COPYRIGHT James Alan Bickford 999

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1 DIRECT MEASUREMENT OF PARTICLE BEHAVIOR IN THE PARTICLE-LAGRANGIAN REFERENCE FRAME OF A TURBULENT FLOW A dissertation submitted by James A. Bickford In partial fulllment of the requirements for the degree of Master of Science in Mechanical Engineering TUFTS UNIVERSITY September, 999 Adviser: Chris B. Rogers

2 COPYRIGHT James Alan Bickford 999

3 ABSTRACT DIRECT MEASUREMENT OF PARTICLE BEHAVIOR IN THE PARTICLE-LAGRANGIAN REFERENCE FRAME OF A TURBULENT FLOW James A. Bickford Particle laden turbulent ows occur in a wide range of engineering and scientic applications. A particle's trajectory is governed by its own inertia, external forces such as gravity, and the characteristics of its uid neighborhood. The uid neighborhood is in turn coupled to the particle's trajectory which is inuenced by the aforementioned factors. This non-linear dependency is what makes the prediction of particle behavior so dicult. In order to accurately model particle motion, one must be able to model turbulence with statistics from the particle-lagrangian (p-l) reference frame. These measurements were made using a hybrid experimental/numerical method that allowed velocity measurements to be taken from within the p-l reference frame. Atwo axis traverse emulated the path of a virtual particle suspended within a fully developed turbulent water channel by actively controlling its velocity with feedback from an attached two component Laser Doppler Velocimetry (LDV) system. A coupled Digital Particle Image Velocimetry (DPIV) system enabled vorticity measurements to be taken from the particle's reference frame. Using the hybrid technique, turbulent statistics were acquired at a range of particle response times, gravitational levels, and Reynolds numbers. Changes in the measured energy spectra, integral scales, uctuating velocity components, and other eects were directly observed while varying particle and gravitational parameters. An increase in gravity was found to decrease correlation times and add high frequency energy in the particle spectra due to the \crossing trajectories" eect. Particle inertia was found to generally increase the measured integral scales while gravity tended to decrease them. Using DPIV data, iii

4 the Kolmogorov time scale of the ow was estimated along with measurement of a decrease in the average uid vorticity associated with preferential concentration { the non random distribution of particles created by random motions of the uid. The data acquired can be applied to future improvements in turbulence models. iv

5 ACKNOWLEDGEMENTS I would like to thank the National Science Foundation for its role in funding the following research. I also owe my advisor Chris Rogers a debt of gratitude for providing me with the opportunity to work in an environment that I'll always remember for both its academic and social sides. I would also like to thank the numerous individuals that have assisted in this project including Jim Homan and Vinny Maraglia who helped in multiple capacities, and all the members of the TUFTL group who made graduate school a superb experience. I should also specically thank Dave McAndrew, Becca MacMaster, and AJ Bettencourt for their work on the project. The many conversations with Judy Segura and John Eaton from Stanford University were also instrumental to the success of the project. Finally, I would like to thank my parents for their support through my years at Tufts. v

6 Contents Introduction 2. Overview Applications Approach Background Information Experimental Setup 3 2. Experimental Capabilities Water Channel System Channel Structure Water Entrance and Conditioning Region Flow Development Region Test Section Water Collection and Recirculation System Traverse System Laser Doppler Velocimetry System Quasi-Numerical Simulation Setup Digital Particle Image Velocimetry System Illumination System Camera System Triggering System vi

7 2.6.4 Simultaneous LDV and DPIV Summary QNS Methodology Particle Following Methodology Sources of Error LDV Uncertainty Traverse Uncertainty Single-Point Eulerian Channel Proles Stationary Eulerian Measurements Moving Eulerian Measurements Single-Point QNS Measurements Overview Velocity Autocorrelations Fluid and Particle Energy Spectra Observed Phenomena Fluid and Particle Integral Time Scales Anomalous Drift Velocities Conditionally Sampled Fluid Behavior Fluid Fluctuations from the p-l Reference Frame Estimation of Vorticity DPIV Methodology DPIV Uncertainty Spurious Correlation Peak Identication Detection Algorithm Accuracy of Approach vii

8 7.3.3 Vector Replacement Techniques Vorticity Calculations Estimation of Kolmogoro Microscale Vorticity Measurements in the p-l Reference Frame Conclusion 7 8. Summary Comparison to Previous QNS Study Final Remarks A QNS Velocity Autocorrelations with Variable Gravity B QNS Velocity Autocorrelations with Variable Inertia 24 C QNS Energy Spectra with Variable Gravity 37 D QNS Energy Spectra with Variable Inertia 5 E Conditionally Sampled Fluid Behavior Plots 63 F Autocorrelation and Energy Spectra of Unevenly Sampled Data 82 G Detailed Channel Schematics 84 viii

9 List of Figures. Mount Saint Helens after the May 98 eruption which suspended tons of particulate material into the atmosphere Research and exploration of Mars will be aected by the behavior of particles transported in the Martian environment. (Image by Pat Rawlings) Positions of a slice of galaxies relative to our own. Distances are dependent upon the true value of the Hubble constant and instead are given as measured recession velocities. (Da Costa et. el. [9]) Tradeo between measurable parameters and turbulence complexity for QNS,DNS and experiment Channel and traverse setup overview Narrow channel schematic DPIV setup: (a) LDV probe (b) vertical traverse (c) Magnum laser (d) ES- digital camera (e) di-chroic lter QNS methodology overview QNS concept owchart QNS algorithm owchart Particle Velocity vs. Fluid Velocity (St k = 44) Particle Velocity vs. Fluid Velocity (St k = 8) Particle Velocity vs. Fluid Velocity (St k = 2) ix

10 3.7 Particle Velocity vs. Fluid Velocity (St k =4) Mapping locations Eulerian streamwise velocity autocorrelations. (Re=33) Eulerian streamwise velocity autocorrelations. (Re=66) Eulerian streamnormal velocity autocorrelations. (Re=33) Eulerian streamnormal velocity autocorrelations. (Re=66) Stationary Eulerian uid spectra Velocity autocorrelations of the moving Eulerian cases Streamnormal uid autocorrelation (Re=66 p = 25ms) Streamwise particle autocorrelation (Re=66 p = 5ms) Streamnormal particle autocorrelation (Re=66 p = 25ms) Streamnormal particle autocorrelation (Re=66 Drift = %) Streamnormal uid autocorrelation (Re=66 Drift = %) Streamnormal particle energy spectra (Re=66 p = 5ms) Streamnormal uid energy spectra (Re=66 p = 5ms) Streamnormal particle energy spectra (Re=66, Drift = %) Streamnormal uid energy spectra (Re=66, Drift = %) Streamnormal uid energy spectra (Re=33, Drift = %) Streamnormal Fluid Integral Time Scales (Re=66) Smoothed Streamwise Fluid Integral Time Scales (Re=66) Measured streamwise uid integral scales vs. S g. (Re=66) Measured streamwise uid integral scales vs. S g. (Re=33) Measured streamnormal uid integral scales vs. S g. (Re=66) Measured streamnormal uid integral scales vs. S g. (Re=33) Measured streamwise uid integral scales vs. St k. (Re=66) Measured streamwise uid integral scales vs. St k. (Re=33) x

11 6.9 Measured streamwise uid integral scales vs. St T. (Re=66) Measured streamwise uid integral scales vs. St T. (Re=33) Measured streamnormal uid integral scales vs. St k with linear ts. (Re=66) Measured streamnormal uid integral scales vs. St k. (Re=33) Measured streamwise particle integral scales. (Re=66) Measured streamwise particle integral scales. (Re=33) Measured streamwise particle integral scales vs. St T. (Re=66) Measured streamwise particle integral scales vs. St T. (Re=33) Measured streamnormal particle integral scales. (Re=66) Measured streamnormal particle integral scales. (Re=33) Measured anomalous drift velocities. (Re=66) Measured anomalous drift velocities. (Re=33) PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = ms, Drift = %) PDF of streamnormal uctuating velocity component PDF sampled on streamnormal acceleration. (Re=66, p = ms, Drift = %) Eect of particle inertia on PDF mean shift. (Re=66, jaj > :75 ) Quadratic ts for all measured PDF shift acceleration levels. (Re=66) Streamwise uctuating component as a function of drift and time constant. (Re=66) Streamnormal uctuating component as a function of drift and time constant. (Re=66) Streamwise uctuating velocity component versus particle time constant. (Re=66) Streamnormal uctuating velocity componentversus particle time constant. (Re=66) xi

12 6.29 Streamwise uctuating velocity component versus drift velocity. (Re=66) Streamnormal uctuating velocity component versus drift velocity. (Re=66) Streamwise uctuating velocity component versus particle time constant.(re=33) Streamnormal uctuating velocity componentversus particle time constant. (Re=33) Streamwise uctuating velocity component vs. drift velocity. (Re=33) Streamnormal uctuating velocity component vs. drift velocity. (Re=33) Spatial Correlation function (m; n) DPIV uncertainty by window dimension (From McAndrew (998)) Correlation that typically produces a bad vector Ratio of bad vectors identied for a given value Typical central vector region before cleaning Vorticity behavior as noise (bad vectors) increases Sample vector eld obtained by correlating subsections of two 8x8 pixel images Vorticity estimates for one run. ( p = 25ms Drift = %) Estimated vorticity inthep-l reference frame for QNS Drift=% Estimated vorticity inthep-l reference frame for QNS Drift=5% Vorticity measurements in the p-l reference frame at % and 5% Drift Vorticity dip after cleaning and vector replacement Estimated number of images required vs. error tolerance xii

13 8. Comparison of streamwise uid velocity autocorrelations (Re = 66; p = 5ms; S g = ) A. Streamwise particle autocorrelation (Re=66 p = 25ms) A.2 Streamwise particle autocorrelation (Re=66 p = 5ms) A.3 Streamwise particle autocorrelation (Re=66 p = ms) A.4 Streamwise uid autocorrelation (Re=66 p = 25ms) A.5 Streamwise uid autocorrelation (Re=66 p = 5ms) A.6 Streamwise uid autocorrelation (Re=66 p = ms) A.7 Streamnormal particle autocorrelation (Re=66 p = 25ms) A.8 Streamnormal particle autocorrelation (Re=66 p = 5ms) A.9 Streamnormal particle autocorrelation (Re=66 p = ms) A. Streamnormal uid autocorrelation (Re=66 p = 25ms) A. Streamnormal uid autocorrelation (Re=66 p = 5ms) A.2 Streamnormal uid autocorrelation (Re=66 p = ms) A.3 Streamwise particle autocorrelation (Re=33 p = 25ms) A.4 Streamwise particle autocorrelation (Re=33 p = 5ms) A.5 Streamwise particle autocorrelation (Re=33 p = 2ms) A.6 Streamwise uid autocorrelation (Re=33 p = 25ms) A.7 Streamwise uid autocorrelation (Re=33 p = 5ms) A.8 Streamwise uid autocorrelation (Re=33 p = 2ms) A.9 Streamnormal particle autocorrelation (Re=33 p = 25ms) A.2 Streamnormal particle autocorrelation (Re=33 p = 5ms) A.2 Streamnormal particle autocorrelation (Re=33 p = 2ms) A.22 Streamnormal uid autocorrelation (Re=33 p = 25ms) A.23 Streamnormal uid autocorrelation (Re=33 p = 5ms) A.24 Streamnormal uid autocorrelation (Re=33 p = 2ms) xiii

14 B. Streamwise particle autocorrelation (Re=66 Drift = %) B.2 Streamwise particle autocorrelation (Re=66 Drift = 5%) B.3 Streamwise particle autocorrelation (Re=66 Drift = %) B.4 Streamwise uid autocorrelation (Re=66 Drift = %) B.5 Streamwise uid autocorrelation (Re=66 Drift = 5%) B.6 Streamwise uid autocorrelation (Re=66 Drift = %) B.7 Streamnormal particle autocorrelation (Re=66 Drift = %) B.8 Streamnormal particle autocorrelation (Re=66 Drift = 5%) B.9 Streamnormal particle autocorrelation (Re=66 Drift = %) B. Streamnormal uid autocorrelation (Re=66 Drift = %) B. Streamnormal uid autocorrelation (Re=66 Drift = 5%) B.2 Streamnormal uid autocorrelation (Re=66 Drift = %) B.3 Streamwise particle autocorrelation (Re=33 Drift = %) B.4 Streamwise particle autocorrelation (Re=33 Drift = 5%) B.5 Streamwise particle autocorrelation (Re=33 Drift = %) B.6 Streamwise uid autocorrelation (Re=33 Drift = %) B.7 Streamwise uid autocorrelation (Re=33 Drift = 5%) B.8 Streamwise uid autocorrelation (Re=33 Drift = %) B.9 Streamnormal particle autocorrelation (Re=33 Drift = %) B.2 Streamnormal particle autocorrelation (Re=33 Drift = 5%) B.2 Streamnormal particle autocorrelation (Re=33 Drift = %) B.22 Streamnormal uid autocorrelation (Re=33 Drift = %) B.23 Streamnormal uid autocorrelation (Re=33 Drift = 5%) B.24 Streamnormal uid autocorrelation (Re=33 Drift = %) C. Streamwise particle energy spectra (Re=66 p = 25ms) C.2 Streamwise particle energy spectra (Re=66 p = 5ms) C.3 Streamwise particle energy spectra (Re=66 p = ms) xiv

15 C.4 Streamwise uid energy spectra (Re=66 p = 25ms) C.5 Streamwise uid energy spectra (Re=66 p = 5ms) C.6 Streamwise uid energy spectra (Re=66 p = ms) C.7 Streamnormal particle energy spectra (Re=66 p = 25ms) C.8 Streamnormal particle energy spectra (Re=66 p = 5ms) C.9 Streamnormal particle energy spectra (Re=66 p = ms) C. Streamnormal uid energy spectra (Re=66 p = 25ms) C. Streamnormal uid energy spectra (Re=66 p = 5ms) C.2 Streamnormal uid energy spectra (Re=66 p = ms) C.3 Streamwise particle energy spectra (Re=33 p = 25ms) C.4 Streamwise particle energy spectra (Re=33 p = 5ms) C.5 Streamwise particle energy spectra (Re=33 p = 2ms) C.6 Streamwise uid energy spectra (Re=33 p = 25ms) C.7 Streamwise uid energy spectra (Re=33 p = 5ms) C.8 Streamwise uid energy spectra (Re=33 p = 2ms) C.9 Streamnormal particle energy spectra (Re=33 p = 25ms) C.2 Streamnormal particle energy spectra (Re=33 p = 5ms) C.2 Streamnormal particle energy spectra (Re=33 p = 2ms) C.22 Streamnormal uid energy spectra (Re=33 p = 25ms) C.23 Streamnormal uid energy spectra (Re=33 p = 5ms) C.24 Streamnormal uid energy spectra (Re=33 p = 2ms) D. Streamwise particle energy spectra (Re=66 Drift = %) D.2 Streamwise particle energy spectra (Re=66 Drift = 5%) D.3 Streamwise particle energy spectra (Re=66 Drift = %) D.4 Streamwise uid energy spectra (Re=66 Drift = %) D.5 Streamwise uid energy spectra (Re=66 Drift = 5%) D.6 Streamwise uid energy spectra (Re=66 Drift = %) xv

16 D.7 Streamnormal particle energy spectra (Re=66 Drift = %) D.8 Streamnormal particle energy spectra (Re=66 Drift = 5%) D.9 Streamnormal particle energy spectra (Re=66 Drift = %) D. Streamnormal uid energy spectra (Re=66 Drift = %) D. Streamnormal uid energy spectra (Re=66 Drift = 5%) D.2 Streamnormal uid energy spectra (Re=66 Drift = %) D.3 Streamwise particle energy spectra (Re=33 Drift = %) D.4 Streamwise particle energy spectra (Re=33 Drift = 5%) D.5 Streamwise particle energy spectra (Re=33 Drift = %) D.6 Streamwise uid energy spectra (Re=33 Drift = %) D.7 Streamwise uid energy spectra (Re=33 Drift = 5%) D.8 Streamwise uid energy spectra (Re=33 Drift = %) D.9 Streamnormal particle energy spectra (Re=33 Drift = %) D.2 Streamnormal particle energy spectra (Re=33 Drift = 5%) D.2 Streamnormal particle energy spectra (Re=33 Drift = %) D.22 Streamnormal uid energy spectra (Re=33 Drift = %) D.23 Streamnormal uid energy spectra (Re=33 Drift = 5%) D.24 Streamnormal uid energy spectra (Re=33 Drift = %) E. PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = 25ms, Drift = %) E.2 Streamnormal uctuating velocity component PDF sampled on streamnormal acceleration. (Re=66, p = 25ms, Drift = %) E.3 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = 5ms, Drift = %) E.4 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = 5ms, Drift = %) xvi

17 E.5 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = ms, Drift = %) E.6 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = ms, Drift = %) E.7 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = 25ms, Drift = 5%) E.8 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = 25ms, Drift = 5%) E.9 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = 5ms, Drift = 5%) E. PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = 5ms, Drift = 5%) E. PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = ms, Drift = 5%) E.2 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = ms, Drift = 5%) E.3 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = 25ms, Drift = %) E.4 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = 25ms, Drift = %)... 7 E.5 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = 5ms, Drift = %) E.6 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = 5ms, Drift = %)... 7 E.7 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = ms, Drift = %) xvii

18 E.8 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=66, p = ms, Drift = %).. 72 E.9 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 5ms, Drift = %) E.2 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 5ms, Drift = %) E.2 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 8ms, Drift = %) E.22 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 8ms, Drift = %) E.23 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 2ms, Drift = %) E.24 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 2ms, Drift = %) E.25 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 5ms, Drift = 5%) E.26 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 5ms, Drift = 5%) E.27 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 8ms, Drift = 5%) E.28 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 8ms, Drift = 5%) E.29 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 2ms, Drift = 5%) E.3 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 2ms, Drift = 5%) xviii

19 E.3 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 5ms, Drift = %) E.32 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 5ms, Drift = %) E.33 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 8ms, Drift = %) E.34 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 8ms, Drift = %)... 8 E.35 PDF of streamwise uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 2ms, Drift = %) E.36 PDF of streamnormal uctuating velocity component sampled on streamnormal acceleration. (Re=33, p = 2ms, Drift = %).. 8 G. AutoCAD of backward facing step and narrow channel (a) G.2 AutoCAD of backward facing step and narrow channel (b) xix

20 List of Tables 2. DPIV hardware specications Characteristics of the channel mapping (Re=33) Characteristics of the channel mapping (Re=66) Eulerian statistics in the streamwise (u) and streamnormal (v) directions Moving Eulerian statistics in the streamwise (u) and streamnormal (v) directions Primary test matrix with St k for each case studied Results of Kolmogorov uid time scale measurements Vorticity measurement parameters xx

21 NOMENCLATURE Roman Symbols dt d p F HP F LP F P F S f(m; n) g g(m; n) n Re H Re p R ii () S g S ii St k St T T T k T:I: u u u u U U diff Time interval Particle diameter High-pass frequency lter setting Low-pass frequency lter setting Frequency shift from tracer particle Shift frequency DPIV image subsection Acceleration due to gravity DPIV image subsection Sample size Reynolds number based on step height and inlet velocity Reynolds number based on particle diameter Temporal uid velocity autocorrelation Gravitational Stokes number Energy spectra Stokes number based on the Kolmogorov uid scale Stokes number based on the integral uid scale Integral time scale Kolmogorov time scale Directional turbulence intensity Streamwise velocity component Fluctuating streamwise velocity component Friction velocity Mean uid velocity Streamwise velocity deviation from overall mean xxi

22 ~U f Fluid velocity U inlet v v v d V diff Inlet centerline velocity Streamnormal velocity component Fluctuating streamnormal velocity component Drift velocity Streamnormal velocity deviation from overall mean ~V p Particle velocity x y z Streamwise direction Streamnormal direction Desired condence interval Greek Symbols i t fg (m; n) f p f p Bad vector cleaning threshold Time Interval Dissipation Kolmogorov length scale DPIV cross-correlation function Wavelength Fluid viscosity Fluid kinematic viscosity Fluid density Particle density Standard deviation Fluid time scale Particle time constant! Vorticity xxii

23 Additional Symbols ME p,l Moving Eulerian reference frame Particle Lagrangian reference frame xxiii

24 Direct Measurement of Particle Behavior in the Particle-Lagrangian Reference Frame of a Turbulent Flow

25 Chapter Introduction. Overview The behavior of dilute particles in a two-phase ow has many exciting scientic and engineering applications. The path of a dense, moderately sized particle suspended within a turbulent carrier uid is governed by motion of the uid surrounding the particle. In order to accurately model the particle motion, one must be able to estimate statistics in the particle-lagrangian reference frame. Analytical models can be improved and updated with such data. This investigation attempts to address some of these points and act as a stepping stone for advances that will eventually improve the models of these ows..2 Applications The study of turbulence and particle-laden turbulentows has important applications in a wide range of areas. The length scales that characterize the turbulence occur from smallest sub-millimeter range to the largest which can be measured in light years. Applications range from common earth based engineering situations, to processes on the planetary scale, and nally to astrophysical phenomena that could potentially be 2

26 responsible for the universe's current structure. A large number of environmental and engineering applications are aected by the physics that embodies particle laden turbulentows. The knowledge of particle transport is important when trying to predict the dispersion of particles suspended in the atmosphere. This has important implications when considering the motion of radioactive particle released into the atmosphere following a nuclear accident, predicting the motion of ash and debris after a volcanic eruption (gure.) or asteroid/comet impact, understanding the behavior of airborne pollutants, and numerous other natural processes. Figure.: Mount Saint Helens after the May 98 eruption which suspended tons of particulate material into the atmosphere. Future and past space missions will be greatly enhanced with added knowledge of particle behavior. Understanding the motion of particles within space station components is critical to guarantee the safety and success of operations in Earth orbit. In addition, only an incomplete understanding of particle transport exists within the 3

27 Martian atmosphere (gure.2). In order to completely understand and decipher past and future remote sensing observations of the planet, a fuller appreciation of turbulent particle transport that sculpts the surface is required. The design of surface landers and other ground based equipment will be aected by the knowledge of dust suspension in the atmosphere. The inltration of dust into sensitive components, degradation of optical and other hardware from dust abrasion, and other eects could potentially jeopardize the safety and success of future missions. The study of particulate transport in this thesis can be a stepping stone to resolving some of these issues. Figure.2: Research and exploration of Mars will be aected by the behavior of particles transported in the Martian environment. (Image by Pat Rawlings) The transport of particles by turbulence can occur on scales measured in light years. It has been suggested [6] that turbulent ow and preferential concentration is responsible for the creation of the Sun, Earth, and solar system from the primordial 4

28 planetary nebulae that made up our region of the galaxy over 5 billion years ago. Even more astounding is the universe as a whole which appears to have a structure that looks remarkably similar to the structures formed from preferential concentration. Without preferential concentration, the initial uniformity created after the Big Bang might never have formed into its current large scale structure (gure.3). 9 Galaxy Position (km/s) Position (km/s) Figure.3: Positions of a slice of galaxies relative to our own. Distances are dependent upon the true value of the Hubble constant and instead are given as measured recession velocities. (Da Costa et. el. [9]).3 Approach A three-pronged approach is being used bytufts University to study particle behavior in turbulent uid ows. The traditional, and most common method before the advent of computers, involves running experiments with real particles and real turbulence. These experiments use real turbulence of any complexity that can be created, but 5

29 are sometimes hindered by the inability tomeasure certain parameters. In addition, dicult measures, such as ying the experiment in a weightless environment, must be taken to isolate variables to be studied. Direct numerical simulations that solve the Navier-Stokes equations are also used. This method has the advantage that all parameters are exactly known but are limited by the complexity of the turbulence that can be simulated. Computational time is costly and prohibitive for real turbulence with a wide range of scales. A third hybrid experimental/numerical technique developed at Tufts University lls in the gap between the experimental and DNS methods. This hybrid Quasi Numerical Simulation (QNS) method is the primary focus of this thesis investigation. As shown in gure (.4), this technique extends the parameters that can be measured into regions inaccessible to traditional computer and experimental studies. As will be discussed further, QNS allows direct measurement of uid and particle statistics to be taken within the particle-lagrangian (p-l) reference frame of a turbulent ow. Additionally, vorticity measurements can also be made by coupling a DPIV system to the QNS hardware. 6

30 Turbulence Complexity Measurable Parameters Figure.4: Tradeo between measurable parameters and turbulence complexity for QNS,DNS and experiment. The QNS method works by using a control loop to update a traverse to emulate the path of a virtual particle in a turbulent ow. Instantaneous two-component velocity measurements in a fully developed turbulent water channel provide the feedback for the control loop. Velocity measurements are made with a laser doppler velocimetry (LDV) probe attached to the traverse with the LDV measuring volume becoming the virtual particle when the control loop is active. Complete details on the operation is included in the following chapters. Overall, the three methods used at Tufts are complimentary. They all have a parameter region that overlaps, allowing each method to be compared for conrmation, while extending the overall reach into realms not possible with any individual method. 7

31 .4 Background Information Practical interests have signicantly advanced knowledge of gas-particle ows. The 994 Freeman Scholar Lecture [7] gave an excellent overview of the current state of knowledge in the area, following over 5 years of research by numerous individuals. A combination of gravity and a particle's interaction with the surrounding turbulent uid dictate the path of a given particle through a turbulent eld. The trajectory of a particle aects the region of the ow the particle passes through, which in turn aects the trajectory. The non-linear coupling between the two eects previously limited work to a very fundamental level. Tchen [8] was the rst to present a complete transport equation for a particle within a uid. The equation was rather limited, namely by the fact that it required the uid and particle paths to coincide. In real situations, a particle has inertia causing it to lag behind the uid. The particle response time characterizes how fast the particle reacts to a uid change. This value, p = pd 2 p 8 ; (.) which is the time required for the particle's velocity to reach(,=e) of a uid velocity change, is a function of the particle density ( d ) and diameter (d p ), along with the viscosity () ofthecarrier uid. This expression is valid for a small (d p << ), rigid sphere in a Stokes ow which requires, Re p = vd p < : (.2) with v as the uid velocity relative to the sphere, d p as the sphere diameter, and as the kinematic viscosity of the carrier uid. 8

32 Maxey and Riley [3] rened and advanced the work of Tchen and others to produce the latest version of the transport equation. The new expression, 6 d3 p dv p p dt = 6 d3 p( p, f )g, 3d p (V p, U f, 24 d2 pr 2 U f ) + 6 d3 p DU i f Dt, 2 d3 p f( dv p dt, DU i Dt, 4 d2 pr 2 U f ), 3 Z d 2 d3 p dt ( (V I, U i, 24 d2 pr 2 U i ) q d (t, ) (.3) accounts for a number of physical phenomena to produce an accurate model for particle motion in a uid. It includes the eects of particle/uid lag which Tchen neglected. The entire expression relates the change in particle inertia to a gravity force term, a Stokes drag term, a uid pressure force, an \added mass force" which is the force required to ll the void left behind the particle moving through the uid, and the Basset history force. Solving the equation of motion is complicated by the fact that it is a coupled non-linear expression. The particle trajectory is needed to compute particle drag, and the drag is required to compute the particle trajectory. The drag must also be computed along the particle path, which in general does not coincide with the uid path. When the particle density is restricted to be orders of magnitude greater than the uid density, equation (.3) reduces ~ V = p ( ~ U f, ~ V p ), ~g: (.4) Particle acceleration then becomes a function of particle inertia ( p ), relative uid velocity from the particle-lagrangian reference frame ( ~ U f, ~ V p ), and external acceleration (g). The external acceleration, generally gravity, manifests itself as a drift velocity, or free fall velocity, v d = g p ; (.5) 9

33 when in a stationary uid. An estimate of how a particle will react to local turbulence can be made with the Stokes number. The ratio, St = p f (.6) characterizes the behavior of a particle in a given ow with the term f dened as the Kolmogorov uid time scale or alternatively as the integral time scale depending upon the situation. Particles with very small Stokes numbers almost exactly follow the ow while those with very large Stokes numbers almost completely ignore the turbulence. Intermediate values for Stokes number are the most dicult to predict. A similar estimation of the eects of gravity can be made with the gravitational Stokes number [6], S g = v d u pl f : (.7) For values of S g less than one, gravity should have very little eect on particle behavior. When S g increases, the eects of gravity starts to dominate and the particle is less responsive to the turbulence. The strongest gravitational eect is the \crossing trajectories" phenomena rst recognized by Yudine [22]. The \crossing trajectories" eect is created by an external body force such as gravity acting on a particle. As a particle is pulled through turbulence (drift velocity), the particle correlation time decreases as the particle is drawn through uid neighborhoods. This loss of velocity correlation was further studied and conrmed by others including Wells and Stock [2] who varied the external force using charged particles and regulating an external magnetic eld. In cases where the drift velocity was greater than the uctuating uid components (S g > ), the crossing trajectories eect was found to dominate particle dispersion. With increasing v d, the particle sees the related `continuity eect' which follows from the particles drifting through uid regions faster than the eddy decay time, producing a negative dip in the velocity autocorrelation of the lateral direction (Csanady [4]). Particle dispersion

34 is decreased in this direction as correlations are decreased when particles see opposite velocities from its reference frame as it drifts from one side of an eddy to the other. Modeling is further complicated by preferential concentration, an eect where turbulent uctuations produce non-random distributions of particles within a ow. Eaton and Fessler [5] reviewed and discussed the case where Stokes numbers based on the Kolmogorov time scale are on the order of and particles preferentially concentrate. These intermediate sized particles interact with the turbulent structures and are \ung out" of high vorticity regions and concentrate in regions of high strain. Rouson and Eaton [7] investigated preferential concentration of particles in a fully developed water channel using direct numerical simulation. Strong evidence including particle clustering was found in the simulated channel comparable to the system used for this thesis. Kulick [8] also directly observed preferential concentration in a fully deloped air channel. Most recently, Fallon [] studied preferential concentration in a variable gravity environment and showed direct evidence of preferential concentration at St k = O(). Recent computer simulations by Coppen [3] have shown preferential concentration eects to span Stokes numbers over two orders of magnitude (St k =:, ). The rst QNS work by Ainley [], used a system similar to that in this investigation. Using an older particle advancement scheme discussed in Chapter (3), he studied the eects of both particle inertia and gravity on particle dispersion. Fluid velocity elds in the particle-lagrangian reference frame were found to be only weakly aected by particle time constant but strongly aected by gravity. He developed a model to predict particle dispersion for Stokes numbers of order and small drift velocities. McAndrew [4] was the rst to use the updated particle advancement scheme with the QNS method. He extended the work of Ainley to a backwards facing step with particle dispersion measurements at four time constants and a Re=, with no

35 gravitational drift. McAndrew extended the system of Ainley to include a coupled QNS and Digital Particle Image Velocimetry system which allowed vorticity estimates to be made in the particle-lagrangian reference frame. He found a relationship between particle size and vorticity sampled on particle motion through uid structures (ejection or entrainment in the recirculation zone). The DPIV system described by McAndrew and used in this investigation was based upon a technique described by Willert and Gharib [2]. Two successive digital video images are broken into subsections, transformed into frequency space and convolved before being transformed back to real space where a gaussian peak is t to the correlation map to obtain sub-pixel accurate uid velocity displacements. The velocity eld can then be dierentiated to calculate the vorticity of uid regions. This thesis investigation improved on previous studies by signicantly extending the parameters studied. Data analysis was also extended with an updated autocorrelation code and additional studies on uid and particle statistics. The capabilities of the DPIV system were also extended to improve onthe logistics of operation as well as automation of many features. 2

36 Chapter 2 Experimental Setup 2. Experimental Capabilities The Tufts University Fluid Turbulence Lab (TUFTL) has a unique system capable of taking measurements in the particle-lagrangian reference frame of a turbulent ow. The system consists of a two component laser doppler velocimetry (LDV) system mated with a two axis traverse. The LDV system takes instantaneous uid velocity measurements within a water channel. The traverse velocity is actively controlled based on feedback from LDV measurements allowing the traverse to emulate the path of a virtual particle within the turbulent ow. Fluid velocity measurements taken in the virtual particle's reference frame are made with the LDV system as well as with digital particle image velocimetry (DPIV). A pulsed diode laser illuminates a two-dimensional slice of the ow for imaging. The DPIV system is capable of capturing the motion of tiny seed particles within the ow with uid velocities determined by cross-correlating image subsections to determine particle displacements from one image to the next. The resultant vector eld computed for the area around the virtual particle can then be used to estimate vorticity in this region. 3

37 2.2 Water Channel System The experimental setup consists of two individual water channels which are both serviced by a single two axis traverse for experimentation. The backwards facing step provides a shear layer for research investigation while the narrow channel provides a fully developed channel of uniform ow. Since the twochannels share certain common components, only one may be operated at a time. All experiments performed for this thesis were done in the narrow channel. Figure 2.: Channel and traverse setup overview Channel Structure The two channels sit next to one another with the smaller step channel slightly lower and forward and the traverse oset. All share a common parallel alignment. The step channel is considerably smaller and can be completely serviced by the traverse. The narrow channel's size limits traverse movement to the test section. The channels are both placed on an Aluminum Speedrail TM structure allowing the systems to be rigidly coupled and the narrow channel to pass over and not obstruct 4

38 a passage way. Additionally, space is provided to house experimental sub-systems beneath the channel support structure. The narrow channel consists of ve primary sections which transport water for the experiment. The test section has a width of 2h(4cm)andaheight of 25 h (5 cm). Except for the PVC pipe used to re-circulate the water, all tank sections are made of /2" plexi-glass with separate sections joined with rubber gaskets and stainless steel nuts and bolts which can be disassembled for cleaning and maintenance. Each section is an integral part of the system and prepares or re-circulates water for reuse in the primary test section. There are ve primary sections that form the channel system: () water entrance and conditioning region, (2) ow development region, (3) test section, (4) water collection region, and (5) the recirculation system. Figure 2.2: Narrow channel schematic Water Entrance and Conditioning Region The entrance region collects returning water from a 5 cm PVC pipe and reduces large-scale non-uniformity in the ow before entering the development section. Large turbulent structures that enter are rst broken up with two honeycomb structures 5

39 that serve as ow straightners. The two honeycomb structures measure 5 cm deep with individual diameters of.25 cm. Upon exiting the honeycomb straightners, the water enters a fth order polynomial contraction where the ow is contracted from the entrance region's width of 3 cm to the development section's width of 4 cm. Before nally passing into the development section, the ow passes through standard porch screen which provides a pressure drop and serves to reduce ow non-uniformities from the top to the bottom of the channel Flow Development Region The ow development section has the same cross-section as the test section and ensures that the turbulent ow is fully developed before entering the region where data is taken. In this type of narrow channel it was found [2] that 4 h (28 cm) was required for the ow to become fully developed. Data runs for this thesis generally started 2 h (4 cm) downstream with data collection beginning at approximately 2 h (42 cm) on account of the time and distance needed to initially accelerate the traverse Test Section There is no dened boundary between the development and the test sections. The channel has over 3 h (6 cm) of uninterrupted test space created by joining two large plexiglass sections with glue at a butt joint. The joint is nearly invisible and does not adversely eect LDV or DPIV measurements. The limitation on usable length mainly exists on the limits of traverse travel within the fully developed region. Most tests only needed to use a small center sub-section to acquire the needed amount of data. These data runs generally used less than 5 h (3 cm) of channel length. 6

40 2.2.5 Water Collection and Recirculation System After exiting the test section, the water enters a small collection region before entering the PVC pipes which re-circulate the water. Contained within the collection region is afranklin.5 Hp submersible pump motor with a 2 blade, 8 cm diameter Michigan propeller. The three phase motor is controlled by a Parametrics Parajust controller which can adjust the centerline water velocity between 8 and 7 cm/s (Re = 6, 4; ). The water exits the collection box via a 2 cm PVC pipe. The large diameter of the pipe is used to accommodate the propeller which extends within the pipe. The piping eventually narrows to 5 cm to return the water to the water entrance and collection box. The piping makes several turns in order to pass over and around a passage way located beneath the development section of the channel. 2.3 Traverse System A two axis traverse serves as the carrier for the LDV probe which act as a virtual particle when coupled together in a control loop. The streamwise traverse is a Parker- Daedal extrusion with 55 cm of travel which is driven with a Parker-Digiplan ML- 345 servo drive with encoder. A Parker-Digiplan BL75 amplier powers the motor which is geared down 5: with a Hauser gearbox. Attached to the streamwise traverse is a ball screw traverse with 45 cm of travel which moves in the streamnormal direction of the ow. The streamnormal traverse is driven with a Compumotor N73FR servo motor powered by an Compumotor Apex sinusoidal analog drive. The amplier drive units are run in velocity mode and take direct velocity commands (as a voltage) from the QNS control system. Each amplier individually compensates for variations to maintain the requested velocity. The response of the system was found to be as small as 2 ms. 7

41 A Compumotor At645 motion controller was also also used to run stationary and moving eulerian cases. Three dimensional proles were obtained by attaching a third 35 cm of travel Parker-Daedal traverse to the streamnormal axis. The z-traverse was driven by a Compumotor SM233AE servo motor powered by a Compumotor TQ torque drive amplier. The At645 provided an automated system to control all three traverse axes simultaneously. Velocity measurements from the LDV system were still limited to two components though measurement points could exist in threedimensional space. 2.4 Laser Doppler Velocimetry System Instantaneous two component single pointvelocity measurements were made by a TSI laser velocimeter system. A 6 Watt Spectra-Physics Stabilite model 26 argon-ion laser was aligned with a TSI model 92 ColorBurst multicolor beam separator which split the laser beam into two single wavelength components. The blue ( = 488:nm) and green ( = 54:5nm) beams were then sent to the model 923 TSI ColorLink mulitcolor receiver where each beam was split and one of each color was Bragg shifted. The four beams were then coupled to a ve component ber optic cable and sent to the beam focusing probe which was rigidly attached to the vertical traverse. The fth ber optic cable returned the reected light from the measuring volume of the LDV system. A TSI IFA-75 digital burst correlator converted the optical signal to a voltage using a photomultiplier tube and was digitally ltered and processed returning a doppler frequency corresponding to the uid velocity. A number of lenses could t on the end of the focusing probe, each lens had a beam separation of 5 mm. A lens with a focal length of 25 mm was used for all measurements resulting in a measuring volume of 23m normal and 647m in the direction to the crossing beams. This measuring volume is two orders of magnitude 8

42 smaller than the smallest length scales in the ow, insuring that the tracer particle paths are linear within the volume. 2.5 Quasi-Numerical Simulation Setup The control loop to simulate the virtual particle was controlled by a Macintosh Quadra 95 NuBus computer. The computer provided a Labview front end user interface for the system and also logged LDV data for each run. During each run a National Instruments NB-DSP-23 digital signal processing (DSP) card took control of the computer's bus. The DSP was coupled with a National Instruments NB-DIO-32F digital board and a NB-AO-6 analog board. Upon taking control of the computer at the beginning of a particle following run, the DIO board would communicate with the IFA 75 to read instantaneous uid velocities and update the traverse velocity with signals sent through the analog board. A 33 MHz Texas Instruments microprocessor on the DSP integrated the transport equation and updated the traverse velocity to match the acceleration computed by the transport equation for a given velocity measurement. Chapter (3) gives complete details on the operation of the system. 2.6 Digital Particle Image Velocimetry System Unlike the single point measurements of LDV, Digital Particle Image Velocimetry (DPIV) was used to look at uid velocities in a two dimensional region of interest. The velocities in this region, generally around the virtual particle, could then be used to estimate vorticity. Section (7.) discusses the DPIV algorithms while the paragraphs below detail the hardware system. 9

43 2.6. Illumination System The two dimensional region of interest was illuminated by a sheet laser ( = 67nm). The seed particles used for the LDV system also served the DPIV system where laser light would sidescatter o these tracer particles. The particles' motion was captured by a digital video camera aligned perpendicular to the laser sheet and ow. These video images were post-processed to determine the velocity eld in the camera's reference frame { which also is the particle-lagrangian reference frame when the camera is attached to the traverse system. A Lasiris Magnum 75 mw line generating diode laser provided DPIV illumination. A wavelength of 67 nm was used so the 488 nm and 5 nm light from the LDV's argon-ion laser could be ltered using a di-chroic lter which reected the illumination light but allowed the LDV beams to pass. The Magnum laser also had the capability tobepulsed at up to khz. The Magnum laser was placed at the top of the vertical traverse, allowing the laser to travel and move with the system as it is following the virtual particle. In this way, the laser was always providing illumination in the region of interest surrounding the virtual particle. The laser light was reected by a rst surface mirror at an angle of 45 degrees in order to reect the light and provide an illuminated plane passing through the measuring volume of the LDV yet perpendicular to the camera system. Figure (2.3) demonstrates the laser working in this manner. A cylindrical lens placed in front of the Magnum laser reduced the sheet width providing brighter illumination and higher contrast in the desired area Camera System Images were acquired with a Kodak ES- Megapixel digital camera interfaced with an Imaging Technologies IC-PCI frame grabber. A dedicated Pentium Pro 2 MHz computer with 28 MB RAM could grab up to 8 8x8 images at 3 Hz without 2

44 dropping frames. Table (2.) gives the specications of the camera system. Frame Grabber Imaging Technologies IC-PCI 4 MB onboard memory AM-DIG-XHF-HS digital daughter card Image Mill drivers for Labview Camera Kodak ES. 8 horizontal x 8 vertical pixels 3 fps 8 bit pixel depth Frame exposure: 25 s to fully open Dimensions: 2." x 2.7" x 6." C-Mount lens thread Triggered acquisition with TTL trigger input TTL output for light strobing Kodak Interline chard coupled device (CCD) 9 micron square pixels 6 % ll ratio with microlens architecture Computer Pentium Pro 2 MHz 28 MB RAM GB hard drive space WinNT 4. Table 2.: DPIV hardware specications 2

45 2.6.3 Triggering System At higher Reynolds numbers, the uid uctuations are severe enough to produce smearing in images as particles move a considerable distance within a single =3 th s frame. In addition to smearing, large displacements produce larger uncertainties in DPIV correlations. The camera was run in a \frame stradling" mode during these circumstances in order to minimize these eects whichwould produce erroneous DPIV correlations. Frame straddling mode allowed both the duration of illumination per frame and the time between exposures to be adjusted to optimum levels. Frame stradling involves running the camera in continuous mode at 3 Hz but exposing the CCD only a short fraction of the frame by strobing the illumination source. The strobing process works on image pairs where a short light pulse illuminates a short period towards the end of one frame and another short period at the beginning of the next frame. The end eect is two short exposures which minimizes smearing with a user selectable delta time which can be adjusted to optimum levels. The vice of this method is a loss is time resolution between vector elds since DPIV correlations can only be performed on image pairs rather than each image. Reducing the exposure time during each frame also costs image quality with a reduction in contrast due to the reduced exposure time per frame. For these reasons, frame straddling was only used when absolutely needed. Otherwise, standard 3 Hz image capture was used. A dedicated computer controlled the camera and illumination when triggering was used. A MacADIOS II data acquisition board with a 2 bit A/D converter placed in a Macintosh IIci computer was connected to the Pentium Pro computer used for image acquisition. Upon initiation of the grabbing sequence, the Pentium Pro sent a trigger to the Macintosh which initiated the trigger sequence. The Macintosh then waited for a pulse the camera sends just prior to capturing a frame. When the strobe pulse was received, the Macintosh built a pulse train corresponding to the illumination sequence 22

46 which was then sent to the Magnum laser. The system was tested to insure that the pulse train properly synced with the image pairs Simultaneous LDV and DPIV Since vorticity measurements were desired in the region around the virtual particle being simulated, the DPIV imaging area had to include the LDV measurement volume. In addition, the plane to be imaged had to be normal to the LDV system so the velocity direction measured by the LDV would correspond to the velocities in the DPIV vector eld. To accomplish this and keep the camera view normal to the illumination plane, a di-chroic lter was used so the LDV and DPIV regions would overlap. Figure (2.3) demonstrates the di-chroic lter in operation. The lter is placed at a 45 degree angle relative to the LDV and illumination planes. The blue and green LDV beams pass through the lter and operate normally. Above 6 nm light is reected by the lter consequently reecting the red laser light used for DPIV illumination. The camera, which is held in place above the lter, looks at this reection which isa view that corresponds to the region around the LDV beams. 2.7 Summary This chapter has given an overview of the hardware system used for experiments performed in this thesis. Overall, the hardware system provides a turbulent water channel which is used by the traverse and LDV system to emulate the motion of a particle while recording data in the particle-lagrangian reference frame. The following chapters provide fundamental explanations of the theory behind the operation of the devices summarized here as well as validation of system performance and the data acquired. 23

47 Figure 2.3: DPIV setup: (a) LDV probe (b) vertical traverse (c) Magnum laser (d) ES- digital camera (e) di-chroic lter. 24

48 Chapter 3 QNS Methodology 3. Particle Following Methodology Understanding the physics and behavior of particles is essential to properly model particle motion in turbulent ows. A key ingredient in understanding such behavior is the knowledge of particle and uid statistics from the reference frame of the particle { or the particle-lagrangian (p-l) reference frame. The Tufts University Fluid Turbulence Lab (TUFTL) developed a system to emulate virtual particles within a real turbulent ow allowing uid velocity measurements to be made from within the p-l reference frame. Water within a water channel in essence solves the Navier-Stokes equations using real physics, while particle motion is governed by the particle transport equation. A simplied form (see section.4) of the ~ V = p ( ~ U f, ~ V p ), ~g; (3.) where ~ U f is the uid velocity and ~ V p is the particle velocity, is solved to determine particle accelerations which is simulated using a traverse system which follows the path of the emulated particle. A laser Doppler velocimetry probe (LDV) attached to the traverse takes measurements from what is now the p-l reference frame. Additional devices, such as a camera can also be attached to take data as well. 25

49 Figure 3.: QNS methodology overview. The system relies on a control loop using a discretized version of equation (3.) running continously on a DSP card to take feedback from the LDV system and update the traverse acceleration to match the virtual particle. The discretized form of (3.) has the term V (n+) p, V n p t = p (V p, U f ) n, g (3.2) (V p, U f ) n (3.3) which corresponds to the velocity measured by the LDV probe. Equation (3.2) can be solved yielding V (n+) p = t p U n measured + V n p, tg (3.4) which is the velocity sent to the traverse. This equation is solved continuously within the control loop in order to emulate the virtual particle with the traverse. The emulated particle's location corresponds to the measuring volume of the LDV probe from which the value of term in equation (3.3) is determined. 26