components of the digital image processing system are shown in

the flow diagram of Fig.3. The hologram H is reconstructed by illumina

tion with a continuous parallel beam from a He-Ne-laser, which simulates


the reference beam. The optical information contained in the reconstructed

image I is scanned by the video camera K, and transmitted to the digitizer

D. Here, the signal is transformed into digital information and is stored in

the digitizer frame memory, in form of an array of 512 x 512 picture elements

(pixels) of 8 bits. This means that each picture appears as a pixel matrix

in which the colour of each pixel can be represented by one of 28 possible

grey tones. The digitizer is directly connected to the host computer C by a

16 bit bus interface, allowing for fast communication. The processing of the

digitized picture is then carried out by the host computer using the digitizer

frame memory interactively, for pixel allocation. In order to visualize the

information actually stored in the digitizer frame memory, this produces a

continuous RGB (false color: red, green, blue) output signal which can be

observed on the graphics monitor M.

H Hologram

~ Laserlight

M

• Nozzle

spray Image

Fig.3. Digital image processing system.

Scanning of Holographic Images

K

c

Personal Computer

The pictures to be scanned are obtained from single or double pulsed

holograms. They represent a three-dimensional (3-D) image corresponding

to a "frozen" scene of the spray, as shown in the scheme of Fig.5. In this

scheme, rand z are axis symmetrical, cylindrical coordinates with origin


o at the nozzle outlet. Single pulsed holograms contain information about

the geometry of the spray, the break-up of the liquid sheet, and the droplet

distribution in the control volume. Complementary, double pulsed holograms

provide information about the droplet velocities and trajectories. A typical

picture obtained from a single pulsed hoiogram is presented in FigA.

FigA. Typical picture of the reconstructed spray flow.

Due to the fact that the video camera can only record two-dimensional

(2-D) pictures, for example in the optical plane X-Z, it must be focussed step

wise along the depth coordinate Y in order to process the 3-D holographic

image, as suggested in Fig.5. In this manner, the 3-D holographic image is

transformed into a series of many 2-D video pictures. In order to achieve a

one-to-one relation between the spatial coordinates r, z of the holographic

image and the planar coordinates X, Z of the video pictures, it is necessary

to adjust, as accurately as possible, the optical distance between the picture

plane (focal plane) being scanned and the camera sensor, so that each picture

can be correlated with a value of the depth coordinate Y (normal to the pic

ture plane). This is best accomplished by carrying out the focussing process

by moving the camera itself rather than adjusting its objective. For this re

alization, it was also necessary to provide a very good alignment between the


optical axis of the camera lens system and the direction (depth coordinate

Y) of the reconstructed object beam of the hologram. By forcing the optical

axis of the camera to coincide with the Y-coordinate, lateral displacements

of a pixel with coordinates X, Z on the camera sensor do not occur while

the camera moves along the Y -coordinate. Consequently the droplets can be

identified with an excellent repetibility.

r I lzj

SPRAY ANGI.£

OPTICAL FROM n-tE POINT OF VIEW OF n-tE VlDEOCAMERA

Fig.5. Description of the Spray Flow and Example of the Scanning a holo-

graphic reconstruction

In order to control the position of the camera and to measure the values

of the Y -coordinate, the camera was mounted on a traversing mechanism

(as schematically indicated in Fig.3) with a linear resolution of 2 Ilm which

allows the repositioning of the camera within a relative error of 10 Ilm over

a distance of 200 mm. The traversing mechanism itself is supported by a

coordinates table with 5 degrees of freedom to facilitate the alignment.

Criterion for Automatic Focussing

In scanning holographic images of spray droplets with a video camera,

single droplets have to be selected from the 2-D image in the camera sensor as

the camera is moved stepwise through the reconstructed holographic image.

For a sequence of video pictures of the same droplet imaged at different, but

very narrow, focal distances, the best focussed picture has to be identified.


When this succeeds, the corresponding droplet can be selected. After that,

the droplet is measured and classified. In order to apply systematically this

selection criterion to all droplets in the holographic image, an automatic

focussing procedure was developed. This new procedure was calibrated by

applying grey value gradient operations as proposed by Haussmann (1979).

First a typical area of interest of the image is scanned and analyzed.

From this analysis one determines the grey level of the background, of the

speckle noise and of the structures to be evaluated. The pixel size is selycted

according with the droplet size in order to have enough pixels for the digi-

tal drop representation. With this information a typical droplet is defined

throughout the pixels

g(i,j), g(i - k,j - 1), g(i + l,j - k), g(i + k,j + 1) and g(i -l,j + k)

as shown in Fig.6. 9 means the grey value, i,j, defined positive, mean the

coordinates in the pixelmatrix and k is a counter (k = 1,2, ... ) which points

on rows or columns of the pixelmatrix starting on i,j. If there are structures

of the image matching the droplet definition, they are initially selected as

droplets and checked whether they are sharply focussed or not. This can be

realized by varying the counter k, building the grey value gradients \l 9 of

the form

\l _g[i+1,j-(k+l)J-g(i+l,j-k) 9 - 1':1itl.j (1)

with the coordinate increments 1':1i, 1':1j 2: 1

and by comparing the gradient value with a previous calibrated threshold.

If the value of the gradient is bigger than the threshold, it is assumed that

a sharply focussed conture is reached. The remaining directions are also

analyzed in the same manner. For visualization of the focussing process,

the contour pixels of the droplet image are enhanced by setting their grey


value equal to the maximum grey value (gmax = 255). On the contrary, if

the threshold is bigger than the gradient, it implies blurred edges and the

structure (blurred droplet image) is filtered out.

Once the sharply focussed droplets along the entire image have been

identified, the areas defined by the enhanced contours are measured by count-

ing the amount of pixel contained in the closed contour. The corresponding

center points of those areas which satisfy a given circularity criterion are

calculated and the information stored. This very simple algorithm works one

order of magnitude faster than that by Haussmann (1979) and can be easily

adapted for other applications. The Haussmann's method can be applied

to calibrate the value of the threshold as described by Chavez & Mayinger

(1990).

irt= I I

1

I I I

I

I

i+1,j-k t I I ~

I I I I I

1",-,> i •. .. , I li,j I

:--+ +k,j+1

:.' T I I I I I I I ,

I I I j·:1,j+k I I ! I I I 1 I I

k=1,2,3 ...

Algorithm for fast

droplet identification

Fig.6. Pixel array for droplet definition in video pictures taken from holo-

graphic reconstructions.

The circularity analysis mentioned above is performed in two steps as

follows: In the first step the area being analysed, is assumed to be formed by

a pixel ensemble p(i,j) in which the coordinates (i,j) vary from i!:,in to


i~ax and from j::'in to j~ax (the index P is used here only to identify

pixels of the ensemble p(i,j)). With the height (j~ax - j::'in) and width

(i~ax - i!'..in) of the ensemble p(i,j), the parameters ~max and ~min are

defined in such a manner, that ~max corresponds to the larger dimension,

height or width, and Amin to the smaller dimension. Next, the quotient

CI = ~min (2) ~max

provides information about the slenderness of the area. For example if

CI ~ I the area corresponds to a pixel lane but not to a circular body.

More information about the area itself can not be obtained from CI because

it is not area-sensitive (to understand this notice that for an area correspond

ing to two circles of diameter d, drawn symmetrical to a vertical line one at

the side of the other, CI results CI = d/2d = 0.5; but if the line of symmetry

line is rotated 450 it results CI = 1. That means that CI can only inform

when an area is far away to match a circular body).

In the second step, the area Al defined by the pixel ensemble p(i,j) is

compared against the area A of a circle of diameter d = ~max. In this case,

the quotient

C2= Al A

(3)

is very sensitive with respect to the pixel area A, but it underestimates

strongly the circularity of pixel ensembles possessing a pic (e.g. of the form

of a raindrop), which are very common in digitized images of particles. In

order to compensate this drawback we propose to use the quotient CI of

Eq.(2) if the amount of pixels in the ensemble p(i,j) is smaller than 25 and

the quotient C2 for larger areas as a circularity criterion. By testing these

criteria with typical pixel arrays of droplet images, we selected a lower limit

of 0.4 of the circularity factor, for which the images are still considered as


droplet images. This value includes the effect of an aspect ratio of 0.7 for

rectangular pixels (For square pixels, a lower limit of 0.6 is recommended).

EVALUATION ROUTINES

The images to be processed are of the same kind as the droplet zone

of the photograph presented in FigA. They contain a collection of spots

of different grey-values, ranging from 0 = black to 255 = white, which

represent the spray droplets, and a given fine grain pattern forming a noisy

background (speckle noise). This speckle pattern is produced by the diffuse,

coherent illumination used to record the hologram.

Although single and double pulsed holograms are of the same nature

and their evaluation is similar, we decided to configur two different evalua

tion routines. They are: the routine EINZEL, which evaluates single pulsed

holographic images for which accurate measurements of the drop size are of

essential importance, and the routine DOPPEL, which evaluates the images

obtained from double pulsed holograms. For this last case, the size of the

droplets does not need to be recalculated, the routine DOPPEL is dedicated

to identify spot couples which are originated by droplets imaged at two suc

cessive positions. The distance between these successive positions represents

the droplet velocities.

The Routine EINZEL

The processing of images obtained from single pulsed holograms involves:

the separation of the droplet images from the background, identification of

the sharp focussed droplets, measuring their projected areas, and the evalua

tion of their equivalent diameters and center points with respect to a reference

frame. All these operations, from the image capture by the video camera to

the final result, are carried out by the program EINZEL. It consists in a


series of digital filters and gradient operators selected from standard image

processing libraries and our own algorithms developed for measuring, cali-

bration and data handling.

The application of the focussing criterion described earlier constitutes

the kernel of the program. A brief description of its principal features can

be summarized as follows:

1. Initially, the camera is situated in the middle of the traversing mech

anism, and its objective is adjusted with the help of a calibration hologram,

so that the focal distance coincides with the center of the holographic image

to be evaluated (the origin of the depth-coordinate Y is set here). Then,

the camera is moved away from the holographic image until it disappears

completely. Here the initial point Y i of the image processing is set. From

this point, the camera will be driven stepwise towards the hologram, so that

the focal plane, corresponding to the focal distance of the camera objective,

will be moved through the 3-D holographic image.

2. An image (A) taken by the video camera is stored in the frame mem

ory of the digitizer and the grey-value of the noisy background is evaluated.

By a simple adjustment of the gain and the offset of the image contrast, the

grey-value scale is shifted so that the noise is filtered out or reduced to a

minimum, without modifying the grey value gradients of the image.

3. The image A is smothed using an average filter of the form

3 3

g:j = ~ L .E 9 (i + 2 - k,j + 2 - I) . mkl . f(i,j) k=l 1=1

with the kernel

(4)


and f(i,j) as a function which plays the role of a spatial

pass band filter

and the resulting image is named B. After that, the focussing criterion is

applied and the spot contours enhanced. All parts of this last picture con

taining grey-values from 2 up to 253 are filtered out applying a contrast

enhancement operation (binarization). The resulting image is named C.

4. The images Band C are superimposed using the boolean operation

AND. Image parts in C produced accidentally during the processing are

filtered out. This automatically assures the authenticity of the information

being processed. The resulting image is named D.

5. The equivalent diameters and centers of those spots (droplets) pos

sessing a circularity larger than 0.4 are evaluated and stored for later data

processing. And finally, the videocamera is moved a step of 0.5 mm towards

the hologram and the complete routine is repeated until the whole depth of

the holographic image is scanned.

Examples of representative stages of the image processing are presented

in form of photographs in Fig.7. Here, The photographs 1 to 3 show the fo

cussing process. The nozzle is included for better orientation. 1 represents an

original view from the holographic reconstruction, 2 the gradient extraction

and 3 the binarization. An enlargement At of the droplet zone of picture 1

is showed to illustrate the process of noise filtering and the final result in the

lowest picture of the right column.

To assure a high confidence of the measurement, the method was cali

brated by evaluating an hologram of glas pearls which were previously mea

sured with the help of a microscope. The pearl images were measured at

different enlargements. For each enlargement, the pixel size was calibrated

by using an optical grid. As spected, the binary pearl images represented


by 6 to 10 pixels presented relative large deviations of the order of 7% with

respect to their calibrated size. For image representations with 40 or more

pixels, the deviation was smaller than 1 %.

The Routine DOPPEL

From double pulsed holograms one obtains information about velocity

and trajectory of the droplets. The holograms represent a conglomeration

of spot couples, in which each couple represents a spray droplet imaged at

two successive positions corresponding to the times t = tl and t = t 2, where

t2 - tl = 6.t is the time interval! between the two exposures used to take the

hologram. b.t can be adjusted by the ruby-laser electronics between 1 - 800

/ls.

In order to evaluate the double-pulsed images, the routine DOPPEL was

developed. In this case, where the computation of the drop velocities is the

objective, the correct identification of the center point of the drop images

becomes more important than the measurement of the drop sizes. Here, the

drop images can be first expanded by using a fuzzy mask, allowing a rapid

identification of both spot partners of a couple when they lie in different

focal planes. Summarizing, the task of the program DOPPEL consists in

identifying the spot couples from the pictures taken by the video camera,

in measuring the distance between the center points of the two successive

droplet images, and in computing the droplet trajectories related to the space

coordinates in the injection volume.

Representative stages of the image processing are presented in form of

photographs in Fig.S. In this figure, (1) represents the source image, (2) the

fuzzy mask, (3) the image after noise filtering, (4) the identification of spot

couples and (5) the evaluation of the image.


Fig.7. Representative steps of the image processing of a single pulsed holo

gram of the Rl13 spray.

1) Original image, 2) smoothing and gradient extraction and 3) bi

narization. The right picture column shows an enlargement Al of

the droplet zone of picture (1), its noise filtering and the final droplet

identification.

This figure appears in color on p. viii


Fig.S. Representative steps of the image processing of a double pulsed holo

gram of the RIl3 spray.

1) Original image, 2) fuzzy mask, 3) thresholding, 4) identification of

the particle couples, and 5) final evaluation. The middle point of the

screen has the real coordinates r = 14.70 mm, and z = 25.19 mm; 29

particle couples were found from which a mean velocity v = 2.6 mls was obtained.

The most important task of the program DOPPEL consists in finding

the spots corresponding to the two successive positions of the droplets. In

order to perform this, the subroutine VEL was developed. It consists of two

modules: a spatial frequency analyser and a measuring algorithm. For the

description of the subroutine performance, the photograph A of Fig.9 will be

analyzed. Without regard to color or form and with the previous assumption

that picture A was obtained from a double pulsed holographic reconstruction,

and that the elements of A represent droplets which are falling down between

a guessed angle of ±45° with respect to a vertical line, the first module of

VEL has to recognize automaticly the two positions of each droplet. First

the coordinates (si, sj) of the spot center points S and the vectorial distance

a between each two center points are calculated as illustrated in picture B

of Fig.9. The amount N of possible distances a obeys the combinatoric law


( NS) Ns! N = 2 = 2! (Ns - 2)! '

wherein N s means the amount of center points S.

and

From the vectors a, the scalar arrays

-1 an f3n = cos lanl

(5)

(6)

(7)

are defined which describe the spatial distribution of the vectors an. Now

the operation N

F(f3) = L < f3n . f3 > , n=l

where the product < f3n . f3 > is defined by

< f3n . f3 > = {I if f3 = f3n ± b o else

(8)

with b meaning an arbitrary given tolerance, transforms the spatial distribu-

tion f3n into the frequency distribution F(f3). F is represented in Fig.9D as

a normalized frequency distribution f(f3) = F(f3)/ Fmaz with the maximum

frequency F maz as the norm.

With the information of this first result, a second operation of the form

of Eq.(8) with the distances an as the independent variable is carried out.

This is N

F(a) = L < an' a>, (9) n=l

Analogous to Eq.(8) F is represented in Fig.9E as the normalized frequency

distribution f(a) = F(a)/Fmaz . The diagrams D and E of Fig.9 show a

preferential direction f3p and distance ap respectively, which when written as


v = ap sin j3p i + ap cos;3p j (10)

where i and j mean unitary vectors in the direction of the coordinates i and

j, represent a mean velocity vector v of the drops in the droplet swarm. This

mean velocity will be used by the second module of the subroutine VEL in

order to find out the real droplet velocities. For this realization the magnitude

ap of v is incremented by the tolerance ±0.2 ap and its corresponding angle

;3p is also incremented by the tolerance ±;30. ;30 can be varied between 7 and

150 allowing for strong variations of the droplet trajectories. The picture

C of Fig.9 shows the result of applying the subroutine VEL on picture A .

- 1.e ~ u ~ 0.8

~ 0.5

Ci: 0,4 ~

§ ~ is 0.0

• • • • •

•

" '"

~ ... .

c

'I

" "

" , 1 .,

!l

. • •

· I •

1

,., 1/ fI.1 II

~ • • •

~-

Z -so .oW ..JO -20 -10 0 10 20 30 40 so

Angle {J to the vertical

'l~ • . " •

'.0 .. ~-., 1

1 1 .. 1

1 ,

" .. 100 3)0 300 0&00

Distance a, Pixel

Fig.9. Steps for recognition of spot couples corresponding to two successive

positions of droplets using module 1 of the subroutine VEL.

A) Source image, B) calculation of the spot center points and the vec·

torial distances between each two center points, C) identified couples.

D) and E) Frequency analysis.

The scheme in Fig.l0 illustrates the working method of the second module of

VEL. Herein V; and j3i mean the real magnitude and direction of the velocity

corresponding to the imaged positions of the droplet.

138

droplet Image B


-;---..___ Cosillcn 1

-±-o.2IVIjJ; v. /~.:.:O~- v: ~P _ possible ~ pcsition 2

~ identified .' posItion 2

./ I secanc

:- {3o / ~~:t

Fig.lO. Working method of the module 2 of the subroutine VEL.

The complete evaluation of a double pulsed hologram IS carried out

similarly as described for the case of single pulsed holograms.

RESULTS

As an example of the applicability of the discussed evaluation technique,

results of the evaluation of a large series of holograms (160) are presented

in Figs.ll to 14. These results are part of a study dedicated to the charac-

terization of sprays when liquid is injected into a condensable environment

at high reduced pressures (Pr = pIPeri,). In this case, the holograms corre

spond to the situation in which subcooled liquid refrigerant R1I3 (Trifluo-

rotrichloroethane) is injected into an atmosphere formed by its own saturated

vapour considered to be in repose. The experiments were systematically car-

ried out at stationary conditions in the thermally insulated autoclave de-

scribed earlier where the vapour atmosphere could be prepared and kept at

constant pressures of 0.10, 0.15, 0.20, 0.25, 0.40, 0.60, 0.80, and 1,00 MPa.

For each vapour pressure, five experiments were carried out at liquid mass

flow rates of 0.8, 1.37,2.0,2.72, and 3.86 gls which correspond to moderate

up to high Reynolds numbers (100 ~ Re ::; 3500). A pressure-swirl nozzle

of 0.6 mm bore diameter was used as an atomizer. For each experiment,

four holograms (two single and two double pulsed holograms) were taken,


covering two spray zones: near the nozzle and 55 mm downwards from the

nozzle. A comprehensive discussion of the physics of the experiments and

results were presented in reference (Chavez, 1991).

190

180 %' Po MPa

10T-______________ -----------.

~ 9+-__________ ~ ____________ ~ ~

170 - 0 , 10 0,10 '" 0,40

C

"'" !e 0,15 <- 0,60

160 < 0,20 0,60 ;; g " 0,25 1.00 @ 150

ii 140 =~ + 6 ~ 130 i 0 a ~~ 120

~ 110 -::;

100

= 6+-______________________ ~~

': 7+-__ ~------~--~2~~------~--~~y " <,..-.~ 6+-__ ~ ______ ~~~~--~--------~ '" :!. _T~ c ~ 5+-__ ~~~~~~--~----~----~ c /~ is :/ S 3+--,~A ________ ___

~ 2+-,-.-1 ________ ___

lL-~O~~--~~~~~~~

Po MFa

0 0,10 040

0.15 0.60

0,20 0,80

0.25 1.00

0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

Mass Flow Rate m,9/s MassFlowRate m,9/s

Fi~.11 Mean drop me d as a fwae\ion of the mus 80w rate M of the RII3- Fi,.12. Mean drop velocity v &II a function of the m&U fiow rate M of the

spray LDjeet.ed i.nto it. own MJ.urate<! vapour at different vapour preI- RIll-spray injected into ita own Nlurated vapour at. differe:t\t vapour

60

70

= 60 :S III 50 OJ ~

40

>- 30 ~ !E 20

10

I

R~ -;: -S,;'~I---·-~. ! < " ~ jl /v,v ]-". ---- --'----

"~' Po MPa

i'W--~ 0,10 -' 0,40

0,15 .:. 0,60

E " i i ~ 0,20 0,80 I

rr, - 0.25 C 1.00

0,0 0,5 , ,0 , ,5 2,0 2,5 3,0 3,5 4,0

Mass Flow Rate m, 9/S

E g 7 N

...J

.r::. ~ 4 III

...J

~ 2+-~~--~~~~~~~==~~~~ -"' ro III as

0.5 1,0 1,5 2.0 2.5 3,0 3.5 4,0

Mass Flow Rate m, 9/S

Fi~.13. Break-up ~h La u a functioD 01 the mus flow rate M of the Fig.14 Spray ~le 0 as a functiOD of ~ mus 60wrate M at the RIll-spray

Rll3-spray injected into ita 0Wtl a.1a.rated vapour at di1l'erent YapOW'

.",..,.,..,

The diagrams of Figs. 11 and 12 summarize, in form of arithmetic mean

values of drop diameters and velocities, a great amount of measuring data

(position, size and velocity of about 2 000 000 droplets were stored). The

pressure Pv of the vapour environment is plotted as a parameter. The dia-

gram of Fig.ll shows an asymptotical decrease of the mean drop diameter

when the liquid mass flow rate is increased. According to Frazer & Eisenklam

(1956), the increase in the inertial forces, which depend on the flow rate, has

a desintegrating effect upon the liquid sheet of the spray near the nozzle.

This leads to the production of droplets with a smaller diameter. Figure 12


reveals that the mean drop velocity increases almost linearly with increasing

liquid mass flow rate. In this case, the effect of varying the environmental

pressure is clear. The mean drop velocity diminuishes when the vapour pres

sure increases as would be expected since the vapour density and viscosity

also increase when the saturation pressure of the vapour increases.

As commented above, the measurement of the liquid sheet of the spray

is very important for spray characterization. The present evaluation method

is an ideal tool to perform that measurement as well. From processed images

of the nature of Fig.4, it is very easy to characterize the liquid sheet.

The liquid sheet geometry can be represented by its break-up length Lz

and its corresponding angle Q at this length. The diagrams of Figs. 13 and 14

present the measurements of Lz and Q, respectively, as functions of the mass

flow rate M. The pressure Pv of the vapour environment is plotted again as

a parameter. The diagram of Fig.13 shows the typical decrease of the break

up length when the mass flow rate is increased. The vapour pressure works

like a reduction factor upon the shape of the curves. The diagram of Fig.14

illustrates the strong influence of the vapour pressure Pv upon the spray angle

Q. At higher vapour pressures, the resulting force in the radial direction,

responsible for the formation of the hollow cone, diminuishes because of the

higher resistance of the vapour against the liquid flow. This observation

agrees with the theory of Frazer &. Eisenklam (1956) for increments of the

vapour pressure below a given limit. IT the vapour pressure is incremented

above that limit, the vapour-cored vortex in the swirl chamber of the nozzle

diminuishes and tends to disappear. This leads to a stagnation of the flow at

the nozzle outlet producing a high turbulent zone. In this zone, the high sheer

stress produces a sudden atomization of the liquid without formation of any

liquid sheet. The discharged liquid occupies more volume than if it would be

discharged as a liquid sheet. For this reason the spray angle increases again.


For the studied nozzle a vapour pressure of 0.25 MPa corresponded to the

minimum spray angle. At pressures higher than this, the spray angle behaves

contrary as expected, it increases proportional to the vapour pressure.

From the observation of Figs.ll to 14, we can conclude that the vapour

pressure has only a weak influence upon the drop size, but it is quite impor

tant with regard to the geometry of the spray and the droplet distribution

in the injection volume.

Uncertainties

The main source of uncertainty of the measurement method lies in the

pixel representation of circular objects (droplets), specially when these ob

jects contain less than 10 pixels (independent of the absolute pixel size). By

setting the resolution of the area measurement method to 5 pixels, a maxi

mum error of ±3% was obtained by comparing a circular area with a pixel

ensemble in which the amount of pixels was varied between 6 and 40 pix

els. In this work, the smallest drop images contain 6 pixels (¢> 60 J1.m) and

the largest ones 148 pixels (¢> 300 J1.m). For larger objects or structures, the

error is less than 1%. The reduction in the uncertainty of this hologram eval

uation method as compared with other evaluation methods reported earlier

by the authors (Chavez &. Mayinger (1988»), represents about one order of

magnitude (earlier ~ ±17%, this work ±3%).

Other uncertainty sources such as optical aberrations due to the holo

graphic method are of minor importance for droplet sizing in a large droplet

collective. They become important only if the shape 0 a droplet is to study

in detail. A discussion of these kind of errors can be found in standard texts

on optical holoe:ranhv.

142

CONCLUSION


The use of personal computers in the evaluation of pulsed laser holo

grams of particle fields constitutes a very important tool. It permits the

immediate analysis of the holograms in the same optical laboratory. This

will surely contribute to engage more researchers to take more frequent ad

vantage of the excellent properties of the holographic techniques in the study

of dispersed flows.

Acknowledgements - The authors wish to thank the Deutsche Forschungs

gemeinschaft (DFG) for the financial support for this study

REFERENCES

Chavez, A. & Mayinger, F. (1988), Single- and double-pulsed hologra

phy for the characterization of sprays of refrigerant R113 injected into its

own saturated vapour, Proc. 1st World Conf. on Exp. Heat Transfer, Fluid

Mechanics and Thermodynamics., Dubrovnik, Yug., eds. R.K. Shah, E.N.

Ganic, and K.T. Yang, pp. 848-855.

Chavez, A. & Mayinger, F. (1990), Evaluation of pulsed laser holo

grams of spray droplets using digital image processing., Proc. of the 2nd

IntI. Congress on Particle Sizing, Ed.: E. Dan Hirleman., Tempe, Arizona,

pp. 462-471.

Chavez, A. (1991), Holografische Untersuchung an Einspritzstrahlen -

Fluiddynamik und Warmeiibergang durch Kondensation - Dissertation,

Technische Universitat Miinchen.


Frazer, R.P. & Eisenklam, P. (1956), Liquid atomization and the drop

size of sprays, Trans. Instn. Chern. Engrs., Vol. 34, pp. 294-319.

Haussmann, G. (1979), Digitale Bildverarbeitung an dreidimensionalen

Hologrammrekonstruktionen, Dissertation, U niversi tat Gottingen.

Haussmann, G. & Lauterborn, W. (1980), Determination of size and

position offast moving gas bubbles in liquids by digital 3-D image processing

of holographic reconstructions, Applied Optics, Vol. 19 No. 20, pp. 3529-35.

Ligthart, G & Groen, C.A. (1982), A comparison of different autofocus

algorithms., Proc of the IEEE, PATREC 82 Vol.2 pp. 597- 602.

Schafer, M. & Umhauer, H. (1987), Realization of a concept for the

complete evaluation of double pulse holograms of particulate phases in flows,

Particle Characterization vol. 4 pp. 166-174.

Trollinger, J.D. (1975), Particle field holography, Optical Engineering, vol

14, pp. 470-481.

CHARACTERIZATION OF SAVONIUS ROTOR WAKE USING

IMAGE PROCESSING TECHNIQUES

J.Massons, Jna.Gavalda, J.Escoda, x.Ruiz and F.Diaz

Lab. Fisica Aplicada. Dept. Quimica. Univ. Barcelona.

Tarragona, Spain

ABSTRACT

This paper analyzes the generation of the wake of a static

savonius wind machine for Re=4200 using chronophotographic flow

visualization and digital image analysis of the pictures. This

study is carried out by determining the evolution of the

geometric parameters characterizing the wake, together with the

streamfunction and vorticity distributions of the flow.

INTRODUCTION

The consciousness about the limited character of the

fossil supply has led to considerable attention being focussed

on the utilization of renewable energy sources, one of which is

wind energy. The characterization of the Savonius wind machine

has been a subject of considerable interest and the analysis of

the response of the machine under different wind conditions and

design parameters has been realized by several authors [4,9].

However, Ii ttle experimental information exists on the flow

field generated by the Savonius rotor in spite of the influence

of the rotor aerodynamics in its general performance [7].

In the present study, image processing techniques have

145


146 J. MAS SONS ET AL.

been applied to the chronophotographic visualizations of the

Savonius wake generation. In recent years, there has been

considerable interest in these techniques [2,3,5,6,8]. This

characterization is carried out by determining the evolution of

geometric parameters governing the wake which are obtained from

the distributions of streamfunction and vorticity.

EXPERIMENTAL DETAILS

The flow investigated corresponds to the transient wake

generated by a Savonius rotor when the angle between the wind

direction and the main plane of the rotor is 90°. The

visualizations were conducted in a towing water tank [8] and

the Savonius model was made of two half cylinders of

transparent plexyglass of D=6 cm in diameter, with an overlap

of 3 cm (see Figure 1). The Reynolds number referred to the

Savonius diameter is 4200 [1]. Magnesium powder of 2-5 ~m in

diameter was used as particle tracers for conduct the

visualizations. Using a chronophotographic technique [6], the

particle motions projected on the observation plane were

recorded in photographic film at regular time intervals after

the start of the motion. A Nikon FM2 camera, equipped with a

Nikon MD12 motor drive, is used for take the visualization

recordings. Synchronism between the start of the movement and

SA VONlUS ROTOR WAKE USING IMAGE ANALYSIS PROCESSING TECHNIQUES 147

o

Fig. 1. Sketch of the Savonius model

step 1

step 2

Fig. 2. Algorithm used in the obtention of the binary image

148 J. MASSONS ET AL.

the taking of the photographs was achieved by using a Nikon

MC-4 remote-control cable. The illumination of the observation

plane was achieved by means of two movie lights, each of 1000

w, located on both sides of the tank and collimated by 5 rom

wide sl its. The time corresponding to each photograph was

normalized by the time taken by the potential-steady flow in

covering a distance equal to the Savonius diameter (t*=tU~D,

where Uo is the free-stream velocity in the final steady

regime). The exposure time used in the visualizations was 1/8

s, an interval in which it may be allowed that the velocity

field of the flow does not evolve to any appreciable extent.

IMAGE PROCESSING TECHNIQUES

The digital processing of the pictures has, as main

objective, the localization, isolation and measure of each

trace contained in the photograph, in order to extract the

whole flow field.

The photographic recordings containing the paths traced by

the magnesium particles were digitalized in a Panasonic camera.

Further, the digital processing of the images was carried out

in a VIDAS/IPS equipment. This process involves the

SA VONIUS ROTOR WAKE USING IMAGE ANALYSIS PROCESSING TECHNIQUES 149

homogenization of the lighting and the shading correction.

These processes were achieved by combining the original image

with the resulting from applying a low pass filter to the

former. After this, a standard histogram normal ization was

realized in order to increase the dynamic range of grey level

values in the image. The increasing of the contrast between the

traces and the background was achieved applying a filter to the

image which simultaneously tended to heighten the contrast

between the traces and the background and soften their outline.

This algorithm produces a high-pass filtered image from the

difference between the original image and the result of a low

pass filtering. Adding this result to the original image gives

a contour enhancement in which the slope of the transition

between different phases gets steeper. Mathematically the

action of the filter used can be written as follows

fEMPHAS (X, Y) c, fORIGINAl (X, Y) - C2 flO't/PASS (X, Y)

where fEMPHAS (X, Y) corresponds to the grey value of the pixel of

coordinates (X, Y) in the enhanced image and the subscrips

ORIGINAL and LOWPASS refer respectively to the original and

low-pass filtered image. The low-pass filter is realized by

means of a 15x15 square matrix. C, and C2 are parameters of the

filter which are adopted as 2 and 1, respectively.


Once optimum image quality was achieved, binarization was

carried out selecting a suitable threshold of grey level to

discriminate unambiguously traces from the background. This

election will condition the amount of traces automatically

identified. A first election of this threshold was obtained

using otsus's method (10). Typically, this assumption produces

a highly noisy image in which the background is sometimes

incorrectly classified as a trace. For surmount this fails in

the classification process, a second binarization is realized,

chosing a greater threshold as Gn=(GO+3x255)/4, when Go is the

threshold obtained from otsu's method. In this case no

background noise is generated, al though a reduction of the

trace length, and a division of some traces occurs. Using the

algorithm depicted in Figure 2, a new image was generated,

being coincidental with the one obtained from the lower

threshold level, but without background noise. This algorithm

produces a selective dilatation of the structures contained in

the image binarized with the higher threshold level (image

without background noise), limited by the size of the

structures of the other binarized image. This was accomplished

using standard "DILATE" and Boolean "AND" operators. The test

realized for controlling the procedure is based in the

application of a XOR operator. The process is stopped when no

modifications are introduced in the new iterated image.

Finally, the traces clearly uncorrelated with the


neighboring ones or the undetected traces are treated manually.

The proportion of automatically reconstructed traces is of

about 75%. Once the image is in correct binary form, each trace

is measured.

As an example of the trace identification process, Figure

3 shows the filtered image (inverted) for a typical

visualization of the flow and the image containing the traces

automatically identified, as well as the traces extracted from

the picture.

The velocity field was obtained from the particle path

distribution and, using an interpolation process, the velocity

vectors were rearranged in the mesh points of a grid which

covers the whole image. The distributions of streamfunction and

vorticity are computed from the previously determined velocity

field, allowing the temporal evolution of the main geometric

parameters characterizing the wake to be determined.

RESULTS AND DISCUSSION

Figure 4 shows, as an example, a set of visualizations

corresponding to the Savonius wake generation between t*=0.57

and t * =9 . 6 . A photographic recording was taken every t * =0. 5,

although only the results at time intervals of 3 are presented


., b)

0)

Fig. 3. Exemple of the trace identification process a) filtered image (inverted) b) Final binary image c) Traces detected from the pictur.e

01 bl

01 0)

Fig. 4,. Visualizations of the flow a) t"=0.57 b) t"=3.58 c) t*=6. 59 d) t*=9. 60


here. To smooth out as far as possible the difficulties arisen

from the lack of information in certain zones of the image, the

present evaluations were obtained by taking into account the

information contained in a set of five preselected series of

visualizations obtained under the same experimental conditions.

The visualizations obtained show that at the beginning of

the process the flow is potential. Later, the flow separation

produces the formation of two small vortices downstream of the

rotor, giving a closed wake and, after t*=4, a double Karman

vortex street formed by alternate vortices takes place.

It must be stated that during the transient phase the

first vortical structure is generated in the gap between the

blades and an alternate one is formed in the lower part, rear

the blade. The inner flow of the rotor drifts downstream to the

vortex generated in the rotor gap, preventing its growing. For

this reason, when we refer to the first vortex, we consider the

vortex formed in the lower part of the rotor.

Figure 5 shows the streamfunction and vorticity

distributions for a set of selected photographs. These

variables were normalized according to

.* * (a)

where Uo is the free-stream velocity and D is the Savonius


Fig. 5. Streamfunction and vorticity distributions

Fig. 6. Definition sketch of the geometrical parameters analyzed

0-0----

0

ajD bj

0 Do

1 14 16

t*

Fig. 7. Time evolution of the longitudinal and lateral position of the center of the two first vortices shed from the rotor

SAVONIUS ROTOR WAKE USING IMAGE ANALYSIS PROCESSING TECHNIQUES 155

diameter. It is important to note that the vorticity of the two

alternate vortices is clearly different, as a consequence of

the geometry of the rotor.

The above mentioned distributions allow to determine the

evolution of the first pair of vortices shed from the rotor,

concretely the longitudinal and lateral positions of the vortex

center, b, a (see Figure 6). Figure 7 shows the evolution of

both variables with respect to the non-dimensional time t*. The

analysis of this figure demonstrates that the two vortices

behave differently. The first one shows a reduced lateral

evolution with a constant and maintained displacement with

respect to the wake axis of around b/D=-O. 5. The lateral

evolution of this vortex is limited to the lower part of the

wake by the inner flow of the rotor, whereas the second,

generated in upper blade when the recirculating flow incides,

shows an important deflection towards the wake axis. The

longitudinal evolution of both vortices are quite similar,

showing a maintained and regular increasing which becomes more

intense at the moment of the shedding. This is due to the fact

that, after shedding, the wake tends to fill the whole non

potential flow area, showing a growing velocity of A (a/D)=At*/2.

156 J. MAS SONS ET AL.

CONCLUSIONS

In this work an analysis of the generation process of the

wake of a Savonius rotor is carried out using

chronophotographic flow visualization complemented with digital

image processing techniques. This study has shown that the

transient phase is relatively brief and that a wake

characterized by the presence of a double Karman vortex street

is formed after t*=4. It was also observed that the time

evolution of the pairs of vortices formed in the wake is

clearly different, due basically to the influence of the inner

flow of the rotor.


REFERENCES

1. Gavalda,Jna.: Analisi estadistica dels vents a Catalunya i estudi de rotors eblics d'eix vertical. Thesis. University of Barcelona, (1989).

2. Hernan, M.A. and Jimenez,J.: Computer analysis of a highspeed film for a turbulent mixing layer. J.Fluid Mech., 119, (1982).

3. Hesselink, L.: Digital image processing in flow visualization. Ann. Rev. Fluid Mech., 20, (1988).

4. Khan, M.: Model and prototype performance of a Savonius rotor windmill. Wind Engn., ~, (2), (1978).

5. Kobayashi,T., Yoshitake,Y., Saga,T. and Segawa,S.: An improved image processing technique for determining twodimensional flow with reverse flow. In Millet,M.L., Kim,J.H. and Heidrick,T.R. (ed) , Int. Symp. on Physical and Numerical Flow Visualization. FED, vol.22, 39, (1985).

6. Massons,J., Gavalda,Jna., Diaz,F. and sole,Ll.: Image processing of cylinder wake generation. Phys.Fluids, A, ~, (8), (1989).

7. Massons,J., Gavalda,Jna., Ruiz,X. and Diaz,F.: Image analysis of the wake generated by a Savonius rotor. Wind Engn., il, ( 6), ( 1988) .

8. Massons,J. Ruiz,X. and Diaz, F.: Image processing of the near wakes of stationary and rotating cylinders. J. Fluid Mech, 204, (1989).

9. Modi,V.J., Roth,N.J. and Pittalwala,A.: Blade configurations and performance of the Savonius rotor with applications to an irrigation system in Indonesia. J. Solar Engn., 105, (1983) .

10. otsu, N.: A threshold selection method for grey-level histograms. IEEE Trans. Man Cybern. SMC-9, 62, (1979).

An Application of Image Processing Methods to Determine the Critical Shear Stress in Sewer Systems

L Introduction

V. Baier, W. Bechteler, S. Hartmann

Institute for Hydraulic Engineering University of the Armed Forces Munich

Werner-Heisenberg-Weg 39, D-8014 Neubiberg, Germany

At the Institute for Hydraulic Engineering at the University of Armed Forces in Munich/Germany

we are developing a measuring device to determine the critical shear stress in sewer systems. Sediments like sand and organic material are transported by waste water into the sewer systems where

they settle at those locations where the flow velocity is decreasing. Settled sediments narrow the profile of the waste water channels, reduce storage capabilities and therefore cause difficulties to calculate the actual efficiency of the network. Because nowadays sediments contain pollutants (such as chemicals and heavy metals) they cause corrosion of the channels and, when they are remobilized by increasing discharges e.g. due to heavy rainfalls, the biological section of a sewage treatment plant might be destroyed. Out of these reasons it is of importance to describe the sediment transport mathematically as a basis for simulation models as well as practical actions to clean the channels by water flushing.

The subjective observation of the flow and the determination of the critical shear stress, which is a measure for the erosion of the sediments, should become more objective and reproducible. This investigation has been supported by the German Research Foundation (DFG, Be 738/12).

b Measuring Unit and Experiments

The measuring unit was constructed in the laboratory of our institute. Fig. 1 shows the experimental setup for measurements in a sewer.

Underwater

~/ Video Camera

Rectangular Lamp

cm cm

Fig. 1 Measuring unit to determine the erosion velocity in a sewer system

159


=15cl -.-,l. 6 CI

160 v. BAIER ET AL.

Fresh water from a fire-hydrant enters a rectangular channel and passes a steadying section with different grids. An opening in the metal bottom of the channel is placed on top of the sediments so that its surface is overflowed. The flow through the channel is controlled by the discharge which is adjusted by an inductive flowmeter. On top of the plexiglass cover of the channel an underwater video camera is installed to record the experimental section filled by the sediments. This section is lighted by mirrors focussing the light of two lamps from outside of the channel.

Experiments have been performed in the laboratory as well as in sewer systems using different cohesive and non-cohesive sediments (coarse, middle and fine sand, bentonite, mixtures of sand and bentonite and finally organic material). The discharge was increased step by step until a movement of particles of the sediments was visible. Applying the continuity equation the critical discharge at which the movement began allows to calculate the critical erosion velocity. Three different methods were used to determine the critical shear stress, including the influences of the turbulence, the roughness and the water depth:

- Darcy-Weisbach-law - Ga uckler - Manning -Strickler formula - Karman-Prandtl logarithmic wall-law

For different materials a different behaviour could be observed. For sand two characteristic phases are distinguished : sporadic movement of individual grains and the continuous motion of grains at different places of the sediment. For bentonite a first movement takes place even at small velocities (surface erosion), later on bigger parts of the bentonite are carried away by the flow (mass erosion).

~ Strategies of Image Processing

The underwater video camera is connected to a video-recorder and a Pc. The signals are digitized by an Image-Processing Board in the PC and stored for further analysis.

The basic idea is that any movement of particles that happens between the recording of two sub~ sequent pictures will cause changes of their grey values. These changes can be counted.

Fig. 2 is shows the surface of the examined sediment area (mixture of sand and bentonite) at time steps I and 2 with a time difference of approx. I sec.

Fig. 2: Surface of the sediment at time steps I (left) and 2 (right), time difference approx. I sec.

THE CRITICAL SHEAR STRESS IN SEWER SYSTEMS 161

Fig. 3 demonstrates the difference of the two pictures resulting from a subtraction of their grey values. Every image point (pixel) of this difference picture represents a movement.

Fig. 3: Grey value differences of the pictures taken at time steps I and 2

The grey value changes are counted by the system and divided by the number of pixels of the whole image to obtain the percentage of moved particles.

The speed of subtracting two subsequent pictures is not sufficient for an online-evaluation. Therefore this method is used to analyze pictures from the video tapes while a second strategy allows the online-analysis during the measurements.

The so-calIed XYT -Method is based on the idea that every moving particle passes a certain column in the image and changes the grey value. The video camera is instalIed in such a way that the column of a picture is perpendicular to the direction of the flow. If 512 columns are stored into oneimage the changes over the whole period represent the transported sediment. The time step between two following columns is reduced to 120 ms. If there are just horizontal Jines in the image, no transport took place.

Fig. 4: XYT -image with horizontal structures (no movement)

162 V. BAIERET AL.

Movement on the other hand is represented by vertical structures which are visible at first as soon as transport starts at the critical erosion velocity.

Fig. 5: XYT -image with vertical structures (movement)

~ Problems of the Svstem

Some kind of electronic influences, caused by using different video recording systems, create disturbances, visible as diagonal structures in the images.

Fig. 6: Disturbances in the image caused by electronic influences

These disturbances do have a small grey value width. The elimination of them by defining a grey value boundary above that width is not useful because while examining material with a low contrast grey value changes caused by particle movement below that boundary are lost.


A reference picture at zero discharge also includes these disturbances. If such a reference picture is used for calibration of the system, the disturbances are reduced but not eliminated completely. The use of filter operations (median, etc.) reduces disturbances, too, but the information of the image is changed.

A further method is the use of the Fourier analysis. With the help of that powerful tool the frequencies of the disturbances can be determined and eliminated in the digitized picture. These procedures need much time and computing power, so they only can be used for off-line analysis.

Using the method of the Gauss-Laplacian pyramid disturbances are reduced step by step. At first a low-pass-filter is used and the image reduced to half of its size. The low-pass-filtered image is then subtracted from the original image. The disturbances cannot be seen anymore after the second step.

Beside the disturbances a second problem was the time step between two subsequent images. The PC-based Image Processing System has only two frame buffers to hold images. While the digitization of the video signals is done in real-time, the storage of the digitized image on the PC takes about 3.5 sec. The complete procedure of digitization, subtraction and storage takes at least 6 sec. Within that time period a lot of particles, especially at high flow rates, are moving through the channel without being noticed by the system. Because the duration of the measurement at a certain velocity is limited to I min. only about 10 pictures were saved and analyzed. The results were random and not usable for statistical analysis.

To avoid these problems we extract out of a 512 x 512 pixel image the center of the flow with an area of 256 x 256 pixels. 4 of those quarter images are put together into one frame buffer and then stored on the Pc. That reduces the time step to 440 ms between the quarter images which is an acceptable accuracy for that application.

Fig. 7: 4 quarter images of 256 x 256 pixels combined to one image of 512 x 512 pixels

The result of using the difference strategy is a diagram with the percentage (pixels in the difference image divided by the total number) against the flow velocity. Fig. 8 shows the curves of four different measurements with sporadic movement at 22 cm/s and the beginning of erosion at 26 cm/s.

164 V. BAIER ET AL.

25

20

.-. ~ 15 III ., " ~ OJ ... ~ 10

is

:~ 16 20 24 28

Velocity [cm/a]

Fig. 8: Percentage of moved particles against the flow velocity (material: sand 0.3 - 0.8 mm)

For the XYT -method a statistical analysis is done by calculating the standard deviation of the rows. The value of the median of all standard deviations represents the movement. If the same procedure is done for the columns within the image, the standard deviations and their median are characteristic for the behaviour of the material.

Finally we combined the standard deviations of the rows and columns by calculating the quotient "column over row". That quotient shows a similar characteristic for all examined materials.

°D~-----7------~'D~-----'~5------2~D------~25

Velocity [cm/a]

Fig. 9: Quotient of the standard deviations of columns and rows

The values obtained from the analysis with Image Processing Methods were compared to former investigations based on subjective observations. A well-known represenwtion is the Unsold diagram, a modification of the Shields diagram. The range of the beginning of transportation is given by a


number of curves of constant dimensionless transport intensity. Therefore a constant but weak

movement of particles is possible also below the critical conditions. In Fig. 10 the results of our

measurements and different calculations are marked in the Unsold diagram.

int.ensit:y of transport --------f-------

" Extraoolaloo i " V nach SHIELDS

rI 0):1

Qs + Q" +

() 1('-' ... 10"' • 10-] A 10" o 'Kf~ 0 not measureable

beginning of transport due to subjective observation

data of MANTZ 11977)

• dat:a of \AlHITE,S.J, (1910)

-k c Vo deh ReO

A: Darcy-Weisbach-Iaw

B: Gauckler-Manning-Strickler-formula

C: logarithmic wall-law

I : sand 0.7 - 1.2 mm

2 : sand 0.3 - 0.7 mm

3 : sand 0.2 - 0.5 mm

dso = 0.80 mm

d so = 0.53 mm

d so = 0.32 mm

Fig. 10: Unsold diagram (coarse and middle sand) with measurements and calculations of different

examined materials

2.:. Outlook

So far, the measurements were made in the laboratory and in sewers under idealized conditions using

fresh water and materials with only few organic contents. The difference in using fresh instead of

waste water is not of great importance for the image processing system because the water depth is

only 5 cm and therefore an observation of the surface is possible with waste water, too. However,

the influence of waste water on the one hand and fresh water on the other concerning the critical

erosion velocity has not yet been examined.

The obtained results (see Fig. 10) allow the use of Image Processing Methods as an objective method

to determine the critical she3r stress in sewer systems.

A Three Dimensional Particle Image Velocimetry system and its application

to the measurement of acoustic streaming

M.P. Arroyo* C.A. Greated

Fluid Dynamics Unit, Physics Department,

Edinburgh University, King's Buildings,

Mayfield Road, Edinburgh. EH9 3JZ Scotland. U.K.

Keywords: Particle Image Velocimetry, Acoustic streaming, 3-D flows, Stereoscopy.

* Permanent address: Dpto. Fisica Aplicada, Faculty of Sciences,

University of Zaragoza, Ciudad Universitaria, 50009-Zaragoza, Spain.

167


168

Abstract

M. P. ARROYO AND C. A. GREATPD

A three-dimensional particle image velocimetry (3-D PIV) system has been developed,

based on the concept of stereo-photography. A pulsed laser beam is used to illuminate a plane of

light in the usual manner and this is photographed by a camera, adapted to form stereoscopic

images by the addition of four mirrors, two placed in front of the camera and the other two

between the lens and the film plane. With this arrangement two images are formed simultaneously,

side-by-side on the film. The centre of each image is shifted laterally, as though the camera had

taken each picture from a different position, but with the film and illumination planes remaining

parallel in each case. Both halves of the film are analyzed as with standard particle image

velocimetry (2- D PIV) and the resulting records are combined to give the three velocity

components.

An analysis of the optical arrangement is presented. The arrangement is particularly

convenient in application since it is operated in essentially the same way as a conventional camera

and it can be easily transformed into a standard 2-D PIV system. A comprehensive error analysis

is also included.

The system described has been utilized in the study of acoustic streaming i.e. the mean

flow patterns set up when a fluid is disturbed by an intense acoustic field. In this case, it was the

3-D cellular patterns within a rectangular tube that were measured. Results from these 3-D PIV

studies agree well with theoretical predictions developed from the Rayleigh theory. The system

has also been set up to work as a 2-D PIV. A comparison of the measurements taken both with the

3-D and the 2-D setups is also included.

A THREE DIMENSIONAL PARTICLE IMAGE VELOCIMETRY SYSTEM 169

1 . Introduction

As it is well known [1-5], particle image velocimetry (PlY) usually provides an

instantaneous map of the two in-plane components of the velocity field. If the flow is three

dimensional, the out-of-plane component is a source of error in the measured velocity [6], this

error being directly related to the distance from the point to the axis of the recording system. The

error is due to the dependence of the in-plane projection of the velocity vector on the viewing

direction of this vector and on its out-of-plane component. Nevertheless, it is also possible to use

this effect for measuring this out-of-plane component. The system used here combines PlY with

stereoscopic methods, in which the flow is viewed from two different positions. In general, the

accuracy of the out-of-plane component increases with the distance between the two viewing

directions.

Two different stereoscopic approaches can be used; the optical properties of these have

been condensed elsewhere under a limited range of conditions [4-7]. In the angular approach, the

optical axes of the two recording cameras are no more perpendicular to the flow but they make a

certain angle with the illuminated sheet. In the translational method, the stereoscopic effects are

directly related with the distance between the optical axes of the cameras, which are now

perpendicular to the illuminated sheet. The main disadvantage of the angular method is that it is no

more possible to focus all the flow on the film unless the back of the camera is swung but, in that

case, the recorded images will be affected by distortion. In the translational approach, the common

field of view is quite limited due to the axis distance required for sufficient accuracy. Nevertheless,

the common field can easily be increased by changing the position of the back of the camera in

relation to the lens. So, it is the performance of this translational method which is going to be

investigated here (Section 2).

From a practical point of view, it is important to avoid the use of two different recording

systems, since correlating the two photographs can be a source of error and quite time-consuming

if it is necessary to do it with each pair of photographs. For this reason, we have developed a

setup, which allows us to simultaneously take the two pictures with the same camera, by means of

170

Object plane

8

x i

dx Cf

Lens

plane

L,

M. P. ARROYO AND C. A. GREATED

Image plane

A~"f X,o

g,..l. -------:== ---

.... Jdx, ..,,,-' .---

...... -- 8 .......... ---- 11 ..... --,..,---1!:l%1:'"::-'::.:..:-:-::..::-----------------T-----------. X

Sx,>o

-------------t-------------- ----> z

A _-__ ~ ___ -___ -__ -___ -___ -__ -___ ~ ___ :: __ ~ ___ ~ ~2 S~< 0

_________________ 1 _____________ _

Fig. 1. Stereoscopic translation method

A~T ~o

qJ .. t d~

..... J 82

l~


a mirror system. Once everything is set up, the correlation between the two photographs is

straightforward and can be done automatically. The system, which can also be easily set up to

work as a standard particle image velocimetry (2-D PlY) system, and its performance are presented

in section 3, while section 4 shows some fully 3-D measurements. The accuracy of these

measurements is also compared with that of the measurements taken with the system working as a

2-D system.

2 . Translational method

2.1 Theory

Consider two cameras with their optical axes normal to the laser sheet at distances

&1 and &2 from the centre of the object (Fig. 1). The coordinates of the two lens centres will be

(&j, 0, do), where j=1,2 and do is the object distance. The coordinates (Xj, Yj) (j=1,2) of a point

(x,y,z) in the laser sheets are

j = 1,2

( 1)

where d; is the image distance, and the origin for (Xj, Yj) is in the centre of each lens.

If the particle moves to the point (x+dx, y+dy, z+dz) its new coordinates will be

d; ----(x+dx- 8xj) 4, - (z +dz)

j = 1,2

(2)

172 M. P. ARROYO AND C. A. GREATED

Assuming that z, dz « do, the displacement (dxj,dYj) can be written as

dz dYj = M(dy + -y)

do

j= 1,2

where M is the magnification of the recording system (M=d/do).

-->

(3)

Finally, the displacement vector dr = (dx, dy,dz) can be obtained as

j= 1,2

(4)

where the origin for the coordinates (Xj, Yj) has been taken as the location in each photograph of

the image point, Aj, of the origin A taken in the object. Xjo will then be the distance between the

centre of each image OJ and the respective image origin Aj- That implies that XIO = X20 = Xo and Xl

= X2 = Mx, which simplifies the process. ~

Finally, as usual, the velocity vector V = (Vx, Vy, Vz) will be calculated as

A THREE DIMENSIONAL PARTICLE IMAGE VELOCIMETRY SYSTEM

--> -> dr v=

T

where T is the time interval between two consecutives pulses.

2 .2 . Uncertainty analysis

173

(5)

There are two main sources of error in the determination of the velocity vector with this

stereoscopic procedure: the error due to the recording and processing technique for each of the two

stereoscopic images and the error arising from the process of correlating the measurements taken

from both photographs. In the first case, the measurement errors are dominated by the uncertainty

in the position of the particle images due to the film grain size, in the same way as in standard 2-D

PIV. This uncertainty is independent of the velocity field and is typically about 3 Jlm; this is a

representative value for Kodak Technical Pan developed to 400 ASA.. The second source of error

is due to the inaccuracy in the coordinates of the two measurements we are using for getting the

velocity vector. As we will see, this uncertainty depends on the velocity field itself, namely on its

local gradient and on its out-of-plane component.

Both errors will result in an error on the (dxj, dYj) values we are measuring and this will be

translated to the velocity vector we are calculating. Therefore, let us first calculate the errors in dx,

dy and dz (which will be named o(dx), o(dy) and O(dz) respectively) in terms of the errors in dXj

and dYj (named O(dxj) and O(dYj)). Assuming that the measurements taken from both correlated

images are independent and considering that o(dxJ) = O(dY2), o(dYl) = O(dY2) and OX2 = Ox we

get, from Eq. (4),

I

s: d _ O(.dX[) {I X2}2 u( x)--- +-liM OX2

I

o(dy) = O( .. ~d {I + [o(dX[)y]2 r ,'2M [o(dYI )oxj2 J

(6.a)

(6.b)

174

Fig 2.

Fig. 3.

a)

b)

2

o~~~~~~~~~~~~

o 10

1.0

~ tt,- 0.9

~

j 0.8

0.2

15

H"l

20 25

$= 16' x= 15/Mmm

[=50mm

0.4 0.6 0.8

M

30

1.0

M. P. ARROYO AND C. A. GREA TED

Errors of the out-of-plane component measurements as a function of cp (a).

Maximum error of the in-plane component measurements as a function of M for

several f and a fixed cp (b).

Object plane

Lens plane

Mirror arrangement for the 3-D PlY system

Image plane


o(dz) = O(dX 1) do = o(dxr) _1_ fiM ox fiM tgcp (6.c)

where the errors in OXj, M and do are considered negligible compared to o(dxj) and taking for dy

the mean value of the two obtained from both images.

From Eq. (6.c) we can deduce that the error o(dz) decreases when Ox/do increases, i.e.

when the mean angle of viewing the object from the camera (<\l) increases, since the stereoscopic

effects also increase (Fig. 2a). From eqs. (6.a) and (6.b) it can be seen that the errors o(dx) and

o( dy) will be larger in the outer region of the photograph than in the inner and the overall error will

decrease with increasing ox. For a fixed M and <\I, o(dx) will decrease if the focal length of the

lens increases (Fig. 2.b).

In conclusion, the errors in the velocity field measurements will be decreased by increasing

the angle of viewing <\I and the focal length of the recording lens f. There are some practical

limitations which prevent <\I from being increased indefinitely. First of all, the aperture of the lens

will impose a maximum <\I above which the image will be cut off. Secondly, the optical aberrations

of the lens for large angles of viewing imposes a severe restriction to the maximum <\I allowable.

There are two types of lens which can be used. Planar lenses have no distortion at all but

may have pointwise coma, astigmatism and spherical aberrations. Fish-eye lenses do not have

pointwise aberrations but suffer from a very large distortion. Usually, planar lenses are preferred

for PlV as distortion is a quite annoying aberration to work with and they will also be used for the

stereoscopy method. Aberrations like coma and spherical are greatly reduced by decreasing the

lens aperture but astigmatism will still remain. Nevertheless, astigmatism is not a very annoying

aberration if the camera is focused in the minimum size plane. The main effect of all the

aberrations will be a slight increase in the size of the particles images.

The need to work with low apertures prevents the focal length of the lens from being

increased indefinitely, since increasing it also implies an increasing in the object distance do and


consequently a severe reduction in the amount of light gathered by the optical system. Anyway,

the effect of the focal length in the error is not very significant. (Fig. 2.b).

We have found that, typically, apertures of fIll or f/16 are needed when Mox=50mm,

M=O.7 and do=255mm (angle of working about 160). In that case, we obtain from Eq. (6) (taking

OdYl) = o(dxt}, Mxmax=Mymax=15mm and f=lOSmm) that the error will be

0.71 0(dXl)/M~0(dx)=0(dy)~0.74 O(dxl)/M

0(dz)=2.46 o(dzt}/M = 3.4 o(dx)

(7.a)

(7.b)

This means that the error in the determination of the out-of-plane component is typically about three

times larger than the error in the in-plane components.

To estimate the error in O(dxl) we have to consider the two sources of error we have

previously mentioned. The effect of a misalignment between the two photographs will be included

as an error O(Xl), O(Yl) on the coordinate positioning on the photograph. The effect of the

recording system will be introduced as an error E independent of the coordinates. Therefore, we

may write

(8)

and a similar expression for o(dYl).

Using Eqs. (3) and (5) we obtain

J(dxl) = T[J(VX) + x- 8x J(Vz) + vz] JXI Jx do Jx do (9)

and similar expressions for the other derivatives.

Eq. 8 also holds for 2-D PlY where O(Xl) and o(Yl) are determined by the accuracy of the

system used for displacing the film. This accuracy is typically better than 10 11m. In 3-D PlY,


cS(Xl) and cS(Yl) account for any miscorrelation between the two stereoscopic photographs, which

also includes misalignment. Therefore, it is clear that a good system of correlating and aligning

both stereoscopic photographs is needed to obtain measurements with good accuracy with the 3-D

PlY technique. Such a system is described in the next section.

It is clear from Eq. 9 that the effect of coordinate positioning accuracy on the cS(dXl)

accuracy depends specifically upon the characteristics of the flow being measured. Therefore, it

will have to be calculated for each specific problem under study.

3. Experimental setup

3.1 Description

A system has been devised for avoiding the use of two cameras (fig. 2). The first two

mirrors must be perfectly perpendicular to the illuminated sheet in order to have the optical axis of

the equivalent two lenses perpendicular to the sheet. This setup will also give Ox1 =-Ox2=0x if the

two mirrors are centred with respect to the lens axis. Mirrors must be just half way between the

object and the lens in order to have the light collected from the object by both mirrors passing just

through the central part of the lens. This can be tested experimentally by reducing the lens aperture

and checking that both images are cut in the same way. In that case, the distance between the

mirrors, DI, must be equal to Ox. The adjustment of these two first mirrors will be perfect when

the two displaced images (formed through the mirrors) are symmetrically placed in relation to the

central image (formed directly by the lens) and the three of them are at the same Yposition. It can

also be seen that, with this setup, the distance between the two displaced images (before the second

pair of mirrors) will be 2MOx.

The purpose of the second pair of mirrors is to redirect the light in order that the two

images are formed on the back of the camera, each one in a different half of the film. They must be

perfectly perpendicular to the film. It can easily be deduced from the geometry that the distance

between them must be Dz=MOx-WI4 where W is the size of the film and the distance to the image

plane, P2, where


~ W P2 = - (1 + --)

2 4M8x (7)

In theory, the three images will be focused in the same plane but, in practice and due to the

aberrations, the displaced images are focused slightly nearer the lens than the central one (typically

about 3 mm but this depends on M&). As the central image is brighter than the displaced ones

(about 1.5 times brighter) and it is superposed on them, it must be physically blocked. This is

easily done by putting an opaque object, S, between the first two mirrors, its size and position

being not critical.

The setup is easily converted into a standard PIV by taking out the blockage and covering

the first two mirrors with an opaque object.

3.2 Performance of the technique

The technique does not really need any calibration procedure apart from measuring the

value of Mox and the distance between the two images on the film, but some checks have been

made in order to test that the setup is working properly with the accuracy estimated previously for

the general translational method.

Preliminary tests were first carried out using a test object in the form of flat millimetre grid

This did not show up any significant distortion in the image coordinates, confirming that the

mirrors were of adequate flatness and were correctly aligned. A 25 x 39 mm plate with black

background and glass spheres of about 20 11m spread on it has been used as an object, which will

be 3-D displayed for several known distances. The 3-D PIV is set up with a Micro-Nikkor 105

mm lens working at M=O.65 and fI16. The object and images distances are <10=266 mm and dj= 17

mm respectively. The mirrors are set up to give ox=77 mm, which is about the maximum distance

allowable in this arrangement. Two exposures of the object are recorded on Kodak Technical Pan,

which is developed with D-19 for 5 min at 220C. The in-plane displacements are constant for the

whole object and fixed to dx= 10 1.6 11m and dy=O. The out-of-plane component is also constant in

the whole object but it is set to different values, ranging from -254 11m to 254 11m, in the several


300

200 x x ---E =-'-"

E 100 OJ a OJ u '" 0 P.. '" x :a '0 ~ -100

x + dx ;:I on

'" OJ

6 0 dy -200

x dz

-300 -300 -200 -100 0 100 200 300

dz (11m)

Fig. 4. Measured displacement for its three components versus the real displacement.

• y

z x

Fig. 5. Rayleigh Streaming setup


photographs which have been taken. The accuracy of the displacement is about 5 11m for dx and

dy and about 20 11m for dz.

Either of the two displaced images has been analyzed by a standard Young's fringe method

(8) applied to 16x24 points, and the three components of the displacement have been obtained from

them. Then, their mean value and standard deviation are calculated. The mean value differs from

the theoretical value (Fig. 4) in less than 211m for dy, 5 11m for dx and 20 11m for dz, which agrees

with the accuracy of the displacement system. From the previous uncertainty analysis (Eqs. (8)

and (9)), we obtain that

(11 )

since all the three displacements are constant over the whole photograph. We estimate that the

correlation between both stereoscopic images introduces an error in the coordinates positioning,

O(Xl) and O(Yl), smaller than 0.1 mm. This implies that

o(dx)) '" E '" 3 11m (12)

The standard deviation is about 5 11m in dx and dy while it is about 15 11m in dz, which

agrees with the estimation from Eqs. (7) and (12).

4 . Measurement of acoustic streaming

Acoustic streaming may be described as the generation of non-zero mean motion by a

sound field. A particular form of streaming occurs when an acoustic standing wave suffers

dissipation in the boundary layer generated by a solid wall (Rayleigh streaming).

The most common configuration is which to observe the phenomena occurs when a

standing wave is set up in a tube (9-11). For a rectangular cross section tube the velocity of the

streaming is given by (see coordinates system in Fig. 5)


[ 9 ] . 4D VX=U s -1+ 4F(y)G(Z) Sln;:(X-XO)

3 4D [2J 4D Vy=--us-F(y) G(z)-- (y-ydcos -(x-xo) 8 A 3 A (13)

3 4D [2J 4D Vz =--u -G(z) F(y)-- (z-zdcos -(x-xo) 8 s A 3 A

with

(14)

and us=3am2/8c, where 2Yl, 2Z1 are the dimensions of the cross section of the tube, c is the speed

of the sound, xo is the position of a velocity node, am is the velocity amplitude and A the

wavelength of the sound field. The magnitude of Us can be estimated from the pressure at the end

of the tube, by using the relationship between the velocity and the pressure in a standing wave.

The main two purposes of the present experiment are: to demonstrate the applicability of the

stereoscopic PIV technique to the measurement of the 3-D velocity field in a real flow and to

compare the quality of the 3-D PIV measurements with the standard 2-D PIV ones. This will be

done by comparing the velocity field measurements in both cases with the theoretical expressions

(Eqs. 13-14).

Rayleigh streaming has been produced by introducing sound of frequency 1910Hz into the

tube (length L=625mm and sides 2Yl=22.2 mm and 2Z1=23.4 mm) using a hom loudspeaker with

the horn removed. The tube was sealed with a rubber bung at the other end. The sound field thus

corresponds to the 7th normal mode of the air column and gives a value for the wavelength of

"-=180 mm. The pressure is monitored by a probe microphone (Bruel & Kjaer type 4166 with 2

182

a)

b)

Fig. 6.

M. P. ARROYO AND C. A. GREA TED

Photographs of the z=9.4 mm plane of the acoustic streaming flow near a velocity

node taken with a) a standard 2-D PIV setup and b) the 3-D PIV setup. Both

photographs were taken with 6 pulses of duration 30 ms and separation 300 ms.


mm i.d. probe attachment) inserted through the rubber bung, with an accuracy of 1 dB (11). The

streaming was set up with a pressure of 147 dB (re 20 ~Pa), which gives us=2.6 mmls.

The light from an AT laser (mean power about 4 W) is optically shaped into a sheet of width

~x0==40 mm and thickness ~zo =0.7 mm and transmitted through the tube (Fig. 5), where some

smoke has been introduced into the tube to render the flow visible. An electromechanical shutter is

set to produce pulses of duration 30 ms with a separation of 300 ms. As explained in (11), the

vibrational displacement of the sound field produces streaking of the particle images, which may

even prevent measurements from being taken if too large.. A simple calculation shows that, in our

experiment, the maximum vibrational displacement is around 100 ~m, which reduces to 60 ~m for

x= I 0 mm from the velocity node. Therefore, photographs have been taken near a velocity node of

the standing wave with the stereoscopic setup previously described. The mirror system has been

positioned so as to displace the two images along the tube axis. Several planes have been recorded

with this setup working as a 3-D PlY system and as a standard 2-D PlY system. A print of the

photographs obtained from the plane z=9.4 mm are shown in Fig. 6. Fig. 6.b shows clearly the

effect of the remaining aberrations (mainly astisgmatism) in the stereoscopic system even after

reducing the recording lens aperture to fI16: the particle images are more blurred and not as

contrasted as in the 2-D PlY photograph (Fig. 6.a).

Either of the two images on the 3-D PlY photographs is analyzed in the usual way (11) and

the three components of the velocity are obtained from them. The same analysis is done for the 2-

D PIV photographs and the in-plane components ofthe velocity (Yx,Vy) are obtained from it. In

both cases, measurements have been made up to x=lO mm from the velocity node. A comparison

of the measurements for either the three or the two velocity components with the theoretical

expressions (Figs. 7-9) provides a value of us=2.6 mmls and Us'" 2.7 mmls respectively, which

agrees with the expected value.

The maximum error in dxl due to the out-of-plane velocity component (Vz) for the 2-D

measurements is 1.5 11m. The error due to the uncertainty b(X\) and b(YI) is about 0.7 ~m and is

mainly due to the terms aVx/ax and aYxlay respectively. This, together with the uncertainty of

184

,-, <r.J -E 0 E '-" >< :>

-1

a)

-2

,-, ~ E E 0 '-" >< :>

-1

b)

Fig. 7.

0

0

x=-2.90mm <> x = -0.75 mm c x= 1.35mm

x= 350mm V x= 5.65mm o x= 7.80mm

5 10

y(rrun) 15

+ x=-2.40mm x=-O.25mm

<> x= 1.90mm

+ +

c X= 4.05mm X= 6.20mm

V X= 8.35mm

5 10

y(mm) 15


20

20

Dependence of V x component on Y at several x values. a) Data obtained from

Fig 6a. b) Data obtained from Fig. 6b. The solid lines are the corresponding

theoretical predictions.

A THREE DIMENSIONAL PARTICLE IMAGE VELOCIMETRY SYSTEM

a)

b)

Fig. 8.

a)

b)

Fig. 9.

05 ~ E E 0 ";:; >

-<l5 + x .-0.75 mm

·1.0

10 IS 20

y(mm)

1.0

05 ~ E E- o >. >

-<l5

·1.0

10 " 20

y(mm)

Dependence of Vy component on Y at a) x=-O.75 mm, obtained from Fig. 6a and

b) x=-O.25 mm obtained from Fig. 6b. The solid lines are the corresponding

theoretical predictions.

05

~ E 0 'N' >

-<loS

10

Z<mm)

r-- ...... .,.-. '++

+ x z ..o..25mm

10

y(mm)

IS 20

IS 20

a) Dependence of Vy component on Z at y=4.7 mm, x=O. Data obtained from eight

standard 2-D PIV pbotograpbs. b) Dependence ofVz component on Y at x=-O.25

mm obtained from Fi.g 6b. The solid lines are the corresponding theoretical

predictions.

185


311m due to the recording process, gives a total uncertainty of 3.5 /lm for b(dxI) which means an

error of 15 /lmls in the velocity measurements. Therefore, the overall uncertainty is I % of the

maximum for Vx and Vy measurements with the 2-D PlV setup.

For the 3-D measurements, the uncertainty in dxl due to b(XI) and b(Yl) is governed by the

terms dVx/dx and dVx/dy in the same way as for the 2-D PlY. Since b(Xl), b(y!) are 10 times

larger than they are in 2-D PlY, the uncertainty is now 7 11m. This gives an overall error for dxl

(Eq. 8) of 7.5 11m, which means an error of 5.5 11m in Mdx. Therefore, the overall accuracy is

1.5% full scale for the Vx measurements, 5% full scale for the Vx measurements and 5% full

scale for the Vz measurements (Eq. 7.b) taken with the 3-D PIV system. The error in dYI is 4.2

11m as Vy-gradients are smaller than Vx-gradients. This implies an error of 1 % full scale for Vy.

Comparison of the Vx measurements (Fig. 7) shows that the 2-D PIV provides more

measurements than 3-D PlY due to the better quality of the particle images on the 2-D PIV

photograph. The deviation of the measurements from the theoretical curves is slightly higher for

the 3-D PlY, in agreement with its slightly lower accuracy. Comparison of Vy measurements

(Fig. 8) also show a very good agreement with the theoretical predictions and similar deviations for

both cases. Finally, a comparison is made between the dependence of Vy and Z and the

dependence of V z on Y (Fig. 9). Each of the Vy measurements has been obtained from a different

photograph and we see that they agree well with the theoretical predictions. The Vz measurements,

taken from the z=9.4 mm plane (Fig. 9.b), show more random deviations from the theoretical

curve than the Vx and Vy measurements corresponding to a smaller accuracy in that component.

5. Conclusion

A new technique which combines particle image velocimetry and stereoscopy has been

described. Two stereoscopic PIV images are taken simultaneously with the same camera by means

of a mirrors system. Both PIV images are analyzed with the usual techniques. The three velocity

components are obtained by correlating the in-plane measurements from both images. The

simultaneous recording of these two images with the same camera makes this correlation very


straightforward. In performance, the error in the out-of-plane component is about three times

bigger than the error in the in-plane components, which remains the same as in the usual 2-D PIV

system. The technique has been demonstrated in the study of Rayleigh streaming flow in a

rectangular tube. Comparison between 3-D PIV and 2-D PlV photographs taken from the same

planes with our set up shows that the lower quality of particle images in the 3-D PIV photographs

decreases the dynamical range of the technique even though the accuracy of the measurements is

similar in both cases for the in-plane components.

Stereoscopic PIV offers new possibilities for the investigation of 3-D flows as it allows the

three components of the velocity to be obtained simultaneously. It has the advantage of being quite

similar to conventional PIV, which makes its implementation quite straightforward.

188

REFERENCES


1. Meynart, R.: Instantaneous velocity field measurements in unsteady gas flow by speckle

velocimetry. Appl. Opt. 22 (1983) 535-540.

2. Arroyo, M.P., Yonte, T., Quintanilla, M. and Saviron, J.M.: Particle Image Velocimetry

in Rayleigh-Benard convection: photographs with a high number of exposures. Opt. and

Lasers in Engng. 9 (1988) 295-316.

3. Adrian, RJ.: Multi-point optical measurements of simultaneous vectors in unsteady flow -

a review. Int. J. Heat and Fluid Flow 7 (1986) 127-145.

4. Lourenco, L.M., Krothapalli, A. and Smith, C.A.: Particle Image Velocimetry. In:

Advances in Fluid Mechanics Measurements. Lecture notes in Engineering 45. Berlin:

Springer Verlay, (1989) pp. 127-199.

5. Dudderar, T.D., Meynart, R. and Simpkins, P.G.: Full-field laser metrology for fluid

velocity measurements. Opt and Lasers in Engng. 9 (1988) 163-199.

6. Jacquot, P. and Rastogi, P.K.: Influence of out-of-plane deformation and its elimination in

white-light speckle photography. Opt. and Lasers in Engng 2 (1981) 33-55.

7. Gauthier, V. and Riethmuller, M.L.: Application of PIDV to complex flows:

measurements of the third component. In: VKI Lectures Series on Particle Image

Displacement Velocimetry. Brussels, March 1988.

8. Gray, C. and Greated, C.A.: The application of Particle Image Velocimetry to the study of

water waves. Opt. and Lasers in Engng. 9 (1988) 265-276.

9. Lord Rayleigh (1896): In Theory of Sound Vol. 2, Art. 352, Dover Publications, 1945

reissue.

10. Lighthill, MJ.: Acoustic streaming. J. Sound Vib. 61 (1976) 391-418.

11. Sharpe, J.P., Greated, C.A., Gray, C. and Campbell, D.M.: The measurement of

acoustic streaming suing particle image velocimetry. Acustica 68 (1989) 168-172.

A CAMERA FOR MEASURING DENSITY, SIZE AND VELOCITY OF

RISING AIR BUBBLES AND WATER VELOCITY IN A BUBBLE

PLUME

Christoph Hugi, Andreas Mueller

Institute of Hydromechanics and Water Resources Managment

Swiss Federal Institute of Technology

CH-8093 Zurich, Switzerland

INTRODUCTION

Bubble plumes are installed for aeration in several lakes in Switzerland. These lakes are

eutrophic due to an excessive growth of algae in summer. The oxidation of this organic

material depletes the oxygen of the lower layers of the lakes. In order to decrease the algae

growth, attempts have been made to reduce the supply of phosphorus into the lake, which is

a fertilizer for the algae, and to increase the supply of oxygen in the water.

In winter, when these lakes are not stratified, compressed air is released at the bottom of the

lakes to create a plume that intensifies the natural convection. Oxygen of the air is then

mainly dissolved at the free surface and transported to greater depth by the return flow. In

summer, when the lakes are stratified, the oxygenation is performed by a discharge of small

bubbles of oxygen at the bottom which are dissolved in the hypolimnion.

189


190 C. HUG! AND A. MUELLER

The water flow in a bubble plume is driven by the drag of the bubbles, a force which can be

described in an integral model as a mean buoyancy. A special property of this buoyancy is

that it is not fixed to the fluid but can migrate due to the slip velocity of the bubbles. The

mean buoyancy is given by the void fraction which can be calculated when the discharged

gas volume per second is smeared out into a volume. This volume is given by the product of

the plume area and the path-length of the bubbles per second. This path is proportional to the

sum of the water velocity in the plume and the slip velocity.

PURPOSE OF THE INSTRUMENT

In an effort to model bubble plumes in a laboratory tank (Mueller et al, [5]) instrumentation

had to be designed to quantify bubble plume experiments. Among these, bubble size and

void fraction (bubble density) are important because they enter the process in several ways.

(1) The slip velocity function, which depends on the diameter of the bubble, is different in

laminar, transitional or turbulent flow conditions (Habennan and Morton,[4]). (2) The gas

volume and therefore the diameter of the bubbles change with decompression, and loss or

gain of gas by solution and dissolution. (3) The rate of gas exchange strongly depends on

bubble size. (4) The slip velocity effects the density and therefore the buoyancy of a given

gas discharge.

Bubble size measurements were reported by Barzewski [1], who used a device, developed

by Todtenhaupt [8], that sucks the air of a bubble through a capillary tube at constant rate

and detennines optically the duration of the air stream. Reimann [6] tried to determine

bubble size and rise velocity from the variation of the conductivity in the bubble water

mixture. Galaup and Delhaye [3] developed an optical instrument for measuring void

fraction and average rise velocity based on two glass fibres which are optically connected

when the probe tip is located within a bubble. Phase-Doppler measurements allow bubble

size and bubble velocity specially at sizes below 200 pm to be detennined (e.g. Saffrnann et

al. [7]).

A CAMERA FOR MEASURING DENSITY, SIZE AND VELOCITY OF RISING AIR BUBBLES 191

The present paper reports a direct, visual technique to determine bubble size, bubble

velocity, void fraction and water velocity. This technique was utilized in the design of a

bubble camera. The method yields a size distribution of the bubbles in a small measuring

volume, an estimate of the void fraction, and the absolute rise velocity of the bubbles. Water

velocity is measured by tracking suspended particles in the water.

PHYSICAL IDEAS

Optics and image recording

Bubble size and velocity can be determined, if sharp images with clearly defined edges can

be produced. This allows the diameter and the displacement of individual bubbles during an

interval of time to be measured. Short illumination times are needed to freeze the motion.

The density is then estimated, by the average volume of gas in the bubbles seen in the field

of view of the instrument. The bubbles used in the experiment reported here have diameters

between 0.1 and 1 mm. They are spherical and have a slip velocity between 40 and 250

mm/s. Similarly the displacements of particles in the size range of 100 pm during the same

time interval are proportional to the flow velocity.

Light is totally reflected at the water air interface when the angle of incidence, <p, exceeds

48.7 0. For the central plane of a bubble, Fig. 1 shows a sketch of rays originating from

parallel incident light beams. The central beam penetrates the bubble. Beams with angles of

incidence between 0° and 48.7° become gradually more refracted. Beams with <p > 48.7°

are totally reflected, and only beams passing outside the bubble again reach the sensor.

Good images of bubbles can therefore be produced in parallel back light. They appear as

black circles because all the light beams are blocked except the central beam. When bubbles

are illuminated from the side a reflection is seen which does not show the edge of the

bubble.

192

Fig. 1:

Fig. 2:

C. HUG! AND A. MUELLER

(3)

(41 ============tE~~~~-----:~----121

Sketch of the central plane of a bubble showing (1) the central penetrating beam,

(2) the refracted beams, (3) the totally reflected beams, and (4) beams passing

outside the bubble.

Photograph showing the bubble camera in front of the bubble plume.

A CAMERA FOR MEASURING DENSITY. SIZE AND VELOCITY OF RISING AIR BUBBLES 193

A microscope objective with an aperture of 5 mm was used to form an image with an area of

4 x 2.6 mm to 5 x 3.7 mm on the sensing array of a CCD-camera. The larger dimension

coincided with the vertical direction. The object plane was only 53 mm in front of the lens,

thus avoiding a long light path in the air water mixture which could blur the images. Only

light beams with a divergence of less the ±2.8° enter the aperture of the lens. It follows that

the bright spot in the center of the image is limited to 5% of the bubble diameter. Based on

the sharpness of the edges and the visibility of the central beam a depth of view of 3 mm

with an error of ±25% can be defined.

A strobe light with a pulse duration on the order of 10 )Is is adequate to freeze the motion.

The blur is reduced to 0.025 mm when the velocity is 250 mm/s.

The field rate of the video signal is 50 Hz (CCIRIPAL standard: 625 lines, 50 Hz). The

strobe light was triggered at 100 Hz. This produces a double strobe on each field forming a

double image during the 19 ms integration time of the CCD-camera. The video signal of the

camera was recorded for later analysis.

Statistical Considerations

Typical air discharges, Q ,of the bubble generator in the experiments are, Q = 1000 to 3000 o 0

mm 3 Is distributed over an area, A ,of A = 17'700 mm 2 • The average bubble diameter, d, o 0

is, d = 0.4 mm. The volume, VI' seen by the instrument has a base,~, of 2.6x3 mm and a

height, h, of 4 mm, i.e. ~ = 7.8 mm 2 , VI = 31.2 mm 3 •

The number N of bubbles per unit time passing through the volume VI is given by the total

number of bubbles (per unit time) multiplied by the ratio ~/Ao'

N=-------

A o

13 to 40 bubbles/s

194

Fig. 3:

Fig. 4:

300 nun

§ ;z:==:::== ~

C. HUGI AND A. MUELLER

60 mm

I

I

12 • I I'l I C:;;I

r-r---=---';:- '" 2 .

I I I

Sketch showing the cross section of the bubble camera, (1) object plane. (2)

microscope objective. (3) CCD array.

Photograph of the millimeter grid used for calibration of the camera.


The average slip velocity is about 80 mm/s and the water velocity is 70 mm/s, resulting in a

path of 1.5 mm between strobes. The frame rate of the CCD-camera together with a double

exposure by the strobe light is therefore adequate to record the bubbles and their motion.

EXPERIMENTAL SETUP

The bubble plume experiments are conducted in a 3 x 5.8 x 3 m deep tank:. It has glass walls

on the long sides and contains up to 52 m 3 of water. A Laser-Doppler-Anemometer was

adapted for measuring flow velocities at the centerline of the tank:. The size of the measuring

volume was estimated to be 0.15 mm in diameter and 3 mm in length. Bubbles are produced

by releasing pressurized air through a round porous ceramic plate with 150 mm diameter.

Plyolite powder with a grain size of 60 to 100 )lm and a density p = 1.03 was used to track

the water velocity. The water depth was 2 m.

The bubble camera consists of an Aqua HR 480 CCD-camera which is equipped with a 4: 1

microscope objective covered with a flat glass plate to avoid water contact with the lens. The

entire assembly is encased in a submersible container (Fig. 2,3). The adjustable position of

the camera determined the magnification and the location of the object plane. The distance

between the object plane and the lens is about 2.5 lens diameters. The distance between the

object plane and the housing is more than 1.3 housing diameters. Therefore, the disturbance

of the velocity field in the object plane should be minimal.

A General Radio 1540 Strobolume was used as strobe light, which was triggered at 100 Hz

by an oscillator. The flash lamp was placed outside the glass wall of the tank approximately

1.5 m appart.

A frame counting code was superimposed on the video signal which was recorded on a VO-

5850P Sony Vmatic 3/4 inch tape recorder. The sensitivity of the camera was adjusted to

give maximum contrast. When synchronisation with LDA measurements was required, the

vertical sync of the video signal was recorded together with the velocity time history.

196

Fig. 5:

Fig. 6:

c. HUGI AND A. MUELLER

Double exposures of a bubble and plyolite particles with a time delay of 10 ms

between strobes. The vertical movement of the bubble and the particles is from

left to right.

0

e 0 ., ~

u 0

e :J ll.. '" 0 e 0 ... :J " .D 0

'-

Ul ... 0, 0

0

°0.0

"/ I~-'-

---,. ,,-

( , , J f

( ~

" I " ,'(

I :. J

I . , . ,

I ,.' I (

I .

( . I

, f

I I , / : '" · / I · · I -' " ( 1.:, ..... '

0.2 0.4 0.6 O.B

Bubble Diameter [rnrn] 1.0

H100rT\rn

H400mm

Hl000mm

H1500tTu'n

Cumulative size distribution of bubbles measured at H = 100 mm, H = 400 mm,

H = 1000 mm, and H = 1500 mm.


The camera was calibrated by putting a plexiglass plate with a Imm grid into the object

plane. The image of the grid shown in Fig. 4 was used as reference. The depth of view of the

objective was 3 mm determined by moving air bubbles adhering to a glass tube through the

measuring volume. A criterion to limit the depth of view was the visibility of the bright spot

of the central beam penetrating the bubble (Fig. 5). When the bubbles were out of focus the

bright spot disappeared. This criterion, however, depends on bubble diameter. This is one of

the reasons why the size of the measuring volume cannot be determined better than ±25%.

The images were digitized with a frame grabber, which grabbed every sixth video frame.

Diameters and displacements of bubbles and particles were evaluated from pixel coordinates

measured with the cursor on the screen of the monitor. A total of 125 to 145 images per

point covering a record of about 60 s were used for the analysis.

TABLE 1: Mean values of bubble size, flow velocities and void fraction

H=I00mm H=400mm H=I000mm H=1500 mm

Volume averaged

Bubble Size [mm] 0.43 0.40 0.52 0.52

Number of samples NB 142 133 127 129

Flow velocity [mm/s]

w 24.9 72.9 38.2 63.4

w' 3.74 8.65 15.2 17.9

u 1.12 6.23 2.0 3.34

u' 1.28 4.4 8.57 12.2

Number of samples Np 198 141 151 132

Void fraction ""pip 0.5610-4 1.110-4 1.4 10-4 1.3 10-4

198 C. HUG! AND A. MUELLER

RESULTS

In this section results of series of four measuring points on the plume axis are presented. The

uncertainty of the data is discussed and a physical interpretation is given.

An air discharge of, Q = 1280 mm /s was released at the source, where the bubbles formed at o

distinct locations of the porous plate and rose in chains like CO2 in a glass of fresh beer. The

plume was laminar within the ftrst 500 mm, where the flow was accelerated. The contracting

plume forced the smaller bubbles toward the center of the plume. At higher elevations the

flow became turbulent which leads to instabilities also of the bubble stream.

Bubble size

The cumulative size distributions, F(H,d), of bubble diameters are shown in Fig. 6. At

H=loo mm the distribution of bubble size is very narrow indicating that only one or two

bubble chains passed through the measuring volume. At H=4oo mm the distribution is

shifted to smaller bubble sizes as expected from the contraction of the flow. This convected

small bubbles into the center of the plume. Only at H=IOOO mm and H=15oo mm does the

distribution show a wider spread of bubble size. Decompression, when increasing the

average radius of the bubbles by 5%, was too small to be detected. The volume averaged

bubble diameters are given in table 1.

The outline of the image corresponds to the outline of the bubble, within the following

errors: (a) If parallel backlight is used a beam reflected from the surface within <p=±2.8° can -4

reach the sensor. The uncertainty of the edge is of the order (l-cos<p)R or less than 4.10 R

(R = radius of the bubble). (b) The light source is located 1.5 m away and has a length of

0.15 m. There is, therefore, an uncertainty of the angle of the incident light of tg<p=O.1. The

corrsponding error is again (l-cos<p)R or less than 0.5%. The size measurements was

crosschecked by a measurement of the diameter of two steel spheres (diameter 1.5 an 3 mm).

The measurement was accurate within 4%.


a ,.....,

UJ

"-E E

'--'

>.. "" u 0 QJ > ~ '-QJ >

c ....,

UJ

"-E E

'--'

>.. .... U a QJ > ~ '-QJ >

0 ci ::'

0 ci UJ

0 0 '" 0 ci .. 0 ci '" o

-gO.O

0 ci ::'

0 ci IX)

C! 0

'" 0 ci .. 0 ci '" o

-go.o

Fig. 7:

I

& ~A Lj l-

I I

-25.0 0.0 2:1.0 :10.0

Horiz. Velocity [rnrn/sJ

v v

-25.0 0.0 2:1.0 :10.0

Horiz. Velocity [rnrn/s]

b ....,

UJ

"-E E

'--'

>.. "" u 0 QJ > ~ '-QJ >

d ....,

UJ

"-E E

>.. "" u 0 QJ > ~ '-QJ >

~ 0

::'

0 ci UJ

0 0 .,

0

ci .. 0 ci '" 0

-gO.O

0 ci ::'

0 ci IX)

C! 0 .,

0 ci .. 0 ci '" o

-go.o

0

0

0

-2:1.0 0.0 2:1.0 !50.0

Horiz. Velocity [no no/ s ]

I o 0'00

0; f; ~ : 0 ~

<>0 v

000 00

o 00 O,>~ o~ 0 0

r. ~o 000Qv,. 00

0% ~1 ?~80o .v 0 v 0

~°Q>~O o ~ o 0

o ~ <It)

00

0

I

-2:1.0 0.0 25.0 :10.0

Horiz. Velocity [rnlTl/sJ

Vector Plot of the vertical and horizontal velocity components measured at

(a) H = 100 mm, (b) H = 400 mm, (c) H == 1000 mm, and (d) H = 1500 mm by

particle tracking.

200 C. HUGI AND A. MUELLER

The images on the monitor had diameters between 25 and 75 pixels with an average of 50

pixels. The bubble diameters could be determined within ±2 pixels including deviations

from the spherical shape. The uncertainty of the measurement is therefore ±4% for an

average bubble. Additionally there is a statistical error in the estimate of the cumulative size

distribution function, which is described by the Kolmogoroff test [9], of, LW = 1.36~ NB =

± 11 % with a confidence interval of 95 %.

Flow velocity

The velocities of the particles which were tracked during the observation time of 60 s are

depicted in Fig. 7 a-d. At H=I00 mm the flow showed little fluctuations. It accelerated

remarkably when H was increased to, H=400 mm. The individual velocity samples at

H=I000 and H=1500 mm showed a wide spread indicating that the flow was turbulent. The

mean vertical and horizontal velocities together with its rms are given in table 1.

The displacement x of the particles was measured with an accuracy of flx/x= 6%. This error

translates to an error of the mean velocity of only (~x/x)~ Np ' where Np is the number of

tracked particles. More important is the statistical error of the mean velocity which depends

on the number, n = 60, of independent measurements given by the ratio of the record length

and the integral time scale of the order of 1 s. The error ofw is of given by w'/~ n = w'n.5.

The presence of the camera does not change the velocity significantly. A possible effect of

the camera housing on the fluid velocity at the location of the measuring volume was

checked with the LDA. The velocity was measured with and without the camera present. At

a vertical velocity of 140 mm/s the change of the mean vertical velocity was less the 1 % and

the change of its rms was also of the order of of 1 %. This change was smaller than changes

due longterm plume wandering.

Velocity samples of tracked particles were compared with LDA measurements at the same

location to within a few milimeters (Fig. 8). The absolutes velocities agree within 10%.


Fig. 8:

Fig. 9:

ci 0 -

......, ci en '" "-... E E ci

<D

A ..., U ci 0

.. Q)

> ci '"

ci 0.0 0.2 0.4 0.6 O.B

Time [sJ

Comparison of velocity data measured with the LDA (solid line) with

simultaneous velocities of tracked particles (single points).

r""I 6. fj, 6. H100"",,,"

U o 0 0 H<400rnrn QJ

III "il "il V H1000mm

"-... 0

E ci ~

E 00 L...I

<> <> 0 H1S00mm

V

A a

+'

U 0

0 0

Q) o db > a.

(/) 0

0

~ 0.1 1.0

Bubble radius [mm]

Bubble slip velocity, calculated from the difference between the absolute bubble

rise velocity and a simultaneously measured particle velocity, versus bubble size

at different heights H. For comparison the rise velocity in fluid at rest (solid

line) reported by Haberman and Morton [4] is shown.

202 c. HUGI AND A. MUELLER

Considerable gradients of the velocity field were observed within the field of view (velocity

differences approximately 10%).

Bubble slip velocity

The bubble slip velocity was calculated as the difference between the absolute bubble rise

velocity and a simultaneously measured particle velocity. This result is compared with the

rise velocity in fluid at rest reported by Haberman and Morton [4] (Fig. 9). At all heights the

measured values exceed the Habermann and Morton values by up to a factor of two. The

effect of turbulence on the rise velocity of bubbles is controversial. Increase and decrease of

bubble velocities are reported in the literature (Buchholz [2]).

If there is an effect of turbulence, it must depend on the mechanisms which determine the

pressure distribution at the air-water interface. Bubbles with a diameter of 0.5 mm and a rise

velocity of 150 mm/s have a Re-number of 75. The length of the separation bubble was

visualized to be of the same order as the diameter of the bubble [10]. If a in stationary

turbulent flow stripped part of the recirculating fluid, a reduction of the drag would occur, in

the limit to the Stokes resistance. All the observed slip velocities plotted in Figure 9 involve

a drag force which is greater than the Stokes resistance.

However, the magnitude of the deviations of the measured slip velocities from the Haberman

Morton curve was found to depend on the height above the source. Close to the source,

where the flow is not yet fully turbulent, individual bubble chains may produce larger

transverse velocity gradients than in the developped part of the plume, and the ambient

velocity at the location of the bubble determined by tracking a single particle in the field of

view is less representative in the near source region.

The displacements of the bubbles on the monitor were typically around 200 pixels with a

resolution of ±4 pixels. The uncertainty is therefore less than ±2%.


Void fraction

The volume of gas passing through the measuring volume can be calculated from the volume

of all bubbles divided by the number of video frames and the size of the measuring volume.

Inspite of the poor accuracy, the void fraction measurements given in table 1 agree with the

average density at the source of, flp/p = 0.78 10-4. The measurements reflect the contraction

of the plume in the accelerating zone (H=400 mm). Density stays constant within the margin

of errors up to H=lS00 mm and does not show a decrease proportional to z-S/3 as in a thermal

plume.

The following errors are involved: (1) volume of bubbles: ± 12%/~ NB, where NB is the

number of recorded bubbles, (2) the number of fields: ± 1 %, (3) the size of the measuring

volume: ±2S%, and (4) a statistical uncertainty of the the order (J/~where n is the number

of independent measurements and (J the rms of the density fluctuations. An accuracy for the

mean density of ±30% is a conservative estimate.

CONCLUSIONS

The bubble camera is capable of locally measuring the bubble size, the bubble velocity, the

water velocity with adequate resolution and provides an estimate of the average void

fraction.

204

REFERENCES

C. HUG! AND A. MUELLER

1. Barzewski, B., Neue Messverfahren fUr Luft-Wassergemische und deren Anwendung auf

zweiphasige Auftriebsstrahlen, Mitteilungen lnstitut for Wasserbau, Universitiit Stuttgart,

(1979) Heft 45

2. Buchholz, R., Ein Beitrag zur Aufkllirung der Stromungsstrukturen in Blasensaulen.

Habilitationsschrift, Fachbereich Chemietechnik, Universitiit Dortmund, 1986

3. GaIaup, I.-P., Delhaye, I.-M., Utilisation de sondes optiques miniatures en ecoulement

diphasique gaz-liquide, La Houil/e Blanche 31 (1976) 17 - 30

4. Haberman, W.L., Morton, RK., An experimental study of bubbles moving in liquids,

Proc. ASCE 80 (1954) 379- 427

5. Mueller, A., Grass, E., Wuest, A., Gyr, A., Modelling of bubble plumes in Hydraulic

Modelling, In: Cunge, I.A., Ackers, P. (ed.), Lausanne: Proc. XXII IAHR-Congress

(1987) pp. 348 - 353

6. Reimann, R Methoden zur Bestimmung gewisser Gasblaseneigenschaften in kiinstlich

beliifteten Seen Material und Technik (1985) 126-130

7. Saffmann, M., Buchhave, P., Tanger, H., Simultaneous measurement of size,

concentration and velocity of sperical particles by a Laser Doppler method. In: Adrian

RI. et aI. (ed.), Laser Anemometry in Fluid Mechanics II. Lisbon: LADOAN-Instituto

Superior Tecnico (1986), pp. 85-103.

8. Todtenhaupt, E.K., Blasengrossenverteilung in technischen Begasungsapparaten, Chemie

lngenieur Technik 43 (1971) 336 - 342


9. Van der Waerden, B.L., (1965), Mathematische Statistik, Berlin: Springer Verlag (1965)

1O.Van Dyke, M., An album offluid motion, Stanford: The Parabolic Press (1982)

The Application of PlV to Turbulent Two-phase Flows

D R McCluskey,* C Elgaard,t W J Easson* & C A Greatedt

The University of Edinburgh, * Department of Mechanical Engineering

t Fluid Dynamics Unit, Department of Physics Kings Buildings, Mayfield Road, Edinburgh ERg 3JL,

Scotland, UK.

Abstract

The ability of PlV to determine the flow characteristics of a turbulent air-

flow and a turbulent air-particle flow field is discussed. PlV has been able

to determine the spatial structure of a grid generated turbulent air-flow

and the velocity data from the PlV flow record has proven to be accurate

when compared with LDA measurements. Both velocity and concentration

behaviour of a particle jet have been obtained from the PlV flow record.

Furthermore, the potential capabilities of PlV to simultaneously determine

the behaviour of both phases of an air-particle flow field are discussed in

the context of particles interacting with the coherent structures of a grid

generated turbulent flow field.

keywords: turbulence, 2phase (air-particle) flows, PlY.

207


208 D. R. McCLUSKEY ET AL.

I Introduction.

Particle Image Velocimetry (PlV) has emerged as a powerful experimental tool

which can provide whole flow field information concerning a wide range of fluid

phenomena. This paper will discuss the information that PlV can provide con

cerning the two phenomena, grid turbulence and a two-phase, air-particle, flow

field.

The study of turbulent flows is of interest both to those concerned with the de

velopment of the PlV technique, and to fluid dynamicists who have an interest

in the flow phenomenon itself. Several authors have presented results from the

application of PlV to turbulent flows [1,2,3), but a full discussion of length-scales

and the detection of coherent structures has yet to be raised. Therefore, if PlV

is to be utilised to study such flow fields, its ability to detect coherent structures

satisfactorily must be assessed. However, there is no general consensus concern

ing the definition of a coherent structure: theoretical studies of turbulence have

produced several distinct definitions [4,5,6]. This paper's discussion of a grid

turbulent flow field will loosely define coherent structures as the largest eddies

in the flow field. A quantitative evaluation of the size of structures that can

be resolved using this particular PlV method will be made. This size will be

compared with the typical size of the coherent eddies and the different statistical

turbulent length scales. Furthermore, the accuracy of the PlV velocity measure

ments will be assessed in comparison with the data obtained from laser-Doppler

anemometry (LDA) measurements of the flow field. The structural information

contained in the turbulent flow field is of concern in Section 5 where a particle

jet is issued into this same fully developed homogeneous turbulent flow field.

The interaction of particles with turbulent airflows is of interest in many areas

THE APPLICATION OF PIV TO TURBULENT TWO-PHASE FLOW 209

of fluid dynamics, for example coal combustion, crop spraying, and the dispersal

of pollutants. LDA has been extensively utilised to study many aspects of air

particle flows as it is possible to distinguish between the seeding of the air-phase

and the particle phase using this method. Thus, LDA measurements can provide

mean velocity and turbulence measurements concerning both phases of the flow

field and additionally concentration information concerning the particle phase.

However, as a point measuring technique, LDA cannot be used to provide whole

field information and so cannot be utilised to examine the interaction of parti

cles with large scale coherent structures. Interest in particle-coherent structure

interaction has been raised in the discussion of coherent modelling of two-phase

flows [4J. The results predicted by such models suggest that particles tend to

congregate in streaming regions of the turbulent flow fields between the highly

rotational eddies. Furthermore, the experimental observations of the trajectories

of soap bubbles in a turbulent jet flow [7J indicate that the bubbles were thrown

out of the coherent eddies. The above paper also showed that solid glass spheres

as large as 86p,m in diameter travelling through a turbulent jet were influenced

by the turbulent coherent structures of the jet. As a whole field technique, PIV

offers the potential advantage over LDA of providing valuable information con

cerning particle interaction with coherent structures. However, in order for this

potential to be realised, the PlV technique needs to be developed to distinguish

between the phases of an air-particle flow field and to assess concentration in

formation about the particle phase of the flow field. This study will discuss the

measurement capabilities of PlV regarding these issues using experimental data

concerning an air-particle jet issued into a grid generated turbulent flow field.


2 Experimental Apparatus.

A schematic diagram of the small wind tunnel used to study both grid turbulence

and two-phase flows is shown in Figure 1. Air from a fan is monitored by the

pitot-static tube and then passes through an expansion before encountering a

series of four meshes each of which is followed by a settling section, the last of

these being longer than the preceding three. The air then passes through a 12:1

wind tunnel contraction before encountering the glass test section. The glass test

section has a square cross section with a diameter of 52mm. LDA measurements

in the test section showed that this arrangement produces a uniform air-flow

with a turbulence level of less than 1% . In the study of turbulence, the grids

were placed between the outlet of the contraction and the inlet to the glass test

section, generating a homogeneous isotropic turbulent flow field a short distance

downstream ofthe grid. This air-flow was seeded with corn-oil droplets of 1-2JLm

diameter in order to obtain a PIV record of the air phase.

In the study of two-phase air-particle flows an air-particle jet was injected, via a

5mm bore tube, into the turbulent air-flow. The particle jets were generated by

means of a fan picking up particles from the outlet of a dust-hopper with a screw

feed. Both the screw feed of the dust-hopper and the fan were variable so that

both the injection velocity and particle loading of the jet could be controlled.

Once the air-particle mixture had passed through the test section it encountered

a cyclone separator. The particles were solid glass spheres with a density of

2500kgm-3 and a mean diameter of 76fLm. The behaviour of the air-phase of the

flow field is again determined by seeding the air with corn oil droplets.

THE APPLICATION OF PIV TO TURBULENT TWO-PHASE FLOW

contraction

Figure 1: Schematic diagram of the small wind tunnel rig.

ROTATING POLYGON

WITH MIRRORED FACES

CW LASER

Figure 2: Scanning beam method of illumination.

glass test section

I cyclone

separator

211


3 Recording and Analysis of PIV Flow Records.

3.1 Illumination of the Flow Field.

The scanning beam method [8J is used to illuminate the flow field as illustrated in

Figure 2. The pseudo-light sheet is produced when a laser beam is directed onto a

rapidly rotating multi-faceted mirror. Successive facets reflect the beam through

an arc of ~ , where N is the number of facets on the polygon, thus the laser beam

passes through this arc every T = Fi. seconds, where F is the frequency at which

the mirror is rotating. As this scanning beam illuminates a subsection of the

flow field, each particle within that region of the flow will be illuminated for a

brief period of time as the laser beam passes, and then re-illuminated when the

scanning beam returns to that section of the flow. A full theoretical analysis of

the scanning beam method can be found in reference [8J.

3.2 Automated Analysis of PIV Negatives.

The automated analysis system at Edinburgh is based on the Young's fringe

method and is shown schematically on Figure 3. A collimated ImW Helium-Neon

laser beam probes a small area of the developed PlY-negative, which is mounted

on a micro-translation stage. The resulting Young's fringe power spectrum is cap

tured by a CCD camera and the information is then sent to a micro-computer via

a framestore. Processing the digitised power spectrum using a Fourier transform

produces the auto-correlation of the particle images in the interrogation region.

A typical Young's fringe power spectrum and its corresponding auto-correlation

plane are shown in Figure 4. (The self-correlation or halo peak has been removed

from this figure.) The spacing and orientation of successive particle images in

THE APPLICATION OF PlY TO TURBULENT TWO-PHASE FLOW 213

the area of the interrogation region is directly determined from two-dimensional

position of the signal peaks in the auto-correlation plane, the location of which

are found by the micro-computer. For the purposes of determining particle con

centration and of assessing the quality of the PIV recording, the microcomputer

also evaluates the either the visibility of the Young's Fringes or the volume of

the signal peak. This issue will be addressed in Section 5. The microcomputer

then stores this information and instructs the micro-translational stage to move

the negative to another location. This procedure is repeated until the whole flow

field has been determined. Further information concerning the flow field, such as

vorticity and turbulent statistics, can then be extracted from the analysed flow

field by post-processing software.

4 Grid Turbulence.

The following section will discuss the ability of the PIV technique to resolve

the different turbulent length-scales contained in a grid generated homogeneous

turbulent flow field.

A homogeneous and near-isotropic flow field with a turbulence level of around

6% was generated in the wind tunnel by a square mesh grid with round rods

of diameter d=2.44mm, and grid spacing f = 8.41mm. The resulting flow was

seeded with corn-oil droplets of a diameter of 1 - 2J-Lm. A PIV negative recorded

the turbulent flow field, which was generated on a mean airflow of 2ms-1 , in the

region 22-30 mesh diameters downstream from the grid, where turbulence theory

predicts that the flow is fully developed and isotropic. A typical plot of the unin

terpolated 2-D velocity vectors obtained from the analysis of the PIV negatives is

214

OPTICAL ARRANGEMENT

~!'~~t~%acli~~ micro-translation stage

collim.ating optiCS

ELECTRONIC ARRANGEMENT

D. R. McCLUSKEY ET AL.

Figure 3: Schematic diagram of automatic PlV analysis system.

Figure 4: Typical Young's fringes and the corresponding auto-correlation plane.


shown in Figure 5. The mean velocity has been subtracted in order to identify the

instantaneous velocity vectors associated with the isotropic turbulence. Figure 6

shows the vorticity plot obtained from the velocity vectors shown in Figure 5.

The structural information in the flow field has been clearly identified.

In order to estimate the precision of the PIV measurements, the mean velocity

and turbulent intensity of the air-flow were evaluated from the velocity vectors

shown in Figure 5. The mean velocity was found to be 1.99ms-1 and the turbulent

intensity was calculated to be 6.6%. LDA measurements of the same area of the

flow field found the mean velocity to be 2ms- 1 and the turbulent intensity to

be 6.2%. This demonstrates that the quantitative velocity information obtained

using the PIV technique is similar to that obtained from LDA measurements.

Given that the PIV negative was recorded at magnification of 0.5 and analysed

on a 0.5mmXO.5mm grid the smallest structures in the flow field that this system

can resolve are 2mm. This must be compared to the standard statistical theory

turbulent length-scales; the Kolmogorov scale and Taylor's microscale, which are

defined below, as well as the typical size of the largest coherent eddies in the flow

field.

4.1 Kolmogorov Scale.

The Kolmogorov scale, 'r/, corresponds to the size ofthe smallest dissipative eddies

in the flow field [4J. The value of the Kolmogorov scale, 7], for this grid generated

turbulent flow field is found as follows. The turbulence Reynolds number, Rt) is

u/l Rt =

v (1)

216

E 44 E "--+.J U :J \J (/) (/) o L U o c o

:tJ "en o Q.

24

4

D. R. McCLUSKEY ET AL.

1- O.250m/sl ________ ~,~_'" _, I "'I"~_~ .. """" "" , _____ ..... , " ••• "",,.,.,. #' #' ........ , , I I" , , ,,; .. __ .. "" "~",,,,,,

,,--_ --_ ... , f"' ............. - .. "", I', " ...... ~- .. _ .. ,"'''''''"", ::::?:~~~~~~::::::::~~::::::::~::::=:::f:~::;:=;;;~~~:~~~:::: ~\,~~:~~~~~~~'~:=:=::::::::~:::~~~==~ :!~~~~!:: :;:~~~~~~~=:::: t t \, ...... __ ...... , , , ............... ~ .. _ ~ .... " ............ , \ , \ " \ \ , I I , , II', , •....• t t \ ............. __ ............................. '1. ~ , '" .... ,,_ ... , \ .......... \ 1 , \ , , , I t I , , , , I f I ....... . 11'\' .... ___ ........... , ... " ...... ,'_ ....... , .. ' ..•• , ....... ,' t' t" t' "t" r, , ............. . t '\ ", ___ ........... , \ , ........ \ 'e' , ,. .... , ,1 .•••••••• , \ , , , , , I , t r , I • I ................... ..

~~: ~~~'::~::::~~~~~~ ~ ~ ~ ~:! : : : : : : :1: : : : : : : : : : : : ! ! ! ~ ~ ~ ~ ~ ~ ~ ~ ~ : : :: ~ ~ ~ : : ....... , "" \., " , \. \. \., "','" 'f\ ••..••• 1 •• ~ . ~ ••••••• , ......... "," \ \. , ~ ~ .. , " " " .... .. ... ,," \. , t , , I ,\ ,",', .................... , .... ~ .. _ ~ •••.•••• * ..... ,,"" , , , •• I I , , ••

:~~ll;;: ~:~~~~~~~~::::~::::::::f::::::::::::::: ::~~~~~::::::: :~: . " , , , , , t , , , , I " ................................. _ ..................... __ • I \ \ \ , I I I •• « ... ...

. , , I , , Itt t , , •• \ \ " ................... ____ ... _ •.•...•••..• - - _ .......... , t , \ \ , .......... ..

~ : ; : !: t ~ ~ ~ ~ : : : ~ ~ ~ ~ ~ ~ ": -: ~ ~ ~ ~ ~ : : : : ~ ~ ; = : : ; ; :.;.:-=~!'!-~: : : ~ ~ ~ ~ ~ : : : : : , \ , \ , , I ) II' , "" 40 , I , ttl , •••••• , , , .... _ .... , I I , ,1, , * ...... ',. t , \ , , \. " , I •••

,\'" t "'1/" .. " ... ",11 ..... . " ......... .". .... ,' 1,1, I •• ,. 't. _"\\.\\'" ".

~~~~! ~~:~~~~~-;-;:~~~;::::;:: :;;:::::::: ~: ::::: ~:; ::::: :!~~~~: ~:: ....... ,' ,I' II """, t""" ••••• , . .•••• " t', ,I., ....... " '1' •••.••• ,' ,"

-~~ ~ ~ ~ ; ~ ~ ~ ~:! ~~~ ~ ~ ; ; ~ ~~~~ ~k~~~~~-~-) ~; ~ ;~~~~~~-;'~ ~ ~ ~ ~ ~ : : ~ ~ ~ ~ ~ ___ .. ~ ..... _ .. _ ......... " .............. , .......... _ ........... , 1""''''''''''' 1 ., .. " •........ _ .... , ••••••• __ ..... 1" ,.~"", ............... _ •• 1 """"",, I ••• ,., ..... ..

• ' • f , , • ~ ... ~ 40 ____ .... I' .. •• , 'I."'" .......... I " , II J", I ., \ \, .... ",. ......... ..

.... 1,',.,.', .... __ ....... """ .. <0 ••••• \ .......... ,1111','",\, .......... .......... ..

... 0#", ..... ""-----" .... , ................. ,",11"'" .. ,, ......... _ ... , ..

.. "" .... ,I"I ............. " .. - ............... - ..... ~"',I""' ....... - ..... ...... .. .. "" .... "" ....... , .................... _ ...... ' ... ' ... ',,",""' ............... . , ..... . ... I' \ ", .................. , ...... """",,""" .. ",1 I 1, .. ' .. __ .................. .. .. "" .................. , ........... "", .... """, ........................... ..

180 190 200 210 220 230 240

position downstream of grid/mm

Figure 5: Uninterpolated velocity vector plot of grid generated turbulence which is typically 6%, with the mean velocity of 2ms-1 subtracted. Typical eddies are framed by boxes.

190 200 210 220 230 positon downstream from grid/mm

units 10- 38-1

vorticity abo"" 0.16 rus -. 0.16 OJl - 0.13

Cl.09.. o.u 0.07 - Cl.09 0.00 - D.O? D.03 - D.06 0.01·- D.03

-'()'o1 - 0.111 -0.03 •. -0.111 -0.D6 - .-0.03 --0.07 - ·"().06 -0.09·· '''().07 -0.11 - -0.09 --0.13 - -D.ll -D.15 - -oJ3

vortlclty bel..... -D.15

Figure 6: Vorticity map of a the flow field shown in Figure 5.

This figure appears in color on p. ix.


where u ' is the rms velocity fluctuations, the value of which can be found from

LDA measurements, and

£ is the spacing of the turbulence generating grid which has a value

£ = 8.41mm.

Using the relationship

_ R-3/ 4o "1- t .(. (2)

'fJ is found to have the value "1 = 0.335 X 1O-3m. This length scale is much less

than that the PIV system can resolve.

4.2 Taylor's Microscale.

An estimate of Taylor's microscale, >., is obtained using the above value of 'fJ and

the LDA data as follows:

(3)

. h R U'). WIt ). = 7'

and so an estimated value of >. = 3.8mm is obtained. Length scales of this size

can be resolved by the PIV system.

A further check on the ability of PIV to resolve structures in the flow is made

by comparing the value of Taylor's microscale, >., found using LDA data of the

flow field with the value of>. derived from the PIV velocity measurements of the

flow field. The PlV based estimate is obtained by calculating the longitudinal

correlation function over a row in the velocity vectors shown in Figure 5 and then

fitting a parabola to this curve. (For further information consult reference [9].)


The curve and parabola can be seen in Figure 7. Thus, using this method and

the PlV velocity data, an estimated value A = 4.5mm is found, which confirms

the order of magnitude value of 3.8mm obtained from the LDA data as outlined

above.

• OIII"I"IIIatlou - parabollo fit -axill

O.O+-------------L--"""--:----l

a 1 2 a 4 56? 8 9 W 11 ~ ~ U ffi ffi

r/mm

Figure 7: Longitudinal correlation function calculated over a row of Figure 5, with a parabola fitted to estimate A.

4.3 Large Structures.

Direct observation of the experimentally obtained flow record shows that the

largest eddies in the flow are of a size comparable to the grid spacing, as would

be expected. Typical eddies are framed by boxes in Figure 5.

4.4 Conclusion.

The PlV technique as employed above is capable of resolving the large scale

coherent eddies in the flow field and structures comparable in size to Taylor's


microscale, .x, although the smallest dissipative eddies, comparable in size to

Kolmogorov's scale 1/, are not resolved. Large scale eddies in the flow field are

readily identifiable from visual examination of the velocity vectors in the flow

field. The mean and turbulent velocity information obtained from a single PIV

data set is comparable to that obtained from LDA measurements.

5 Air-Particle Flows.

The ability of the PIV technique to obtain the concentration profile of a particle

jet will be discussed, as will the possibility of using PIV to distinguish between

the air-phase seeding and the particle-phase of a simultaneous PIV recording of

an air-particle flow field.

5.1 Particle Concentration and PIV.

As discussed above, the analysis of PIV flow records involves probing a small

region of the developed negative with a low powered laser beam which produces

a Young's fringe power spectrum. It has been shown [10) that the intensity

pattern of the Young's fringes, when there are two images of each particle, has

the form

l(k) = 2NI h(k) 12(1 + cos[kds + 1))

where N is the number of particles in the interrogation region

1 h(k) 12 is the halo function

ds is the separation between particle images.

(4)


Performing a Fourier transform of this intensity distribution gives the auto

correlation plane, R(r), which has been shown [10] to have the form

R(r) = N[h(r) Q9 h(r)] * [2S(r) + f( -r + ds ) + f(r - ds )] (5)

That is, the height of the signal peaks is proportional to the number of particle

images in the interrogation region. Thus, concentration information is contained

in the auto-correlation plane and so has the potential to be extracted by the PIV

analysis procedure.

Equation 5 is a theoretical expression, which assumes that the particle distribu

tion in the probe beam is random and so can be described by a Poisson distribu

tion; the particle displacements within the interrogation region are all the same;

the full complement of particle images are within the interrogation region and

all particles have been recorded with the same intensity. The distance between

particle images is the same if there is no significant velocity gradient within the

interrogation region and there is little particle turbulence. The assumption that

the full complement of particle images are present within the interrogation region

is valid if there is no out-of-plane motion and no large velocity gradients moving

particles outwith the interrogation region. Recording all particle images with

the same intensity is easily achieved when using the scanning beam method of

illuminating the flow field [9], if the particles are spherical.

Figure 8 shows a PIV recording of the particle phase of an air-particle jet which

is injected into an airflow. The injection velocity of the jet is 7.5ms-1 and the

mean velocity of the airflow into which the jet is injected is 10ms-1 • A velocity

vector map of the particle flow field obtained from PIV analysis is shown in

Figure 9. The injection velocity of the jet has been subtracted from the velocity


vectors in order to accentuate the changing behaviour of the particles as they

travel downstream from the injection nozzle. Clearly, particles at the outer edges

of the jet accelerate more rapidly than those in the jet's centre-line.

From examination ofthe photographic print ofthe PIV negative on Figure 8, it is

evident that concentration information is also present in the PlV flow record. Fig

ure 10 compares the measured concentration profile of the particles 50mm down

stream from injection, obtained by visual examination of the PlV flow record,

with the height of the signal peak in the auto-correlation plane which was ob

tained by the automated PIV analysis system. This comparison shows that the

theoretical relationship, given by Equation 5, between the height of the signal

peak in the auto-correlation plane and the number of particles in the interro

gation region holds well, except when there are only a few (3 or 4) particles in

the region of the interrogating beam. This discrepancy is not significant in com

parison with other particle concentration measuring techniques. Thus PIV can

determine the concentration profile, as well as the velocity vectors, of the particle

jet.

5.2 Two-Phase PIV Measurements.

In order to assess whether PIV could simultaneously record both phases of a two

phase air-particle flow, PIV negatives of an air-particle jet injected into a seeded

turbulent airflow were captured. A photographic print of such a PlV recording

is shown in Figure 11. The turbulent airflow has the same mean velocity and

turbulen t characteristics as described in Section 4. Clearly, the particle phase

images and the air-phase seeding images are readily distinguished; both by their

image diameters and their image intensities, with the air-phase seeding having a


Figure 8: Photographic print of a PlV recording of a particle jet.

E E 48-

.......... -+J U

-6 en 32-en o L.. U

~ 16-o

~ o a.

- 2.25m/s

... ,,--t , /1" ~ ;~~, ~~ ~~ ___ _ ""/~" ~, -~-"",~,~~~",~- ~~

II"""""",~,--~,-~--,----,,~--~---II' ',11""""""",_, ___ ,, ________________ _ If I , "'I""""'i; ____________________________ _

II """"",~~ _________________________________ _

" -----------------------------------------------L\" \ \"~"""~----------------------------------

,\\\ \\"""""""'~~-------~---------------,\\\,\"'-""'-"'-----------,------------~ , , "' ... "----,----,--------..... , ............. , ....... ,

... " , ....

position downstream of injection/mm

Figure 9: Velocity vector map of the particle flow field shown in Figure 8.

THE APPLICATION OF PlY TO TURBULENT TWO-PHASE FLOW

significantly lower image intensity than the particle phase.

6 Conclusions.

223

The PIV technique as employed resolves the large scale coherent eddies in a

grid generated turbulent flow field and structures comparable in size to Tay

lor's rnicroscale, .x, although the smallest dissipative eddies, comparable in size

to Kolmogorov's scale, 7], are not resolved. The mean and turbulent velocity

information from the PIV data set is comparable to that obtained from LDA

measurements. When using PIV to investigate air-particle flows, the phases of

the flow field can be distinguished from one another, using either image intensity

or image diameter, and concentration information about the particle phase of

the flow is obtained, as with the LDA technique. These developments of the PIV

measurement methods offer a potentially powerful method of investigating the in

teraction of particles with large scale turbulent coherent structures. Work is now

underway on methods of separating the two phases by a thresholding technique.

7 Acknowledgements.

The authors would like to thank the Science & Engineering Research Council

(Process Engineering Division) and Dantec Electronics for sponsoring the work

described in this paper. The contributions of Dr C Gray for the development of

the automatic PIV analysis System, D J Skyner for its upgrading, and T Bruce

for the use of the post-processing software are all appreciated. D Anderson is

thanked for his workmanship in constructing the small wind tunnel rig.


... . . e.

0° OJ . . ~'+n~~~~~~~~~~~~.~'~.~~

w ~ ~ Z M ~ ~ ~ M

p<IIiIltlon aero .. duct/mm

Figure 10: Comparison of observed concentration and PlV concentration information.

Figure 11: Photographic print of a PlV recording of an air-particle flow field

THE APPLICATION OF PlY TO TURBULENT TWO-PHASE FLOW

8 References.

225

1. Adrian, R. J.: Statistical properties of PIV measurements in turbulent flows,

Laser Anemometry in Fluid Mechanics, vol III, (1988).

2. Kompenhans, J. and Hocker, R.: Investigation of turbulent flows by means

of Particle Image Velocimetry, 5th Int. Symp. on Flow Visualisation, Prague,

Czechoslovakia, (21-25 Aug., 1989).

3. Liu, Z-C., Landreth, C. C., Adrian, R. J. and Hanratty, T. J.: High resolution

measurement of turbulent structure in a channel with Particle Image Velocimetry,

Experiments in Fluids, 10, (1991), pp.301-312.

4. Hunt, J. C. R., Wray, A. A. and Moin, P.: Eddies, streams and convergence

zones in turbulent flows. Center for Turbulence Research, Proceedings of the

Summer Program 1988, (1988).

5. Hussain, A. K. M. F.: Coherent structures - reality and myth, Phys. Fluids,

26(10), (Oct. 1983), pp. 2816-2850.

6. Mumford, J. C.: The structure of laege eddies in fully developed turbulent

shear flows. Part 1. The plane jet, J. Fluid Mech., 118, (1982), pp.241-268.

7. Perkins, R. J., Ghosh, S. and Phillips, J. C.: The interaction between particles

and coherent structures in a plane turbulent jet, Advances in Turbulence 3, Ed.

A. V. Johansson and P H Alfredsson, Springer-Verlag, Berlin, (1991).

8. Gray, C., Greated, C. A., McCluskey, D. R. and Easson, W. J.:An analysis of

the scanning beam PIV illumination system, Meas. Sci. €3 Techno., 2(8), (Aug.,


1991), pp. 717 -724.

9. Tennekes, H. and Lumley J. L.: A first course in turbulence. MIT Press,

(1973).

10. Hinsch, K., Arnold, W. and Platen, W.: Flow field analysis by large-area

interrogation in Particle Image Velocimetry, Opt. and Lasers in Eng., 9, (1988),

pp. 229-243.

A Critical Analysis of the Particle Image Velocimetry Technique as Applied to Water Waves.

P. A. Quinn, D. J. Skyner, C. Gray, C. A. Greated, W. J. Easson

Fluid Dynamics Unit, University of Edinburgh.

Abstract

The general uncertainties inherent in Particle Image Velocimetry (PIV) are initially considered in outline. The manifestation of these uncertainties in the Young's fringe analysis method are identified and discussed, along with the errors arising specifically in water wave studies. One particular application of PIV, the measurement of velocity fields under breaking waves incident on a plane, gently sloping beach, is then considered in detail. The results from this study are presented and the particular problems associated with the use of PIV in the measurement of these flows addressed. Finally, all of the errors in the PIV process relevant to the study of wave kinematics on sloping beaches are summarised.

The combined systematic and random errors relative to the maximum velocity in the flow, for PIV measurements of water waves are shown to vary mainly with particle displacement gradient across a small interrogation region on the negative. With a low displacement gradient of about 0.01 the combined error varies from 1.2% to 2.0%, depending on the variation of further systematic errors, such as photographic distortion. With an extreme displacement gradient of 0.05, the combined error varies from 5.1% to 5.9%. These two displacement gradients correspond to typical and extreme cases for the described study of waves on beaches, with the higher gradient occurring at shallower water depths near the breaking point.

1 Introduction Particle Image Velocimetry (PIV) is a well established fluid flow measurement technique and has been used at Edinburgh for a number of years. If this technique is to provide reliable, quantitative velocity data routinely, then an accurate estimate of the inherent uncertainties is essential. A full assessment of the errors and limitations inherent in the measurements involves consideration of the general aspects of the method, of the implementations used, and of the particular problems associated with the flow being studied.

Knowledge of the internal kinematics of waves breaking on sloping beaches is an important element in the understanding of coastal processes, such as beach erosion. PIV offers the possibility to measure the kinematics in more detail than before, and

227

F. T. M. NieuWSladt (ed.), Flow Visualization and Image Analysis, 227-245. © 1993 Kluwer Academic Publishers.

228 P. A. QUINN ET AL.

such studies are currently being undertaken at Edinburgh. This paper attempts to make thorough assessment of all errors and limitations present when PIV is used in this application. In the following sections all of the uncertainties involved in different aspects of the PlY process are considered in detail.

2 An Overview of the PIV System used at Edinburgh. PIV is a two stage measurement technique involving first the photographic recording of the displacement of small seeding particles in the flow, and second the point-bypoint analysis of the developed negative. There are several different methods used to perform these two functions [1]. The systems used at Edinburgh are now discussed, and comparison is made, where appropriate, with other methods.

2.1 The Scanning Beam Illumination System One of the most common methods of producing the pulse illumination required for PlY is to use a system of cylindrical lenses to fan out the incoming laser beam, which may be recollimated by a parabolic mirror depending on the size of the flow. The pulsing is controlled by a mechanical or electro-optical chopper. The main drawback of this technique is the reduction of laser power when the beam is spread out, especially for large scale flows. This problem does not arise with a scanning beam system as it is the unexpanded laser beam which is scanned through the flow.

In most of the experiments at Edinburgh the beam of a CW 15W Argon ion laser is scanned into the flow to be measured. The scanning system used for the water wave studies is shown in figure 1. The laser beam enters from the bottom left and is deflected upwards by a beam steering mirror. It is collimated by two achromatic lenses and hits an octagonal spinning mirror. This scans the laser beam along a parabolic mirror which reflects the beam into a vertical pseudo light sheet.

The scanning beam system was first reported by Gray and Greated [3] and further discussed by Gray et al [4]. Because the exposure of each particle depends inversely on the scanning rate, the system is best suited to relatively low-speed flows. In water wave studies, flows of 1ms-1 and magnifications of 10:1 are typical and, in conjunction with a 15W laser, the scanning beam system has been found to provide enough light for all eventualities. This is achieved with scan times between 0.6ms and 8.0ms.

McCluskey et al [8J have use a similar, but higher speed, scanning beam system to measure the velocity of particles transported in a small scale wind tunnel, where the average axial air flow was 8ms-., and the photographic magnification about 1:1. For flow speeds greater than this, greater laser power is needed and pulsed lasers, such as Nd:YAG, in a system using cylindrical lenses to spread out the beam, should be considered.

THE PARTICLE IMAGE VELOCIMETRY TECHNIQUE AS APPLIED TO WAVES 229

o 0, \ Rotating Octagon&! Mirror

Laser

Figure 1: Scanning beam system

2.2 The Analysis System

The PIV analysis system implemented at Edinburgh determines particle displacements on the developed negative by means of the Young's fringe method [Huntley, 1986]. This method is best suited to PIV photographs with a high seeding density, and a carefully controlled range of particle image separations. This is the case for water wave studies. The method is also capable of producing accurate measurements in a relatively short time.

A diagram of the automated analysis system is shown in Figure 2. Here a low power He-Ne laser is used to provide the interrogation beam, of diameter about 1mm. The beam passes through the negative, which is mounted on a computer controlled microtranslation stage. The scalar light field transmitted by the negative is optically Fourier transformed by a lens. This forms the Young's fringes on the CCD array located in the focal plane of the lens. The digitised signal from the CCD camera is Fourier transformed again by a microcomputer to form the autocorrelation plane.

The halo produced by the diffraction of the individual particles is subtracted and self-correlation peak avoided in the search for displacement peaks. The orientation and separation of the displacement peaks are determined, and hence the local velocity. The negative is then moved to the next position where the process is repeated. In this way the whole negative is analysed.

One other method for determining the particle displacement on the negative is by direct image plane analysis. In this method a CCD camera is used to look directly at a small region of the developed negative. Both Fourier transforms, required to form the autocorrelation plane, are carried out numerically. This method will take slightly longer than the Young's fringe system due to the extra time required to perform the first Fourier transform numerically. However, no low-power laser is required, and for applications with low seeding density where particle tracking is a more appropriate


Translator

~----~-------~--------~ ~----4-------~--------~

MkTor

Figure 2: Automated Young's Fringe Analysis System

method, no adjustment to the system is necessary. One way of reducing the analysis time is to perform both Fourier transforms optically. Moraitis [1988] provides a good overview of the various approaches. The use of a Spatial Light Modulator (SLM), looks to be the most promising, and this method is currently being investigated at Edinburgh.

3 Uncertainties Inherent in PIV The velocity at each interrogation point on the developed negative is yielded by the equation:

Where:

_ CM_ v=rs

• s is proportional to the mean particle image displacement in an interrogation region.

• C is a calibration factor which converts s to particle displacements on the film. This factor is specific to the analysis system.

• M is the photographic magnification of the image size to object size.

• T is the time period between successive particle illuminations.

The quantities C, M and T can be directly measured and uncertainties accurately assigned. These errors can be made to be small; down to within .1 % of their associated value.

The main source of error in the velocity measurement comes from the displacement, s. The uncertainties associated with s arise from both the photographic recording of the flow and the subsequent analysis of the developed negative. These are now dealt with separately, in detail.

THE PARTICLE IMAGE VELOCIMETRY TECHNIQUE AS APPLIED TO WA YES 231

3.1 Uncertainties in the Photographic Recording of the Flow Sources of errors inherent in this phase of PIV fall into two categories. The assumption that the seeding particles faithfully follow the fluid motion and the characteristics of the optical recording system. The effect of the former is minimised by appropriate choice of seeding particle.

The characteristics of the optical recording system will be system specific, depending on the configuration and components used, and factors such as optical distortion of the imaged flow, 3-D velocity components in the flow, limited resolution etc. These will combine to provide a net error in the recorded flow. These errors can be divided into systematic and random errors.

• Systematic Errors

- Distortion of image flow field due to:

* Geometric distortion of camera lens

* Refractive index changes between the measurement zone and the camera.

* Perspective. * Out-of-plane velocity components.

• Random Errors

- Distortion of particle images due to:

* Grain noise.

* Adjacency.

* Shrinkage.

The effects of the systematic errors do not change between measurements and, in principle, some of their effects can be measured and accounted for in the final measurements. Some of these errors can be minimised by the choice of high quality photographic equipment and materials.

3.2 Limitations of the Analysis Method The Young's fringe analysis system measures the mean particle displacement within a small localised region of the developed negative. This is achieved by taking the autocorrelation of the amplitude transmission distribution across the local region of film, by means of optical and computational Fourier Transforms. There are several sources of errors arising from this process:

1. The limited quality of the Fourier transforming lens will introduce some distortion to the resulting power spectrum. In addition to this, noise will be introduced by the refractive index and film thickness variations across the negative.

2. Quantisation errors associated with the digitised power spectrum data, and rounding errors in the numerical calculation of the autocorrelation function.

3. The influence of random correlation noise in determining the centroid of the signal peak.


4. The effect of the discrete and random sampling of an effectively continuous flow by the randomly distributed particles.

The first and second points are functions of the quality of the components and their influence can be minimised by appropriate choice of equipment. The third and fourth points are more fundamental in nature and are due to the manner in which the flow velocity information is represented on, and measured from, the recording medium. These last two points have been addressed by means of a Monte Carlo simulation of the PIV process, described in detail by Gray [1989]. The study took into account variations of the parameters involved in PIV, primarily the seeding concentration and velocity gradients across the probe region. A more extensive study, using similar simulations along with an analytic treatment, has been carried out by Keane and Adrian [1990].

3.2.1 The Effect of Seeding Concentration

It was found that there was a rapid decrease in errors as this seeding concentration varied from 3 to 10 particle pairs per interrogation region, but beyond that there would be little significant reduction in errors. This result is similar to that found by Keane and Adrian [1990].

3.2.2 The Effect of Velocity Gradients

The larger the particle displacement the higher the chance of the multiple particle images falling outside the interrogation region. The effect of this is to artificially reduce the seeding density, and introduce a bias to selecting smaller displacements. This is an important effect and causes an error which varies with displacement gradient across the interrogation region. Keane and Adrian [1990] found the systematic bias to introduce errors varying linearly with displacement gradient. The error, relative to the maximum velocity, varies from 0-5% over a range from 0.00-0.08. Here we define the displacement gradient to be the change in displacement across the whole interrogation area. The definition favoured by Keane and Adrian [1990] is half of this.

The effect of a large velocity gradient across the interrogation area is to spread the displacement peak in the autocorrelation plane over a larger area, hence reducing its height. This increases the random error associated with the velocity measurements. The results of our simulation for random errors are plotted in figure 3 against displacement gradient. The relationship is, once again, similar to that of Keane and Adrian [1990]. In addition, the probability of a random correlation peak being selected also increases, introducing an upper limit to the velocity gradient in the flow which will still yield correct velocity measurements. This problem manifests itself by producing vectors of random magnitude and orientation which have no continuity with any of the surrounding vectors, and are easily identified and removed.

3.2.3 Analysis System Calibration

The influence of the random errors listed in section 3.1 cannot be estimated individually but their influence, combined with the sources of errors in the analysis phase, can be estimated by making a series of artificially generated PIV negatives. These can be formed most conveniently by accurately plotting a random pattern of groups of N dots


,......, 2.00 .....

II)

> >< «I

S LW ..... 0

~ '-'

J.. 1.00 0 J.. J.. II)

e 0.50 0

't:! J:: «I p::

0.00 L----''--~~--'--i-:-:--..1.---'---'-----'---'---L---'---l 0.000 0.010 0.020 0.030 O.~ O.OW 0.060

Displacement gradient Figure 3: Random error associated with displacement gradient

where the dot size and their spacing are specified. Photographing the printed output produces a transparency with a known particle displacement. The whole negative is analysed to obtain an estimate of the variation from this known displacement.

A series of negatives with different particle displacements were analysed and the relationship with the measured displacements was linear. The calibration constant was found with an uncertainty of only 0.1%.

4 The Application of PI V to Water Wave Stud-. les

A 9.75m long wave flume has been constructed at the University of Edinburgh, designed with PIV applications in mind. The tank walls and base are made of 25mm thick glass to allow optical access from all sides. The tank was constructed in three sections, the two furthest from the wavemaker being the measurement zone. The system for illuminating a plane in the flow is mounted on rails beneath the tank, so that any part of the 6.5m measurement zone can easily be illuminated. The middle section of the flume is shown in figure 4 The wave maker is a hinged absorbing paddle [Salter, 1982] controlled by a computer which also controls the camera's activation. The illumination of the measurement zone is provided by the scanning beam system described earlier.

4.1 Uncertainties Arising in Water Wave Measurements

4.1.1 Equipment Selection

To minimise the effect of errors high quality photographic equipment is used. The camera used was a Hasselblad 500 ELjM fitted with a Carl Zeiss 80mm, f2.8Iens. The selection of aperture is made by balancing two factors. The wider the aperture the better the diffraction limit of the lens, however this is accompanied by an increase in

234

oIr

water

water

loser bean

P. A. QUINN ET AL.

wave propagation ~

sloping beach

me06(Kement zone

Figure 4: Wave flume with 1:30 sloping beach and illumination area

spherical aberration. Often, setting the aperture one stop down from the maximum gives the best compromise. In this case £4 was selected.

The film used was Kodak T-Max 400 ISO. Development was carried out at 22°C in the normal way with Kodak T-Max developer. Push processing was not necessary with the illumination system and seeding particles used. The quality of the PIV negatives obtained in these, and other studies at Edinburgh, is such that contact printing to provide a positive image for analysis does not improve the measured velocities.

4.1.2 Photographic Distortion

The distortion due to the camera lens can be measured by photographing a regular grid. At the corners of the developed negative the distortion was found to be at worst 1.5%. The effect of the lens distortion and the refraction conveniently oppose each other so that there is a combined distortion which is almost exactly constant with variation in image height. The variation from the mean has a standard deviation of only 0.3%.

One must remember that these errors due to distortion and refraction have their maximum values at the edges of the field of view, and typically the area of interest is placed at the centre of the field of view where these, already small, errors are greatly diminished.

4.1.3 Scanning Beam System Errors

There is a possibility of a small uncertainty in the illumination interval with the scanning beam system, due to subsequent illuminations of a given particle being at different positions in the measurement zone, and hence different phases of the scan cycle. If the averaging areas for each PIV interrogation are small compared to the total sheet length,

THE PARTICLE IMAGE YELOCIMETRY TECHNIQUE AS APPLIED TO WA YES 235

then these errors are very small [Gray,1991]. In this particular application the error is always less than .25%.

A further problem which is more likely to arise with the scanning beam system than with beam expansion is the flatness of the resulting sheet. Variations from a plane result in the effective magnification being different in different region. Typically the sheet is flat to within ±3mm, which results in a .3% error if the camera is positioned 1m away. The effective thickness of the sheet is about 2mm which gives a random magnification error of 0.1%.

4.1.4 Seeding

In the water wave studies carried out at Edinburgh, conifer pollen has been found to be the most successful seeding material. This is N 50j.tm and, when wet, has approximately the same density as the water. It has, if anything, a small tendency to float. Typical rise velocities are of the order of 0.25mms- 1. As typical flow velocities for laboratory waves are around 1ms-1 , this represents an error of about 0.025%. As the accelerations in water waves are of the order g, the acceleration due to gravity, the relative motion of the water and the particles will not deviate significantly from the stated error.

4.1.5 Out-of-plane Motions

Out-of-plane motion can produce errors in the region of about 20% if the out-of-plane motion is of the same order as the in-plane motion [Sinha, 1988]. However, in the application to measuring water wave velocities in a 2-D wave flume this 3-D component is minimal, thereby reducing the error to a negligible amount. This factor was verified by exposing the negative to for a complete cycle of a sinewave. The resulting circular traces on the negative indicated little, if any, out-of-plane movement. Care must be taken to reconsider these assumptions when, for example, measurements are made of the turbulent flow field following wave breaking.

5 PIV Applied to Waves on Beaches Having addressed the considerations general to the measurement of water wave kinematics with PIV, this section focuses on the particular application of waves breaking on sloping beaches. The construction of the beach is outlined and the experiments described. Typical results are presented and an assessment made of the errors and limitations of PIV in this application.

5.1 Beach Construction

To overcome the problem that the laser illumination must pass through the beach, it is made in two longitudinal halves, with a 10mm gap down the middle. This gap is covered by thin transparent plastic, of the type used on overhead projectors (see figure 5). The main support for the beach is provided by two pairs of parallel fibreglass "I" beams, one pair in each of the two measurement zones of the tank. These are held in position at the end of each zone by cross-supports. The "I" beams also hold the interlocking mechanisms for attachment of the beach. The plane beach surface is made

236

Tank Wall

AIr

WatfT

Tark Wall

Figure 5: A section through the Beach.

P. A. QUINN ET AL.

of 10mm thick "Coplast" plastic sheets. The cross-supports can be moved vertically in the tank to allow any slope in a 1:30 to 1:3 range to be selected.

5.2 Experiments

Experiments have been performed on a plane slope of gradient 1:30 with waves of frequency of 1Hz at each of the two wave heights, 32mm and 48mm. The wave heights were measured visually at the foot of the beach. Measurements were made for four phases and at three different positions along the beach, for each of the wave conditions. The measurement positions were at still water level (SWL) depths of 120mm, 100mm and 50mm. The computer, controlling the wave maker, was also used to trigger the camera. In this way the desired phases of the wave were accurately recorded. The Hasselblad 500 EL/M has a film transport time of I"V 2s, so recording consecutive phases from the same wave was not possible. The waves were recorded with a separation of 10.25s.

Although the scanning beam system can produce a sheet 1m in length only 700mm was used in this application due to positioning the camera at a distance of I"V 0.9m to the side of the illumination plane in order to observe the relatively shallow depth of water in greater detail. The photographic equipment was as noted in section 4.1.1. During the experiments the laboratory was completely blacked out, and with the ability to change the water in the tank regularly, the contrast of the illuminated seeding particles to the background is extremely good. Black paper is also attached to the back of the tank to maximise this.

A regular grid was photographed in the illumination plane prior to any measurements to obtain the magnification ratio. The choice of scan time is made by first of all estimating the maximum velocity in the wave by linear theory, and then adjusting this given initial trial measurements. As the shutter speed of the camera is adjustable only in steps it is not possible to match the optimum scan rate with the shutter speed to enable an integer number of scans to be made. So there will be an area on the negative

THE PARTICLE IMAGE VELOCIMETRY TECHNIQUE AS APPLIED TO WA YES 237

................... ~ .. -.-, "':-.---;;",,,;,,, I' / II , I :: ::.:: : , .. v_" / " / / I I I " I, ' , t \ \ \ ,~ ..... ~ ..... ~:==::::==:::::::;::;::;::;;:-.. -;,,,.///111 'I \ \ ", ........ __ ... , , . ..... --........ I • ..... .". ... ~,,~ III , r \'" ", .................. -.. .. _._._. ______ _ ................ _'_' ___ ..... .....,-"''''.,,,,, I , I' \ ", ........ ........ , .... ---..... - •• -.------_-.--.. . ___ ....... ..----.-.-............ ",.,,'.,.' I " \ ,,, ......... ........ - ......... ---------------------.---.. ~---,."" , , , , , \ , ............. ---- ---------------------------.----~" , ",. , , , , " ....... -.. ---- -- - - ------------------_.------- - - - , --- , , "'-------------------------------._----- - - - -- - - - - - , ----------------------------., __ -._o_a_._._·_· ____ - - - - - - - - -

Figure 6: SWL Depth:120mm; Phase: 0.25

, , I , , I \ \ , , , , . , I l \ \ ---~----------------,~ , , \ , , ,. ,. " ... ... .. .... .... ----------------------~~' ,. " , ~ -- ~ - - -- - -- - - -- - -- - - - - - - - - - - , , ,

~----------------------------------------------------

Figure 7: SWL Depth:120mm; Phase: 0.5

with N multiple images and the rest will have N - 1. The shutter speed is chosen so that there are at least three, and ideally four, multiple images recorded. This is also governed by the criterion that the exposure time must be short enough that the flow does not significantly change in that period. For this study a shutter speed of 1/30s was selected. A scan time of 5ms was chosen giving about 33% of the negative with 6 particle images, and the rest with 7.

5.3 Results Some of the results for the waves with wave height 32mm are presented in the form of vector maps (see figures 6-12). Similar quality results were obtained for the other phases at this wave height and for the waves with a wave height of 48mm. The missing vectors in figure 9 are due to air bubbles in the backwash of the previous breaking wave, similarly the area at the front of the crest in figure 12 is due to the highly aerated turbulence formed as the wave breaks.

It is intended to use the results for verification of several boundary integral models and to obtain accurate estimates for integral properties such as the radiation stress and the mean energy flux.


.. .... .",

--_.-._._--' - , ,. ,. I : : ~ ~ ~~ ~~~'S'~,~,::: .... ::-:,:::,~-::-::-::.::::::::;.;::;;::;.:::_;::;;_::;_::;_::;_;;-;_;;-;_=-;;_~_:-;_::-:_ ---~·~-----.,,',.,.,I " ", ............. _________________ _ ----------"", , "", ...... _------------------------------~~", ""-----------------~~------------------"" "'-------------------------------~-------~~'~I\" ___ --------------~~---------------------- "---------------------------- ---------

Figure 8: SWL Depth:l00mmj Phase: 0.25

""', . . , , I , , I \ , , """'- . , , , , I \ \ , , "", .............. . . .... _-""", I , , , I I I \ \ \ ", ................. """- -- . . ......... """"" , , I I

'~I I \ """ ____ ~ __ ~ __________ ~

I I

"-----------~-----------

-::., :::=~::~ , I

==::::::~~" ----.,.."" ----_ ..... , ------" -------". --------,. ------------====--

---------.-----

Figure 9: SWL Depth:100mmj Phase: 0.5

, , , , , , , , .......... _------------------------------------------------------ - - -- - - -- - - --- - - - - - - - --- - - - - -

------------


-------

THE PARTICLE IMAGE YELOCIMETRY TECHNIQUE AS APPLIED TO W A YES 239

, ...... :::.... .... .,.. , , , ''::::=~~~ ~ , . • I \ , ,

~ ~::==::::::::::::~ I t 1 \ ; , I I t I \ , ~ , , • .. --~ I I I \ \ , .. , , , \ .. - :: -~ -, , I I I \ \ ~ .. .. .. ....... _----.",,. -, I I .. - - - :: --- , I I I I \ ~

.. .. .. ::::::==:::::::: , , I ~ - -- ~

, , , \ \ .. .. .. :: ------- - -, , -, I I . t ~ .. .. :: ------ - -, I . . , ---- - ------- - -: . , , - - -- -- - - - -- . , - - - - - - ---- - -


- - - - - .... - - - - .... - -- , I I , , ,

; I I I \ ,

~ : 11 I ,

. . ~

leo Figure 12: SWL Depth:50mmj Phase: 0.5

1~

>. ... 120 -., c: Q) 100 '0

>. 80 ... -..... :0 eo «I

,Q 0 ~ s.. /l,

20

0.000 0.010 0.020 0.030 0.040 O.~ 0.0«10

Displacement gradient

Figure 13: Probability distribution of velocity gradient for the data set


5.4 Errors and Limitations in the Experiments There are several problems which have arisen during recent experiments, many of them specific to the case of waves on beaches. These sub-divide into those associated with PlV, with illumination cost, and the flow field being studied. They can be summarised as follows:

1. Areas of high velocity variation in small water depths.

2. Dynamic range limitations.

3. Attenuation of the laser beam through the water below the beach.

4. Contrast.

5. Light losses due to beam deflection mirrors, the scanning apparatus, the tank walls, and the transparent plastic covering the beach.

6. Wave repeatability for intercomparison of different wave phases.

7. Deformation of the transparent plastic under wave attack.

5.4.1 Velocity Gradients.

It is possible to make an assessment of the systematic and random errors due to velocity gradients in the waves measured, given the general dependencies discussed in section 3.2.2. Figure 13 plots an overall probability distribution for the velocity gradients present in twelve representative flow fields measured in the study. By reference to the systematic errors discussed in section 3.2.2, and figure 3 for the random error, the expected systematic biasing and random variation of the results can be estimated for the whole data set. The bulk of the distribution is centered around 0.01, with associated errors of about 0.6% and 0.3%, respectively. However, it should be noted that in water waves the main areas of interest normally have the highest velocities and velocity gradients.

A particular velocity gradient probability distribution is plotted in figure 14 for one of the waves known to have especially high velocity gradients. In this case the systematic and random errors can be estimated to be about 3.0% and 2.0%, for the extreme displacement gradient of 0.05.

In this type of study the problem of velocity gradients becomes more severe as the water depth becomes smaller, if the photographic magnification remains constant. Experiments were attempted at a SWL depth of 30mm, but this length corresponds only 3 interrogation areas on the film. No reliable results were achieved at this depth, mainly due to the large velocity gradient in the flow, but the substantial amount of aeration in this post breaking region, also causes significant signal drop-out.

5.4.2 Dynamic Range.

The dynamic range of the PIV system also imposes a limitation on the technique. The system at Edinburgh has a dynamic range of '" 10. By using image shifting techniques, the measurable velocity range could be chosen to include zero velocities, with an accompanying increase in errors if the original maximum velocity was still to be resolvable. Image shifting systems are required for measurements where the direction

THE PARTICLE IMAGE YELOCIMETRY TECHNIQUE AS APPLIED TO WA YES 241

160

140

>. .... 120 .. c: Q) 100

" » 80 .... . --:0 80 <II .0 0 40 1-0

Po. 20

0.000

Figure 14: Probability distribution of velocity gradient for an extreme flow field

of the flow is not known a priori, ego for turbulence measurements past cylinders or through grids, etc. However, for the application to waves traveling along a wave flume the direction of the velocity vectors is aligned with the propagation direction under the crest of the wave, the direction of other velocities in the flow are determined by continuity. This type of study, therefore, is particularly well suited to the Young's fringe analysis system as the inherent directional ambiguity in taking the autocorrelation does not pose a problem.

5.4.3 Beam Attenuation through water.

The quality of the water greatly affects the attenuation of the laser beam and the contrast of the illuminated seeding particles. It was found that with clean water the beam intensity was reduced by 35% over the whole water depth of 750mm, but this attenuation increased to about 75% when the water quality was poor. This problem necessitates that the path length in water be minimised and the water to be changed regularly if laser costs are to be kept as low as possible. This will be an important factor in large facilities where water quality is not good, or the water can not be easily changed. However, the underlying requirement for successful PlY photographs is good contrast between the seeding particles and the background, and it is this contrast which decreases with a reduction in water quality.

5.4.4 Light losses from Laser to Wave Flume.

The photographic recording phase of the PlY process requires a high intensity, collimated light source. This is generally provided by a high power laser; in our experiments a CW 15W Argon ion laser is used. Light losses due to optical components in the beam path must be minimised if laser costs are to be kept down. The following reflection coefficients for the optical components in the beam path have been measured by an intensity meter.


Optical Component Reflection Coefficient Beam steering Mirror 95% Beam collimating Lenses 1% & 6% Rotating Octagonal Mirror 80% Parabolic Mirror 73% Tank Wall 80%

The light losses due to the rotating mirror, parabolic mirror and the walls of the tank are quite considerable. However, very little can be done to improve these due to their design. The rotating mirror's surface is diamond machined, not optically coated. Any optical coating would tend to dissociate itself from the octagon under high speed rotation. The parabolic mirror is made by optically coating a long, thin strip of perspex and then attaching this to a machined aluminium parabola. Optically coating perspex is not as successful as coating glass, and bending the perspex into a parabola will cause some small deformation of the reflecting surface. The losses incurred due to the tank walls are due, mainly, to the thickness of the glass. This thickness cannot be compromised due to mechanical loadings. The losses over the beam path mean that, in this case, the intensity of the beam is reduced from 15W to 4.2W at the bottom of the tank. Assuming good water conditions, the intensity of light falling on seeding particles at the crest of the wave will be '" 2. 7W. The transmission coefficient of the transparent plastic coating for the beach is good when clean. However, as the pollen used as seeding material is slightly less dense than the water, it tends to float, over a period of many hours. The seeding that passes under the beach, therefore tends to coat the underside of it. This greatly reduces the transmission factor of the transparent strip. As the beach is long, cleaning the underside of the strip is difficult. In general, the transparent plastic is replaced frequently.

5.4.5 Wave Repeatability.

The slope of the beach in these experiments was 1:30, which is relatively shallow. However, there will be some reflected wave components. Readings were initiated approximately 30s after the wavemaker was started. This was to allow a steady state wave climate to be established in the tank, bearing in mind the wavemaker is of the absorbing type. Measurements were taken at four phases of the wave, each measurement being separated by a period of 10.25s. To estimate the error due to wave repeatability a fifth phase was recorded at the SWL depth of 100mm. As the phase difference between each of the measurements is 0.25 cycles, the first and the fifth measurement should be identical. The maximum horizontal velocity component in each of these two waves differed by only 1.5%.

5.4.6 Deformation of Beach Covering.

The transparent plastic used to cover the gap in the beach necessary for optical access, is overhead projector transparency (in roll form). This is inexpensive, readily available and has good optical properties. It is, however, very flexible - a factor essential for its use on complex beach forms. On a plane slope this flexibility is not required, and is, in fact, undesirable. In the unsupported region that forms the gap up the middle


of the beach the plastic may deform. To minimise this effect it is carefully attached to the beach, with lengths of double sided adhesive tape, maintaining tension across the gap. The deflection should not introduce any significant effect to the laser beam's transmission, as the plastic is very thin, refraction effects appear to be negligible. By examining the wave profiles as they propagate along the beach, there does not seem to be any deformation in the wave front near the middle, suggesting that any deflection of the plastic is small enough not to affect the waves. The observation of how the wave fronts behave while breaking is a good indicator as to how level the beach is across the tank.

5.5 Summary of Errors and Uncertainties The errors and uncertainties identified in the application of PIV to the measurement of wave kinematics on sloping beaches are summarised in the following table. The percentage errors given refer to the typical errors relative to the maximum velocity measured in the particular flow field.

General Area Particular Factor Random Systematic Error Error

Fringe analysis Analysis calibration factor 0.1% Water wave studies illumination interval 0.2%

Photographic magnification 0.3% Photographic distortion 0.0%-0.3% illumination plane flatness 0.0%-0.3% illumination plane thickness 0.1% Scanning-beam time effect 0.0%-0.2% Seeding not following flow 0.1%

Waves on beaches Velocity gradients 0.3%-2.0% 0.6%-3.0% (PIV errors) TOTAL (PIV) 0.5%-2.0% 0.7%-3.9%

Waves on beaches Wave repeatability 1.5% (Other errors)

6 Conclusion

A critical analysis of the inherent errors involved in the PIV technique has been carried out and an overall estimate of their effect has been given.

The combined systematic and random errors relative to the maximum velocity in the flow, for PIV measurements of water waves varied mainly with particle displacement gradient across a small interrogation region on the negative. With a low displacement gradient of about 0.01 the combined error was found to vary from 1.2% to 2.0%, depending on the variation of further systematic errors, such as photographic distortion. With an extreme displacement gradient of 0.05, the combined error varied from 5.1% to 5.9%. These two displacement gradients correspond to typical and extreme cases


for the described study of waves on beaches, with the higher gradient occurring at shallower water depths near the breaking point. In addition to the errors associated with the PIV measurements there was an additional wave repeatability error of 1.5% between each recorded phase of the wave.

The techniques used here, namely the scanning beam illumination and the Young's fringe analysis, have been compared with other methods. The accuracy of the PIV technique has been verified, establishing its importance as a flow measuring tool.

PlY experiments have been carried out on a plane 1:30 slope and the results show clearly the success of this technique in measuring the velocities of breaking waves on plane beaches. The data should provide a new outlook for testing theoretical models and more accurate estimates of integral quantities [Longuet-Higgins, 1986, Stive, 1980, Stive and Wind, 1982].

7 Acknowledgments This work was undertaken as part of the MAST G6 Coastal Morphodynamics research programme. It was funded jointly by Hydraulics Research Ltd., and by the Commission of the European Communities Directorate General for Science, Research and Development under contract N°. MAST 0035C.

8 References 1. Adrian, R.J. (1988) Optical Methods for Measuring Vector Velocity Fields. Part

II - Techniques. von Karmen Inst. of Fluid Mechanics. Lecture series 1988-06.

2. Gray, C. (1989) The Development of PlY for Water Wave Studies. PhD thesis, Physics Department, The University of Edinburgh.

3. Gray, C. and Greated, C.A. (1988) A Scanning Laser Beam System for Twodimensional illumination of Flow Fields. Von Karmen Inst. of Fluid Mechanics Lecture series, 1988-06, pp 53-62.

4. Gray, C., Greated, C.A., McCluskey, D.R. and Easson, W.J. (1991) An Analysis of the Scanning Beam illumination System. 1991. Journal of Physics, Measurement Science.

5. Huntley, J.M. (1986) An Image Processing System for the Analysis of Speckle Photographs. J. Phys. E: Scientific Instrumentation, 19:43-49.

6. Longuet-Higgins M.S. (1975) Integral Properties of Periodic Gravity Waves of Finite Amplitude. Proc. R. Soc. London, A. 342,157-174.

7. Keane, R.D. and Adrian, R.D. (1990), Optimisation of Particle Image Velocimeters. Part I: Double Pulsed Systems. Meas. Sci. Technol. 1, 1202-1215.

8. McCluskey, D.R., Easson, W.J., Greated, C.A. and Glass, D.H. (1989) The Use of Particle Image Velocimetry to Study the Roping in Pneumatic Particle Conveyance. Particles, Particle Systems & Characterisation, 6.

9. Moraitis, C.S. (1988) Optical Processing, Proc. von Karmen Inst. for Fluid Dynamics, Lecture Series 1988-06, March 21-25 1988.


10. Salter, S.H. (1982) Absorbing Wave Makers and Wide Tanks. Proc. Conf. Directional Wave Spectra Applications, ASCE, 185-200.

11. Sinha, S.K. (1988) Improving the Accuracy and Resolution of Particle Image or Laser Speckle Velocimetry. Experiments in Fluids, 6:67-68.

12. Stive, M.J .F. (1980) Velocity and Pressure Field of Spilling Breakers. Proc. 17th Int. Conf. Coastal Eng., ASCE, pp 547-566

13. Stive, M.J.F. and Wind, H.G. (1982) A Study of Radiation Stress and Set-up in the Nearshore Region. Coastal Eng. 6:1-25.

A COLORED METHOD FOR P.I.V TECHNIQUE

J. STEFANINI- G. COGNET-J.C. VILA- B. MERITE- Y. BRENIER

ABSTRACT:

Since the last decade, the P.l.V technique has received an increasing amount of attention. With the progress of the c.c.O and video camera, and the increase of the speed and the capacity of micro-computer, it is possible now to obtain correctly and more rapidly the velocity field of a laminar or turbulent flow.

In this paper, we propose a new method based on a colored technique. Three pulses of differents colours (red, green and blue) are provided with an adjustable time delay. The first pulse is coming from a pulsed laser (rubis) and the others are given by a continous laser (multi-waves argon). The colours separation and modulation of the continuous laser beam are carried out by dichroic plate, filters and electro-optics shutters. Other dichroics plates allow to obtain the differents beams on the same axis. A cylindrical lens creates a laser sheet and a three exposure photograph records the displacement of the particles.

The main advantage of this technique is an easier method to analyse the images. The image of the particles is recorded with a video camera in the computer memory and an image processing software converts the information in three files containing respectively the red, green and blue particles. After a detection and a measure of the characteristics (centre of gravity, diameter, number of pixels) of the particles in each file, a programm developped at our laboratory gives us the velocity fields and the flow direction. Nevertheless, we have had some difficulties to find a film wich has a sufficient reponse for the three wavelengths.

After describing the experimental procedure, some results are presented on a water mock-up of an impinging jet upon a circular flat wall.

247

F. T. M. Nieuwstadt (ed.). Flow Visualization and Image Analysis. 247-258. © 1993 Kluwer Academic Publishers.

248 J. STEFANINI ET AL.

EXPERIMENTAL APPARATUS

dichroic plate dichroic plate

~,.---_+!-__ -I .!!REEr;f ___ ~ __ _

c'-o-n-:-tin-u-o-u-s -w-a--lve t I I

argon laser

polarizer I I

Pockels RUBIS cell ROD

I I I RED

I ·11l"----m-j-rr-o-+r ~ - - - - -#'

'-- RUBIS LASER ~

.,f!o;;-----i He - Ne laser alignment laser

EXPERIMENTAL APPARATUS

dichroic plate

/ig.!

electro optic GREE~ ).

(~~~ ~;,)' p----+D+--------i~ - 1-- - -/fA G B continuous wave polarizer I I

I I BLUE~ I RED

I I

RUBIS OSCILLATOR ----#' .,'lk-----~ He - Ne laser

alignment laser

mirror

/ig.2

A COLORED METHOD FOR PlV TECHNIQUE 249

INTRODUCTION :

Since the last decade, the P.I.V has received an increasing amount of attention. With the progress of the CCD and video camera, and the increase of the speed and the capacity of micro-computer, it is possible to obtain correctly and more rapidely the velocity field of a laminar or turbulent flow.

In this paper we propose a new method based on a colored technique. Three pulses of different colors (red blue and green) are provided and a photograph records the displacement of the particles. The main advantage of this technique consists in an easier method to analyse the images, mainly to obtain the flow direction (no shifting technique ref. 1 ).

After describing the experimental apparatus, the image processing and the problems that we have had, we will present the first results.

EXPERIMENTAL APPARATUS:

To develop this method, two different optical systems have been tested. Schematic diagrams of the two configurations are shown in figures 1 and 2.

The three pulses of differents colors (red, blue and green) are delivered with an adjustable time delay. The first pulse is coming from a pulsed laser (rubis-694 nm) ; the pulse lasts 20 ns and its energy is 300 mJ. In the first system, the two other pulses are coming from a continuous laser (multiwaves argon) ; the separation of the blue and green colors being carried out by a dichroic plate. With the second system, the green and blue beams are coming from two single wave argon laser (514 nm and 488 nm).

In these two configurations, the continuous beams are modulated by an electro-optic light valves, crossed vith polarizers. When we apply a negative voltage (-15 V) to the valve, the light which is lineary polarized passes through the ferroelectric liquid crystal without changing its polarity and is blocked by the polarizer. If the voltage is + 15 V, the light's polarization plane is changed by 90 degrees and the light then passes through the polarizer. We can produce pulses of varying duration.

By using mirrors and dichroic plates, we can obtain the three beams on the same axis. A cylindrical lens creates a laser sheet and a three exposure photograph records the displacement of the particles.

To take the photograph, we use a nikon camera with a medical lens. The field of analysis is 3,6 X 2,4 cm. The choice of the film has been very difficult. We have had many difficulties to find a film which has a sufficient response for the three wavelengths. Three different films have been tested; the KODAK EKTACHROME 800-1600 (developed at 3200) which is a color reversal film and the KODAK EKTAPRESS 1600 AND EKTAR 1000 which are color negative films. Up to now, the choice of the film is not definitively settled, because the negative films have a good sensitivity but a bad response to the rubis wavelength and the positive film presents a better response to 694 nm but a lower sensitivity on all the spectrum width.


ELECTRONIC TRIGGER DIAGRAM

.------ -------------, I I I manual command L _____ _

'--___ --:-___ delay 170 ms Jcamera opening

__ ---.JI4-. --; 1--__ camera command delay suited to NIKON 501

,11.20 ns

--------~n~----------~ Il-----

--------------~ 60 j..Is ~ t:.t ~ 660 j..Is

red pulse

blue pulse

green pulse

IMAGE PROCESSING

picture

R.G.B video camera SONY 3.C.C.D

computer MACINTOSH IIx

fig. 3

Iig.4

A COLORED METHOD FOR PIV TECHNIQUE 251

To synchronize the opening of the camera with the three pulses, we have developped at our laboratory an electronic trigger. A schematic diagram of this device is shown in figure 3.

Another problem that we met was the extinction coefficient of the light valves. The too low ratio coefficient (100/1) generates a light noise which exposes the film. Because of this bad contrast, it was necessary to optimize the electronic device in order to open the camera only during the three pulses.

IMAGE PROCESSING :

The image processing system includes three stages:

- image recording

- image treatment

- velocity calculation

Image recording:

To record the image in the computer memory, we use as it is shown in figure 4, a slide projector (the intensity of light is variable), a screen to project the images and a R.G.B video camera which transfers through three different outputs (corresponding to the three colors) the informations contained in the red, blue and green planes. The camera used is a CCD SONY DXC 325K and the resolution is 500 X 582 pixels.

The computer is a MACINTOSH 2X, composed of :

a ram of 8 M. Bytes a processor of 16 MHz a digitizing card IMAGE GRABBER (digitizing with 8 bits of precision

(256 grey levels) in real time (1/30 s) and a resolution of 768 X 512 pixels).

More, an image printer (CANON FP510) is connected to the computer.

Image treatment:

The objective of the image processing stricly speaking, is to obtain from the photograph three different files containing respectively the coordinates of the red, blue and green spots of the moving particles.

We use the image processing software (OPTILAB) which has been developped by a FRENCH company GRAFTEK on MACINTOSH computer. The image coding is realized on 8 bits.


TRIPLET RESEARCH PROGRAM (ALGORITHM)

VARIABLE : EC - maximum distance beetwen red and blue point ER - maximum displacement error (turbulence)

RED POINT RESEARCH end n° i END of file (XR YR) i = i+1 • CALCULATION OF THE RESEARCH

LIMITS IN THE BLUE FILE [YR-EC, YR+EC1

+ BLUE POINT RESEARCH not

IN THESE LIMITS found

~ found

CALCULATION OF THE PRESUMED COORDINATES OF THE GREEN POINT XG = (XB-XR) + XB YG = (YB-YR) + YB

+ found GREEN POINT RESEARCH

AT THESE COORDINATES IN THE GREEN FILE

~ not found TRIPLET

~ CALCULATION OF THE RESEARCH FOUND LIMITS IN THE GREEN FILE

[YG-ER, YG+ER]

• GREEN POINT RESEARCH not found IN THESE LIMITS found

/ig.5

A COLORED METHOD FOR PlY TECHNIQUE 253

Each of the three different recorded images corresponding to the red, blue and green planes contains, in fact, more speckles than the corresponding colored spots. That is due to the fact that the blue wavelength is closed to the green one. The difference between the blue and the green spots can be seen as a difference of intensity. To obtain an image containing only the blue particles, we subtract the green plane from the blue plane. Then, by using offset and filter functions we eliminate the noise and we obtain a binary image containing the blue spots. Then a "DETECTION" command allows to detect and measure the characteristics of each spot (centre of gravity, diameter, .. .) and to save these informations in an ASCII file. For the green image, we subtract the blue plane from the green one and we apply the same treatments as for the blue spots. For the red image, we have to use the same procedure because sometimes the intensity of light scattered by a large particle is very high and the corresponding spots are nearly white.

Thereafter the three ASCII files are changed in text files by using EXCEL, which is a data processing software. This manual treatment allows to save only the characteristics which are interesting (centre of gravity coordinates). Therefore, we have three text files containing respectively the coordinates of the red, blue and green spots of each particles.

Velocity calculation:

The third stage of the treatment is the velocities field calculation. We have developped at our laboratory a triplet research software in PASCAL language. An algorithm of this software is given in figure 5.

From the three files (red, blue and green) we create three other files which will be sorted according to the increasing values of V (the centre of gravity is known by its coordinates X and V). The blue and green files are built as following. For each V varying from 1 to 512, there is a record which will contain the location of the particle and the number of particles having the same y. If there is no "V" having corresponding point, the position = 513 and the number = O.

The research is based on the red point (first point of the triplet) and is realised knowing a maximum distance (EC) between the red and blue point and a maximun displacement error (ER) (due to the turbulence). These two parameters involve an assessment of the maximum values of the mean velocity and of the intensity of turbulence. From the coordinates (XR, VR) of the red point, we calculate the limits of the research in the blue file :

VMAX = VR+EC VMIN = VR-EC

XMAX = XR+EC XMIN = XR-EC

and we research a blue point in this fieldWhen a point (XB, VB) is founded, we calculate the assumed coordinates of a green point

XG = (XB-XR)+XB VG = (VB-VR)+VB

254 J. STEFANiNI ET AL.

MOCK-UP fig. 6


If there is no point in the green file at these coordinates, we calculate the limits of the research in the green file

YMAX = YG+ER XMAX = XG+ER YMIN = YG-ER XMIN = XG-ER

and we made a research as we do in the blue file,

When the three points are founded, their coordinates are saved and the points are withdrawn, If it is impossible to find three points in the allowed field, the red point is considered as alone and withdrawn, The research stops when all the red points are withdrawn, Then, we can draw the velocities field,

RESULTS:

The results we present are obtained on a water mock-up of an impinging jet upon a circular flat wall. A photograph of the mock-up is shown in figure 6,

The experiments have been carried out for 1 m/s mean velocity at the outlet of the jet, The pulses duration are 300 s for the green and blue colors and 20 ns for the red color and the corresponding energies are 300 J (B and G) and 300 mJ (red),

The film used is the color reversal film EKTACHROME 800-1600 ASA, which is developped at 3200 ASA.

Two exposure photographs are shown in figure 7a and 8a and the velocities fields associated in figure 7b and 8b, We can see that the greatest part of the triplets (15/18) has been identified by our research software in spite of the image quality which is not very good,

In the two examples presented (aproximatively 40 points in each color plane and 18 triplets), the time treatment is of 13 minutes which are splited in :

- Image record (three planes) : 3 minutes (manual treatment) - Image subtraction: 15 seconds/image, doesn't depend on the number of spots - Offset and filter functions : 10 seconds/image, independant of the number of spots - Particle detection: 4 seconds/image, for 40 spots - Saving file : 5 seconds/file - File transformation (EXCEL) : 2 minutes/file (manual treatment) - Image reconstitution : 5 seconds - Velocities field calculation: 10 seconds - Two minutes to open and close the softwares (OPTILAB, EXCEL, PASCAL)

We can estimate that the time processing would be about 2 minutes for such images when all the process will be automatized,


TRIPLETS PHOTOGRAPH (;g.7a


VELOCITIES FIELD fig.7b


TRIPLETS PHOTOGRAPH fig8a


VELOCITIES FIELD fig.8b


CONCLUSION :

These first results are interesting. However, progress are necessary to improve the image quality .The use of pulsed lasers to produce the blue and green pulses will allow to get higher energy pulses than with a continuous laser and to eliminate the light noise. The green pulse can be obtained with a Nd-Yag laser (with an harmonic generator) and the blue pulse, possibly with a dye laser.

Another improvement is the image processing automation. The first results obtained show an image processing which is easy and repetitive (image subtraction, offset, filtering and detection), and that must allow to automize this method,

The main advantage of this technique is the time necessary to obtain the flow map, which could be about 2 minutes for an image, More, with this color technique, we have the flow direction without using shifting technique.

On the other hand, the main drawback of this method is a laser system more important than usual (use of two or three lasers, optic adjustements more difficult), However, the problem can be solved as I said previously by using pulsed lasers, Perhaps, we can suppose that if this method or another colored method allows an easy way of image processing, the specialised company will propose a special three colored pulsed laser,

REFERENCES:

< 1> R.J.ADRIAN Image shifting technique to resolve directional ambiguity in double-pulsed velocimetry. Applied optics, vol. 25, No 21, novembre 1986

Digital PIV applied to flows around artificial heart valves: analysis by autocorrelation.

A.K. Hind and J.R.E. Christy

Department of Chemical Engineering University of Edinburgh

Edinburgh EH9 3JL

SCOTLAND

Received 17 October 1991; accepted in revised form 23 March 1992

Keywords: PlY, digital, autocorrelation, heart, valves

Abstract

A fast, (0.5-2 seconds per interrogation point), direct autocorrelation analysis tech-

nique for particle image velocimetry is being developed to produce velocity and vortic-

ity maps at, and downstream of, tilting disc, artificial heart valves in a pulsatile flow.

An enlarged, multiply exposed photograph of the seeded flow is digitised, the resulting

digital data compressed and an autocorrelation function calculated on subsets of the

data with a carefully formulated algorithm. This approach allows adaptive variation

of the interrogation area to give the maximum dynamic range possible. A flat bed

scanner is used to digitise the whole photograph with a resolution approximately half

that of the recording film: the cost is significantly less than that involved in using a

CCD array. Data compaction reduces storage requirements and markedly increases

computational speed. The calculation algorithm, details of digitisation and prelim-

inary results are given. Little modification of the algorithm is required to perform

cross-correlations.

259


260 A. K. HIND

1 Introduction

In the design of prosthetic heart valves for human implantation it is important to know where areas

of high shear and stasis occur as they may lead to clot formation[ll and deposition[2l. PIV is an ideal

technique to provide such information: it is non-intrusive and should yield sufficiently accurate

velocity vectors throughout any 2D plane of illumination to allow derived quantities (principally

out of plane vorticity and in plane shear-stress) to be calculated[3l.

A pulsatile flow of approximately 1.2 litres/minute at a frequency of 70 beats per minute

around a Bjork Shiley tilting disc, artificial heart valve mounted in a 32 mm i.d. Perspex pipe is

to be investigated. The beam from a 35 mW He-Ne laser, chopped and expanded to form a thin

rectangular light sheet, illuminates a plane of the water flow, seeded with 50J.lm pollen particles,

within the cylinder. The flow is viewed through the Perspex pipe mounted inside a sealed, square,

Perspex duct filled with water: ray-tracing shows this to produce negligible distortion of the virtual

image seen by the camera, although this requires experimental confirmation and some wall flare

does remain. A series of multiply-exposed photographs taken during the pulsation cycle, (up to

6 per second), is to be used to provide information on the time variation of velocity and derived

quantities over the illumination plane. The pulsatile nature of the flow, along with the presence

of the tilting disc, leads to complex, rapidly evolving structures with a wide range of velocities

and appreciable velocity gradients. These structures are difficult to measure by other velocimetry

techniques.

As no existing image analysis system was readily available the option of using either optical

or digital analysis was assessed. A network of SUN SPARC stations, including an HP9000s300

and a Meiko computing surface, was accessible providing a substantial computing base. Thus if

a simple, cheap means of transferring the coded velocity information from the photographic film

to the computer system were available then a digital analysis system would appear attractive.

Rather than buy a 2 dimensional stepping system and CCD camera to optically generate Young's

fringe patterns or magnify particular interrogation points it is easier and cheaper to obtain a flat

bed scanner, compatible with the existing network, with which to digitise the photographic image.

Therefore a digital approach to the analysis of PIV images has been developed in this work.

DIGITAL PIV APPLIED TO FLOWS AROUND ARTIFICIAL HEART VALVES 261

Further, a direct digital analysis system allows for adaptive variation of the interrogation area

to optimise spacial resolution and to achieve a wide dynamic range. Noise can be reduced before

processing by a selection of digital preprocessing steps. There is also potential to select particle

images from a given intensity range within the whole digitised image so as to analyse, without

interference between signals, images from both phases of a two phase How.

An autocorrelation algorithm has been developed for the analysis of the digitised PlY images.

There seems to have been little development of such algorithms in the literature, although once

developed would be readily adaptable for applications where cross-correlations are required. This

is presumably due to the popularity of the Digital Fast Fourier Transform (FFT) analysis method

which for any given situation would be expected to perform faster than direct calculation of the

autocorrelation function[41. The 'time factor' becomes of decreasing importance as available com

puting power increases. Indeed, if the image is transformed to a black and white representation,

removing all grey scale information, it is possible to speed up the autocorrelation to that of the

FFT analysis system with limited reduction in accuracy.

Direct calculation of the autocorrelation function has several advantages over the FFT system.

There is no need to use interrogation areas whose linear dimensions in terms of numbers of pixels

amount to an integer power of 2, noise from the repeating nature of the FFT is absent, there is no

diffraction halo and peaks within the calculated autocorrelation function are sharper[5). To avoid

noise from 'drop out' from image sequences crossing the edge of the interrogation area in the plane

of analysis, the area can be convolved with a slightly larger area. This latter advantage is more

important for low image density PlY recordings.

2 Digitisation of PIV Images

The multiply exposed PlY image in the form of a 35 mm negative is enlarged to produce an A4 sized

photographic print. For an image covering 30 mm x 30 mm on the negative this is approximately a

seven fold linear enlargement; for images produced from the heart valve flow investigation roughly

a ten fold enlargement can be used. A Hewlet Packard, Scanjet Plus flatbed-scanner is used to

digitise the A4 print to a resolution of 300 dots per inch (that of a laser printer) with 256 grey scale

262 A. K. HIND

levels for each dot (see [6] for a similar approach with regard to cross-correlation). This produces

an A4 image with a resolution of approximately 2500 dots by 3500 dots.

A seven fold enlargement of a PIV negative gives a digitised image with each digitisation point

representing 12.1 J1-m on the negative: 8.5 J1-m for a ten fold enlargement. Depending on the aspect

ratio of the image on the negative in relation to that of A4 paper an equivalent film resolution

of 40-60 lines/mm can be achieved. This is at least comparable to the resolution of black and

white films used in other work (Kodak Technical Pan 320 lines/mm[7], T-Max 125 lines/mm[81)

and represents a loss of half the resolution of the film used in this particular study.

Currently the digitisation resolution is better than that of high resolution CCD cameras (2048

* 2048) and the equipment required is more than an order of magnitude cheaper. The need for

photographic processing stages before analysis is however increased compared with the optical FT

analysis of the negative and can not hope to achieve the rapid image digitisation of a CCD system.

The rate at which images can be recorded using a camera with a motor-wind is equivalent to that

of a high resolution CCD. However it is possible to trigger the camera to record data at a set

point in the pulsation cycle. The digitisation procedure adopted here still retains the potential for

optical analysis of negatives if required.

Once digitised the images can easily be adapted for analysis. A range of algorithms are available

to perform any convolution desired on the image; rotation, scaling, selecting particular areas of

the image, contrast enhancement, particle edge detection, etc.. Initially the images, containing

full grey scale information at every point, are typically 5-8 MBytes in size. The size of the data

file is reduced in speeding up its analysis. After reduction to a black and white representation, by

applying a suitable threshold, the data file can be decreased to around 600 KBytes. Using general

data compaction algorithms it is possible to reduce the image file size to around 50 KBytes.

Digitisation itself takes around 1 minute, and with all processing stages applied a black and

white representation of the enlarged photograph is produced and stored on disk in its most compact

form in 2 minutes. The accuracy of the enlargement and digitisation procedure has not yet been

investigated.

DIGITAL PIV APPLIED TO FLOWS AROUND ARTIFICIAL HEART VAL YES 263

3 Digital Autocorrelation

Using the unbiased correlation estimator for an M by N point interrogation area[41, (equation 1),

M-l-kN-l-h

R~,,(k,h)= M~N L L F{m+k,n+h}.F{m,n} m=O n=O

(1)

for k ~ 0 and h ~ 0

approximately (N(N+1) M(M+1»/2 complex calculations are required to calculate all values of

R~" up to R~,,(M-l, N-l). This is only valid if the 256 grey levels in the digitised image are

retained. If the grey levels are treated as being either black or white by applying a threshold then

these calculations are greatly simplified. After applying this threshold two grey levels are produced

that can be stored as 0 or 1: integer multiplication and addition at each step can be replaced by

an integer AND, and an integer addition. This gives quite a marked improvement in calculation

speed. Further, 32 such pixel representations can be held in a typical machine word, a single

integer AND can then effectively perform 32 of the above operations in parallel, with additions

only being performed if the result is non-zero. With judicious algorithm formulation this gives

further increases in speed of calculation.

3.1 Procedure

By the above procedure of application of a grey threshold and compression of digits (0 and 1) into

32 bit words, an interrogation area of M by N, 8 bit integers from a 256 grey scale digitisation

becomes M by N/32, 32 bit integers. Forming an array A with 2M rows and N/32 columns

containing two copies of the M by N image area (Le. rows 1 to M == rows M+1 to 2M) and an

array B with M rows and N/32 columns containing one copy of this image, then the following

procedure will produce the autocorrelation function estimator (ACF) given by equation (1):

• word 1 of row 1 of array A is ANDed with word 1 of row 1 of array B, if the result is non-zero

then the word is shifted left until it is zero, all the bits set in the word are counted as they

reach the high end of the word.

• this is repeated for all other corresponding words in row 1 of both arrays.

264 A.K.HIND

• this is performed for the first M rows of array A.

The summation of all the set bits counted above gives the value for the ACF at point (0,0).

• the same procedure is followed to AND rows 2 to M + 1 in A with rows 1 to Min B respectively.

- the summation of set bits for the first M-l rows of ANDed words gives the ACF at point

(-1,0)

- the summation of the set bits in the last row of ANDed words gives the value of the

ACF at point (M-l,O).

• the l+x to M+x rows of A ANDed with B yield, on summation of the number of bits set

in the ANDs of the first M-x rows the ACF at the point (-x,O ) and the summation of the

number of bits set in the ANDs performed on the remainder of the rows gives the ACF at

point (M-x, 0)

• after performing the calculation for all x up to x = M-l all the words in B are shifted one

bit to the left, carrying those bits on the left of a word to the next word on the left and

introducing zeros to the the right bits of the words on the far right.

• the process is repeated to give the ACF at points (M-I , 1), ... , (I-M, 1). After shifting the

bits in the rows of B another bit to the left the ACF at points (M-I, 2), ... , (I-M, 2) can be

calculated. This is repeated until the ACF at points (M-I, N), ... , (I-M, N) are determined.

• The ACF at the remaining points can be found from those already calculated by the functions

inherent 2-fold rotational symmetry about point (0, 0).

Approximate peak positions can be found in the ACF and the position of the highest non

central peak determined by interpolation or function fitting. The ratios of the height of this peak

to that of the central peak and to that of the second highest peak found, along with the existence

of peaks in the ACF at multiples of the displacement (along a straight line through the origin) of

the detected peak are used to give an estimate of the reliability of the velocity vector.

The method as described so far results in directional ambiguity of the velocity vectors, since

the flows being analysed are complex and rapidly changing with time this would pose a significant

DIGITAL PIV APPLIED TO FLOWS AROUND ARTIFICIAL HEART VALVES 265

problem in the interpretation of the velocity data. It is hoped that introducing a much shorter,

tagging pulse of light into the illumination sequence will not affect the calculation procedure, but,

after the velocity magnitude and direction have been found, will allow the sign ambiguity of the

velocity vectors to be removed. The interrogation area can be searched along lines corresponding

to the direction of the velocity vector to find the tagging image: this will give additional insurance

that the right correlation peak has been used in determining the velocity.

3.2 Computational Performance

Table 1 gives average execution times for various methods of calculating the autocorrelation for

the same interrogation area samples. The direct calculation methods were written entirely in C,

the FFT version using a NAG library routine. These times are without any specialised array

processing hardware. It is clear that digitisation of the image with subsequent data compaction

does permit analysis in comparable times to that of the FFT. The algorithm is also well suited

to be performed with scattered spacial decomposition on a parallel computing architecture where

further speed improvements would be expected.

The nature of the autocorrelation algorithm means that it executes faster for sparsely seeded

interrogation areas ( i.e. where images take up only a small fraction of the interrogation area). For

the image sizes and image densities achieved in initial experiments this would appear to be the

case in our system under present seeding densities. For much denser images the execution time

would be expected to increase while that of the FFT remains constant.

Table 1: Average execution times for various calculation methods to produce the autocorrelation function from a 64 by 64 pixel interrogation area

II Calculation method I Execution time(seconds)

256 grey levels retained in ACF 90 2 grey levels held as 0/1 in integers 16 2 grey levels, 32 held in one integer: - general M by N ACF routine 1.6 - dedicated 64 by 64 routine 0.41

Fast Fourier Transform retaining grey levels 1.7

266 A. K. HIND

3.3 Peak Position to Sub-pixel Accuracy

The autocorrelation function (ACF) of a single Gaussian peak is roughly Gaussian, the position

of the peak can be determined to sub-pixel level by fitting an appropriate function[6] or by some

centroiding process applied to the peak[5]. After applying a threshold, the ACF becomes that of a

single black circle given by equation (2), where r is the radius of the circle and h the offset of the

autocorrelating circle centre from that of the original as shown in figure 1.

Figure 1: Image and autocorrelation image defining hand r

h

(2)

For a series of four such circles of radius 10 units and separation of centres of 30 units the ideal

ACF would be that shown in Figure 2.

The form of the ACF allows linear interpolation, the fitting of straight lines by least squares,

or direct fit of the theoretical function, to be used in order to obtain peak positions to sub pixel

levels. D is defined (Equation (3» to be the image diameter in pixels and must exceed one to

ensure that peaks are well resolved.

D = image diameter on photograph pixel width

(3)

Peaks remain sharp for small image point separations and may be resolved when images are

almost overlapping. The accuracy of peak location should be approximately that given by Equation

( 4),

DIGITAL PIV APPLIED TO FLOWS AROUND ARTIFICIAL HEART VAL YES 267

Figure 2: Graph of ideal autocorrelation function (in pixels) for four dots of diameter 20 pixels with centres separated by 30 pixels as a function of displacement as a multiple of pixel length.

1200

1000

! 800

2" 600

'iJ 400

200

20 40 60 80 100 120

h (linea pixel lJ'lits)

Error in pixels = ± 1 J(DN(P -1))

(4)

where D is defined in equation (3), N the number of separate image sequences and P the number

of exposures in a sequence. This is valid if each measurement of the diameter in a given direction

of an image circle gives an independent estimate of its centre along that direction. This expression

for the error in displacement takes no account of the fact that given the ideal ACF function straight

lines can be fitted to the data at either side of the peak that may give better accuracy of the ACF

about peak centres. For an interrogation area of 64 * 64, containing two, double pulsed images of

seeding particles of diameter 8 pixels, with centres separated by 30 pixels, an error of around 1%

would be expected.

4 Results

It has been shown that autocorrelation or cross-correlation estimators can be calculated directly,

after data has been compacted, at comparable speeds to the conventional FFT calculation methods.

There is some reduction in accuracy, although for larger image sizes loss of accuracy will be reduced

268 A.K.HIND

Figure 3: Entry profile to heart valve test section (Re = 800): PlY image

and will become of less importance as digitisation resolution increases. Whilst accuracy may be

reduced the location of the sharp peaks of the 2 grey scale ACF is much easier to determine than

that of the 256 grey scale ACF. Analysis of images including a 'tagging' dots has shown that these

have little effect on the results, although the algorithm for directional ambiguity removal has still

to be implemented.

Sample results are given for the laminar entry profile to the test section for the heart valve

investigation (Re = 800) Figures 3 and 4, and for the particle phase of a two phase flow, Figures

5 and 6. These are principally to demonstrate the effectiveness of the analysis technique. Future

work will concentrate on more complex flows.

5 Conclusions

A simple, effective, and cheap (provided reasonable computing resources are available) PlY in

terrogation system has been produced. Direct calculation of the autocorrelation function, after

image compaction, has been shown to be of comparable speed to that of the more common FFT

DIGITAL PlY APPLIED TO FLOWS AROUND ARTIFICIAL HEART YAL YES 269

approach although there is some limited loss of accuracy. It has other advantages in removing

the diffraction halo and noise peaks from the repeating nature of the interrogation area seen by

the FFT method. Interrogation areas of any size can be used to maximise spacial resolution and

dynamic range. As computing power increases the algorithm required for this approach can be

simplified and the emphasis on speed will change to a fuller assessment of its performance.

Figure 4: Entry profile to heart valve test section (Re = 800): Velocity vectors

25r-----~------~------._-----.------~

20 ._----- ._-._--- ._---- ._---- ._-- .

. ---- .--------------15 .-. 4 •• .---.--.-- .- .---

.--...-. ---- .----'----._-------------------._------------ ._-----...---10 .-----------------------._---------- .- .---------

,- ._--------- ._------ .-5

OL-----~----~------~----~----~ o 5 ill ~ W 2S

Figure 5: Particle trajectories from a nozzle in rope dispersion investigations: PIV image. Courtesy of Denise McCluskey, Dept. of Mechanical Engineering, University of Edinburgh

270 A.K.HIND

Figure 6: Particle trajectories from a nozzle in rope dispersion investigations: Velocity vectors. Courtesy of Denise McCluskey, Dept. of Mechanical Engineering, University of Edinburgh

10~----~----~----~------~----~-----,

9 .. //.

2

1

OL-____ ~ ____ ~ ____ L_ ____ L_ ____ L_ __ ~

o 5 10 15 20 25 30

DIGITAL PIV APPLIED TO FLOWS AROUND ARTIFICIAL HEART VAL YES

References

271

1. Morton, W.A.; Parmentier, E.M.; Petschek, H.E.: Study of aggregate formation in region of

separated flow. Thrombos. Diathes. Haemorrh. (Stuttg) 34 (1975) 840-854.

2. Hellums, J.D.; Brown, C.H.: Blood cell damage by mechanical causes. In: Hwang, N.H.C.;

Normann, N.A. (eds.), Cardiovascular flow dynamics and measurements. Univ. Park Press

(1975).

3. Adrian, R.J.: Multi-point optical measurements of simultaneous vectors in unsteady flow - a

review. Int. J. of Heat and Fluid Flow 2 (1986) 127-145.

4. Cedenese, A.; Paglialunga, A.: Digital direct analysis of a multi-exposed photograph in PIV.

Exp. Fluids 8 (1990) 273-280.

5. Keane, R.D.; Adrain, R.J.: Optimization of particle image velocimeters. Part I: double

pulsed systems. Meas. Sci. Technol. 1 (1990) 1202-1215.

6. Utami, T.; Blackwelder, R.F.j Ueno, T.: A cross-correlation technique for velocity field

extraction from particle visualisation. Exp. Fluids 10 (1991) 213-223.

7. 'Kodak': 'Kodak' technical pan films. 'Kodak' publication number P-B55.

8. 'Kodak': Technical data on 'Kodak' black-and-white professional films. 'Kodak' publication

number F-32(H).

Mechanics

FLUID MECHANICS AND ITS APPLICATIONS

Series Editor: R. Moreau

Aims and Scope a/the Series The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modelling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advancement. Fluids have the ability to transport matter and its properties as well as transmit force, therefore fluid mechanics is a subject that is particularly open to cross fertilisation with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains.

1. M. Lesieur: Turbulence in Fluids. 2nd rev. ed., 1990 ISBN 0-7923-0645-7 2. O. Metais and M. Lesieur (eds.): Turbulence and Coherent Structures. 1991

ISBN 0-7923-0646-5 3. R. Moreau: Magnetohydrodynamics. 1990 ISBN 0-7923-0937-5 4. E. Coustols (ed.): Turbulence Control by Passive Means. 1990 ISBN 0-7923-1020-9 5. A.A. Borissov (ed.): Dynamic Structure oj Detonation in Gaseous and Dispersed Media. 1991

ISBN 0-7923-1340-2 6. K.-S. Choi (ed.): Recent Developments in Turbulence Management. 1991

ISBN 0-7923-1477-8 7. E.P. Evans and B. Coulbeck (eds.): Pipeline Systems. 1992 ISBN 0-7923-1668-1 8. B. Nau (ed.): Fluid Sealing. 1992 ISBN 0-7923-1669-X 9. T.K.S. Murthy (ed.): Computational Methods in Hypersonic Aerodynamics. 1992

ISBN 0-7923-1673-8 10. R. King (ed.): Fluid Mechanics oj Mixing. Modelling, Operations and Experimental Tech-

niques.1992 ISBN 0-7923-1720-3 11. Z. Han and X. Yin: Shock Dynamics. 1992 ISBN 0-7923-1746-7 12. L. Svarovsky and M.T. Thew (eds.): Hydroclones. Analysis and Applications. (1992, in prep.)

ISBN 0-7923-1876-5 13. A. Lichtarowicz (ed.): Jet Cutting Technology. (1992, in prep.) ISBN 0-7923-1979-6 14. F.T.M. Nieuwstadt (ed.): Flow Visualization and Image Analysis. 1993 ISBN 0-7923-1994-X

Kluwer Academic Publishers - Dordrecht / Boston / London

Mechanics

SOLID MECHANICS AND ITS APPLICATIONS

Series Editor: G.M.L. Gladwell

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design.

1. R.T. Haftka, Z. GUrdal and M.P. Kamat: Elements of Structural Optimization. 2nd rev.ed., 1990 ISBN 0-7923-0608-2

2. J.J. Kalker: Three-Dimensional Elastic Bodies in Rolling Contact. 1990 ISBN 0-7923-0712-7

3. P. Karasudhi: Foundations of Solid Mechanics. 1991 ISBN 0-7923-0772-0 4. N. Kikuchi: Computational Methods in Contact Mechanics. (forthcoming)

ISBN 0-7923-0773-9 5. Y.K. Cheung and A.Y.T. Leung: Finite Element Methods in Dynamics. 1991

ISBN 0-7923-1313-5 6. J.F. Doyle: Static and Dynamic Analysis of Structures. With an Emphasis on Mechanics and

Computer Matrix Methods. 1991 ISBN 0-7923-1124-8; Pb 0-7923-1208-2 7. 0.0. Ochoa and J.N. Reddy: Finite Element Analysis of Composite Laminates.

ISBN 0-7923-1125-6 8. M.H. Aliabadi and D.P. Rooke: Numerical Fracture Mechanics. ISBN 0-7923-1175-2 9. J. Angeles and C.S. L6pez-Cajun: Optimization of Cam Mechanisms. 1991

ISBN 0-7923-1355-0 10. D.E. Grierson, A. Franchi and P. Riva: Progress in Structural Engineering. 1991

ISBN 0-7923-1396-8 11. R.T. Haftka and Z. GUrdal: Elements of Structural Optimization. 3rd rev. and expo ed. 1992

ISBN 0-7923-1504-9; Pb 0-7923-1505-7 12. J.R. Barber: Elasticity. 1992 ISBN 0-7923-1609-6; Pb 0-7923-161O-X 13. H.S. Tzou and G.L. Anderson (eds.): Intelligent Structural Systems. 1992

ISBN 0-7923-1920-6 14. E.E. Gdoutos: Fracture Mechanics. An Introduction ISBN 0-7923-1932-X 15. J.P. Ward: Solid Mechanics. An Introduction. 1992 ISBN 0-7923-1949-4 16. M. Farshad: Design and Analysis of Shell Structures. 1992 ISBN 0-7923-1950-8 17. H. S. Tzou and T. Fukuda (eds.): Precision Sensors, Actuators and Systems. 1992

ISBN 0-7923-2015-8


Mechanics

From 1990, books on the subject of mechanics will be published under two series: FLUID MECHANICS AND ITS APPLICATIONS

Series Editor: R.J. Moreau SOLID MECHANICS AND ITS APPLICATIONS

Series Editor: G.M.L. Gladwell

Prior to 1990, the books listed below were published in the respective series indicated below.

MECHANICS: DYNAMICAL SYSTEMS

Editors: L. Meirovitch and G.£. Oravas

1. E.H. Dowell: Aeroelasticity of Plates and Shells. 1975 ISBN 90-286-0404-9 2. D.G.B. Edelen: Lagrangian Mechanics of Nonconservative Nonholonomic Systems.

1977 ISBN 90-286-0077-9 3. J.L. Junkins: An Introduction to Optimal Estimation of Dynamical Systems. 1978

ISBN 90-286-0067-1 4. E.H. Dowell (ed.), H.C. Curtiss Jr., R.H. Scanlan and F. Sisto: A Modern Course in

Aeroelasticity. Revised and enlarged edition see under Volume I I 5. L. Meirovitch: Computational Methods in Structural Dynamics. 1980

ISBN 90-286-0580-0 6. B. Skalmierski and A. Tylikowski: Stochastic Processes in Dynamics. Revised and

enlarged translation. 1982 ISBN 90-247-2686-7 7. P.C. MUller and W.O. Schiehlen: Linear Vibrations. A Theoretical Treatment of Multi-

degree-of-freedom Vibrating Systems. 1985 ISBN 90-247-2983-1 8. Gh. Buzdugan, E. Mihiiilescu and M. Rade§: Vibration Measurement. 1986

ISBN 90-247-3111-9 9. G.M.L. Gladwell: Inverse Problems in Vibration. 1987 ISBN 90-247-3408-8

10. G.I. Schueller and M. Shinozuka: Stochastic Methods in Structural Dynamics. 1987 ISBN 90-247-3611-0

11. E.H. Dowell (ed.), H.C. Curtiss Jr., R.H. Scanlan and F. Sisto: A Modern Course in Aeroelasticity. Second revised and enlarged edition (of Volume 4). 1989

ISBN Hb 0-7923-0062-9; Pb 0-7923-0185-4 12. W. Szempliriska-Stupnicka: The Behavior of Nonlinear Vibrating Systems. Volume I:

Fundamental Concepts and Methods: Applications to Single-Degree-of-Freedom Systems. 1990 ISBN 0-7923-0368-7

13. W. Szempliriska-Stupnicka: The Behavior of Nonlinear Vibrating Systems. Volume II: Advanced Concepts and Applications to Multi-Degree-of-Freedom Systems. 1990

ISBN 0-7923-0369-5 Set ISBN (Vols. 12-l3) 0-7923-0370-9

MECHANICS OF STRUCTURAL SYSTEMS

Editors: J.S. przemieniecki and G.IE. Oravas

1. L. Fryba: Vibration of Solids and Structures under Moving Loads. 1970 ISBN 90-01-32420-2

2. K. Marguerre and K. Wolfel: Mechanics o/Vibration. 1979 ISBN 90-286-0086-8

Mechanics 3. E.B. Magrab: Vibrations of Elastic Structural Members. 1979 ISBN 90-286-0207-0 4. R.T. Haftka and M.P. Kamat: Elements of Structural Optimization. 1985

Revised and enlarged edition see under Solid Mechanics and Its Applications, Volume 1

5. J.R. Vinson and R.L. Sierakowski: The Behavior of Structures Composed of Composite Materials. 1986 ISBN Hb 90-247-3125-9; Pb 90-247-3578-5

6. B.E. Gatewood: Virtual Principles in Aircraft Structures. Volume 1: Analysis. 1989 ISBN 90-247-3754-0

7. B.E. Gatewood: Virtual Principles in Aircraft Structures. Volume 2: Design, Plates, Finite Elements. 1989 ISBN 90-247-3755-9

Set (Gatewood 1 + 2) ISBN 90-247-3753-2

MECHANICS OF ELASTIC AND INELASTIC SOLIDS

Editors: S. Nemat-Nasser and G.lE. Oravas

1. G.M.L. Gladwell: Contact Problems in the Classical Theory of Elasticity. 1980 ISBN Hb 90-286-0440-5; Pb 90-286-0760-9

2. G. Wempner: Mechanics of Solids with Applications to Thin Bodies. 1981 ISBN 90-286-0880-X

3. T. Mura: Micromechanics of Defects in Solids. 2nd revised edition, 1987 ISBN 90-247-3343-X

4. R.G. Payton: Elastic Wave Propagation in Transversely Isotropic Media. 1983 ISBN 90-247-2843-6

5. S. Nemat-Nasser, H. Abe and S. Hirakawa (eds.): Hydraulic Fracturing and Geother-mal Energy. 1983 ISBN 90-247-2855-X

6. S. Nemat-Nasser, R.J. Asaro and G.A. Hegemier (eds.): Theoretical Foundation for Large-scale Computations of Nonlinear Material Behavior. 1984 ISBN 90-247-3092-9

7. N. Cristescu: Rock Rheology. 1988 ISBN 90-247-3660-9 8. G.I.N. Rozvany: Structural Design via Optimality Criteria. The Prager Approach to

Structural Optimization. 1989 ISBN 90-247-3613-7

MECHANICS OF SURFACE STRUCTURES

Editors: W.A. Nash and G.lE. Oravas

1. P. Seide: Small Elastic Deformations of Thin Shells. 1975 ISBN 90-286-0064-7 2. V. Panc: Theories of Elastic Plates. 1975 ISBN 90-286-0l04-X 3. J.L. Nowinski: Theory ofThermoelasticity with Applications. 1978

ISBN 90-286-0457-X 4. S. Lukasiewicz: Local Loads in Plates and Shells. 1979 ISBN 90-286-0047-7 5. C. Fift: Statics, Formfinding and Dynamics of Air-supported Membrane Structures.

1983 ISBN 90-247-2672-7 6. Y. Kai-yuan (ed.): Progress in Applied Mechanics. The Chien Wei-zang Anniversary

Volume. 1987 ISBN 90-247-3249-2 7. R. Negrutiu: Elastic Analysis of Slab Structures. 1987 ISBN 90-247-3367-7 8. J.R. Vinson: The Behavior of Thin Walled Structures. Beams, Plates, and Shells. 1988

ISBN Hb 90-247-3663-3; Pb 90-247-3664-1

Mechanics MECHANICS OF FLUIDS AND TRANSPORT PROCESSES

Editors: RJ. Moreau and G.lE. Oravas

1. J. Happel and H. Brenner: Low Reynolds Number Hydrodynamics. With Special Applications to Particular Media. 1983 ISBN Hb 90-01-37115-9; Pb 90-247-2877-0

2. S. Zahorski: Mechanics of Viscoelastic Fluids. 1982 ISBN 90-247-2687-5 3. J.A. Sparenberg: Elements of Hydrodynamics Propulsion. 1984 ISBN 90-247-2871-1 4. B.K. Shivamoggi: Theoretical Fluid Dynamics. 1984 ISBN 90-247-2999-8 5. R. Timman, AJ. Hermans and G.C. Hsiao: Water Waves and Ship Hydrodynamics. An

Introduction. 1985 ISBN 90-247-3218-2 6. M. Lesieur: Turbulence in Fluids. Stochastic and Numerical Modelling. 1987

ISBN 90-247-3470-3 7. L.A. Lliboutry: Very Slow Flows of Solids. Basics of Modeling in Geodynamics and

Glaciology. 1987 ISBN 90-247-3482-7 8. B.K. Shivamoggi: Introduction to Nonlinear Fluid-Plasma Waves. 1988

ISBN 90-247-3662-5 9. V. Bojarevics, Ya. Freibergs, E.!. Shilova and E.V. Shcherbinin: Electrically Induced

Vortical Flows. 1989 ISBN 90-247-3712-5 10. J. Lielpeteris and R. Moreau (eds.): Liquid Metal Magnetohydrodynamics. 1989

MECHANICS OF ELASTIC STABILITY

Editors: H. Leipholz and G.lE. Oravas

ISBN 0-7923-0344-X

1. H. Leipholz: Theory of Elasticity. 1974 ISBN 90-286-0193-7 2. L. Librescu: Elastostatics and Kinetics of Aniosotropic and Heterogeneous Shell-type

Structures. 1975 ISBN 90-286-0035-3 3. C.L. Dym: Stability Theory and Its Applications to Structural Mechanics. 1974

ISBN 90-286-0094-9 4. K. Huseyin: Nonlinear Theory of Elastic Stability. 1975 ISBN 90-286-0344-1 5. H. Leipholz: Direct Variational Methods and Eigenvalue Problems in Engineering.

1977 ISBN 90-286-0106-6 6. K. Huseyin: Vibrations and Stability of Multiple Parameter Systems. 1978

ISBN 90-286-0136-8 7. H. Leipholz: Stability of Elastic Systems. 1980 ISBN 90-286-0050-7 8. V.V. Bolotin: Random Vibrations of Elastic Systems. 1984 ISBN 90-247-2981-5 9. D. Bushnell: Computerized Buckling Analysis of Shells. 1985 ISBN 90-247-3099-6

10. L.M. Kachanov: Introduction to Continuum Damage Mechanics. 1986 ISBN 90-247-3319-7

11. H.H.E. Leipholz and M. Abde1-Rohman: Control of Structures. 1986 ISBN 90-247-3321-9

12. H.E. Lindberg and A.L. Florence: Dynamic Pulse Buckling. Theory and Experiment. 1987 ISBN 90-247-3566-1

13. A. Gajewski and M. Zyczkowski: Optimal Structural Design under Stability Con-straints.1988 ISBN 90-247-3612-9

Mechanics MECHANICS: ANALYSIS

Editors: V.I. Mizel and G.lE. Oravas

1. M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustylnik and P.E. Sbolevskii: Integral Operators in Spaces of Summable Functions. 1976 ISBN 90-286-0294-1

2. V.V. Ivanov: The Theory of Approximate Methods and Their Application to the Numerical Solution of Singular Integral Equations. 1976 ISBN 90-286-0036-1

3. A. Kufner, O. lohn and S. Pucik: Function Spaces. 1977 ISBN 90-286-0015-9 4. S.G. Mikhlin: Approximation on a Rectangular Grid. With Application to Finite

Element Methods and Other Problems. 1979 ISBN 90-286-0008-6 5. D.G.B. Edelen: Isovector Methods for Equations of Balance. With Programs for

Computer Assistance in Operator Calculations and an Exposition of Practical Topics of the Exterior Calculus. 1980 ISBN 90-286-0420-0

6. R.S. Anderssen, F.R. de Hoog and M.A. Lukas (eds.): The Application and Numerical Solution of Integral Equations. 1980 ISBN 90-286-0450-2

7. R.Z. Has'minskiI: Stochastic Stability of Differential Equations. 1980 ISBN 90-286-0100-7

8. A.I. Vol'pert and S.1. Hudjaev: Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics. 1985 ISBN 90-247-3109-7

9. A. Georgescu: Hydrodynamic Stability Theory. 1985 ISBN 90-247-3120-8 10. W. Noll: Finite-dimensional Spaces. Algebra, Geometry and Analysis. Volume I. 1987

ISBN Hb 90-247-3581-5; Pb 90-247-3582-3

MECHANICS: COMPUTATIONAL MECHANICS

Editors: M. Stem and G.lE. Oravas

1. T.A. Cruse: Boundary Element Analysis in Computational Fracture Mechanics. 1988 ISBN 90-247-3614-5

MECHANICS: GENESIS AND METHOD

Editor: G.lE. Oravas

1. P.-M.-M. Duhem: The Evolution of Mechanics. 1980

MECHANICS OF CONTINUA

Editors: W.O. Williams and G.lE. Oravas

ISBN 90-286-0688-2

1. C.-C. Wang and C. Truesdell: Introduction to Rational Elasticity. 1973

2. P.I. Chen: Selected Topics in Wave Propagation. 1976 3. P. Villaggio: Qualitative Methods in Elasticity. 1977

ISBN 90-01-93710-1 ISBN 90-286-0515-0 ISBN 90-286-0007-8

MECHANICS OF FRACTURE

Editors: G.C. Sih

Mechanics

1. G.C. Sih (ed.): Methods of Analysis and Solutions of Crack Problems. 1973 ISBN 90-01-79860-8

2. M.K. Kassir and G.c. Sih (eds.): Three-dimensional Crack Problems. A New Solution of Crack Solutions in Three-dimensional Elasticity. 1975 ISBN 90-286-0414-6

3. G.c. Sih (ed.): Plates and Shells with Cracks. 1977 ISBN 90-286-0146-5 4. G.C. Sih (ed.): Elastodynamic Crack Problems. 1977 ISBN 90-286-0156-2 5. G.c. Sih (ed.): Stress Analysis of Notch Problems. Stress Solutions to a Variety of

Notch Geometries used in Engineering Design. 1978 ISBN 90-286-0166-X 6. G.C. Sih and E.P. Chen (eds.): Cracks in Composite Materials. A Compilation of Stress

Solutions for Composite System with Cracks. 1981 ISBN 90-247-2559-3 7. G.C. Sih (ed.): Experimental Evaluation of Stress Concentration and Intensity Factors.

Useful Methods and Solutions to Experimentalists in Fracture Mechanics. 1981

MECHANICS OF PLASTIC SOLIDS

Editors: 1. Schroeder and G.lE. Oravas

ISBN 90-247-2558-5

1. A. Sawczuk (ed.): Foundations of Plasticity. 1973 ISBN 90-01-77570-5 2. A. Sawczuk (ed.): Problems of Plasticity. 1974 ISBN 90-286-0233-X 3. W. Szczepinski: Introduction to the Mechanics of Plastic Forming of Metals. 1979

ISBN 90-286-0126-0 4. D.A. Gokhfeld and O.F. Chemiavsky: Limit Analysis of Structures at Thermal Cycling.

1980 ISBN 90-286-0455-3 5. N. Cristescu and I. Suliciu: Viscoplasticity. 1982 ISBN 90-247-2777-4


flow visualization and image analysis

Documents