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Quantitative Measurement of the Density Gradient Field in a Normal Impedance Tube Using an Optical Deflectometer


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QUANTITATIVE MEASUREMENT OF THE DENSITY GRADIENT FIELD IN A NORMAL IMPEDANCE TUBE USING AN OPTICAL DEFLECTOMETER By PRIYA NARAYANAN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003

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Copyright 2003 by Priya Narayanan

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iii ACKNOWLEDGMENTS I would like to thank foremost my advi sor, Dr. Louis N Cattafesta, for his guidance, support and patience. His continua l guidance and motivation made this work possible. I would also like to express my heartfelt gratitude to my co-advisor Dr. Mark Sheplak for his support. I owe special thanks to Dr. Bruce Caroll and Dr. Paul Hubner for their help during the design of the experimental set-up. I thank all of the students in the Interdisciplinary Microsystems Group, particularly Ryan Holman for his help during data acquisition. I would also like to express my gratitude to my colleagues Todd Schultz, Steve Horowitz, Anurag Kasyap, Karthik Kadirv el and David Martin for thei r help during the course of this project. I would like to thank my undergraduate a dvisor Dr. Job Kurian for his motivation and encouragement. I would also like to th ank my roommates and friends for making my stay in Gainesville a memorable one. Finally, I want to thank my parents and my sister Poornima for their endless support.

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iv TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES.............................................................................................................vi LIST OF FIGURES..........................................................................................................vii ABSTRACT....................................................................................................................... ..x 1 INTRODUCTION........................................................................................................1 1.1 Basic Schlieren Method......................................................................................2 1.2 Optical Deflectometer.........................................................................................5 1.3 Review of an Optical Deflectometer...................................................................5 1.4 Research Objectives............................................................................................8 1.5 Thesis Outline.....................................................................................................9 2 OPTICAL DEFLECTOMETER................................................................................10 2.1 Normal Impedance Tube...................................................................................10 2.2 Theory of a Schlieren System...........................................................................13 2.2.1 Deflection of Light by a Density Gradient ............................................14 2.2.2 The Toepler Method ..............................................................................16 2.3 Sensitivity Analysis...........................................................................................20 3 EXPERIMENTAL SET UP.......................................................................................27 3.1 Basic Schlieren Setup........................................................................................27 3.2 Normal Impedance Tube...................................................................................28 3.2.1 Components ..........................................................................................28 3.2.2 Fabrication of the Test Section .............................................................29 3.3 Data Acquisition System...................................................................................30 3.3.1 Photosensor Module ..............................................................................31 3.3.2 Positioning System ................................................................................31 3.3.3 Signal-Processing Equipment ...............................................................31 4 DATA ANALYSIS....................................................................................................32 4.1 Calibration of the Optical Deflectometer..........................................................32 4.2 Data Reduction Procedure.................................................................................41

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v 4.3 Dynamic Calibration.........................................................................................45 4.4 Experimental Procedure....................................................................................47 5 RESULTS AND DISCUSSION.................................................................................49 5.1 Theoretical Results............................................................................................49 5.2 Numerical Results.............................................................................................52 5.3 Experimental Results........................................................................................55 5.3.1 Measurement of Density Gradient Using the Optical Deflectometer ...55 5.3.2 Measurement of Density Gradient Using microphones ........................63 5.3.3 Comparison of Results Obtained by the Two Methods ........................65 6 CONCLUSION AND FUTURE WORK...................................................................69 6.1 Conclusions.......................................................................................................69 6.2 Future Work......................................................................................................69 APPENDIX A KNIFE-EDGE GEOMETRY.....................................................................................71 B LIGHT RAYS IN AN INHOMOGENEOUS FLUID................................................73 C UNCERTAINTY ANALYSIS...................................................................................75 C.1 Uncertainty in Amplitude..................................................................................75 C.1.1 Deflectometer Method ..........................................................................75 C.1.1 Two Microphone Method .....................................................................79 C.2 Uncertainty in Phase.........................................................................................82 C.2.1 Deflectometer Method ..........................................................................82 C.2.2 Microphone method ..............................................................................84 D DENSITY GRADIENT..............................................................................................85 LIST OF REFERENCES...................................................................................................87 BIOGRAPHICAL SKETCH.............................................................................................89

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vi LIST OF TABLES Table page 4-1 Slope of the calibration curv e at three different locations........................................40 C-1 Uncertainties in various pa rameters for a deflectometer..........................................78 C-2 Uncertainties in various para meters for the microphone method.............................80

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vii LIST OF FIGURES Figure page 1-1 Simple schlieren setup................................................................................................4 2-1 Plane wave tube........................................................................................................11 2-2 Light source in the plane of the knife-edge..............................................................18 2-3 Ray diagram of the schlieren setup..........................................................................21 2-4 Schlieren head with the conjugate plane..................................................................25 3-1 Deflectometer set-up................................................................................................28 3-2 Normal impedance tube...........................................................................................29 3-3 Setup for normal impedance tube.............................................................................30 4-1 Light source in the plane of the knife-edge..............................................................33 4-2 Knife-edge calibrati on of photodiode sensor...........................................................36 4-3 Photodiode knife-edge calibration...........................................................................37 4-4 Photodiode knife-edge calibra tions at three locations..............................................38 4-5 Linear region of the calibration curves at three locations after regression analysis.39 4-6 Calibration curves after th e ground glass was inserted............................................40 4-7 Slope of the calibration curve plotted along the test section....................................40 4-8 Data reduction procedure.........................................................................................44 4-9 Impulse response of the e xperimental photo-detector..............................................45 4-10 Frequency response of the experimental photo-detector at the experimental gain setting.......................................................................................................................4 6 5-1 Comparison of pressure, density a nd density gradient distributions........................50 5-2 Phase variation for a rigid termination.....................................................................51

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viii 5-3 Pressure waves at various phases for R = 1.............................................................53 5-4 Pressure waves at various phases for R = 0.............................................................53 5-5 Pressure waves at various phases for R = 0.5..........................................................54 5-6 Noise floor of the experimental phot o-detector at an operational gain of 0.35.......56 5-7 Noise floor of the experimental photodetector (light-on) for various gains...........56 5-8 Example of photo-det ector power spectrum at 145.4 dB .......................................57 5-9 Example of reference microphone power spectrum at 145.4 dB .............................57 5-10 Example of coherent spectrum at 145.4 dB .............................................................58 5-11 Example of photo-detect or coherence power at 145.4 dB .......................................58 5-12 Magnitude of the frequency response function at 145.4 dB ...................................59 5-13 Density gradient amplitude along the length of the tube at 145.4 dB ......................59 5-14 Example of photo-det ector power spectrum at 126.4 dB .......................................60 5-15 Example of reference microphone power spectrum at 126.4 dB .............................61 5-16 Coherent power of the photo-detector at 126.4 dB ..................................................61 5-17 Example of coherence spectrum at 126.4 dB ...........................................................62 5-18 Density gradient amplitude along the length of the tube at 126.4 dB ......................62 5-19 Phase difference between the photo-detect or signal and the reference microphone signal at 145.4 dB ...................................................................................................63 5-20 Density gradient amplitude along the length of the tube using the microphone method at 145.4 dB ..................................................................................................64 5-21 Density gradient phase along the length of the tube using the microphone method at 145.4dB ...................................................................................................................64 5-22 Magnitude of the density grad ient using the two methods at 145.4 dB ...................65 5-23 Magnitude of the density grad ient using the two methods at 126.4 dB ...................66 5-24 Phase of the density gradient using the two methods at 145.4 dB ...........................67

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ix 5-25 Phase of the density gradient using the two methods at 145.4 dB after the phase correction from the photo detector...........................................................................67 5-26 Phase of the density gradient using the two methods at 126.4 dB after the phase correction from the photo detector...........................................................................68 A-1 Magnification of the source on the screen...............................................................71 B-1 Deflection of a light ray in inhomogeneous test object............................................73 C-1 Error bar in the amplitude of the density gradient using the deflectometer at 145.4 dB ..................................................................................................................77 C-2 Error bar in the amplitude using the microphone method 145.4 dB .......................81 C-3 Comparison of microphone a nd deflectometer method at 145.4 dB ......................81 C-4 Error bar in the phase using the deflectometer at 145.4 dB ...................................82 C-6 Error bar in the amplitude using the deflectometer at 145.4 dB ............................83 C-7 Comparison of microphone a nd deflectometer method at 145.4 dB .......................84

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x Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science QUANTITATIVE MEASUREMENT OF DENSITY GRADIENT FIELD IN A NORMAL IMPEDANCE TUBE USI NG AN OPTICAL DE FLECTOMETER By Priya Narayanan December 2003 Chair: Louis N. Cattafesta Cochair: Mark Sheplak Department: Mechanical and Aerospace Engineering Interest is growing in optical flow-v isualization technique s because they are inherently nonintrusive. A commonly used optical flow vi sualization technique is the schlieren method. This technique normally provides a qualitative measure of the density gradient by visualizing changes in refractive index that accompany the density changes in a flowfield. The “optical deflectometer” in strument extends the sc hlieren technique to quantitatively measure the density gradient The optical deflectometer has been successfully used to characterize highly compre ssible flows. In this thesis, an optical deflectometer is studied that can provide quantitative measurements of the acoustic field in a normal impedance tube. Results of the static calibration performe d on the instrument are presented. The frequency response of the instrument is infe rred using a laser impulse response test. Two-point cross-spectral analysis between the light intensity fluctua tions in a schlieren

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xi image and a reference microphone signal are used to determine the density gradient field in the normal impedance tube. Numerical simulations were obtained for test cases to validate the data-reduction method. In addi tion, the two-microphone method is used to verify the results obtained from the deflectometer. Results of the experiments performed for normal sound pressure levels of 145.4 dB and 126.4 dB for a plane wave at 5 kHz are presented. A detailed uncertainty analysis is also performed. The results were in good agreement with each other except at the density gradient maxima (pressure mi nima) at the higher sound pressure level.

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1 CHAPTER 1 INTRODUCTION Many modern-day research activities invol ve studies of substances that are colorless and transparent. The flow and te mperature distribution [1] of many of these substances are of significant importance (f or example, mixing of gases and liquids, convective heat transfer, plasma flow). As suggested by the proverb “Seeing is beli eving,” suggests, visu alization is one of the best ways to understand the physics of a ny flow. Visualization also aids flow modeling. Hence, over the years numerous t echniques have been developed to visualize the motion of fluids. These flow visualizati on techniques [2] have been used extensively in the field of engineering, physics, medical science, and oceanography In aerospace engineering, flow visualizati on has been an important tool in the field of fluid dynamics. Several flow visualizati on techniques [3] have been used in the study of flow past an airfoil, jet mixing in supersonic flows and acoustic oscillations. Flow visualization can be cl assified as non-optical and opt ical techniques. In the former, seed particles are usually added and th eir motion observed. Th is indirectly gives information about the motion of the fluid itsel f. However, in the case of unsteady flow, these methods [2] are prone to error because of the finite size of the seed particles. Purely optical techniques, on the other hand, are based on the inter action of light rays with fluid flow in the absence of macroscopic seed particles. The in formation recorded is dependent on the change in optical properties of the fluid. One of the commonly studied properties is the variation in refr active index with fluid flow.

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2 Several methods are commonly used to vi sualize refractive-index variation in fluids. Common methods [3] include sha dowgraph, schlieren and interferometric techniques. Since these techniques are non-in trusive, the flow is not disturbed by the measurement technique. Classical optical flow visualization tool s (like schlieren and shadowgraphy) were used in the study of compressible or high-de nsity gradient flows [2] (shock waves in wind tunnels, turbulent flow, convection patt erns in liquids, etc.) Many of these techniques have been extended for the quantit ative study [4] of the fluctuating properties of the flow under consideration. The objective of this thesis is to use one of the classical techniques to detect acoustic waves in a normal impedance tube. Th ese waves, unlike the flow fields in the previous studies, produce very small density gr adients, and there is not a mean flow. If successful, this technique could be used to study various acoustic fi elds. One immediate application is the characterization of compliant back plate Helmholtz resonators [5] at the Interdisciplinary Microsystems Laboratory at the University of Florida using the normal impedance tube. These resonators will late r serve as fundamental components of the electromechanical acoustic liner used for je t noise suppression. A quantitative optical flow visualization technique can be used to study phenomen a (such as scattering effects in the acoustic field) that cannot otherwis e be determined quantitatively using only microphone measurements. 1.1 Basic Schlieren Method Of all the flow-visualization techniques mentioned earlier, shadowgraph is perhaps the simplest. Shadowgraphy is often used wh en the density gradients are large. This technique can accommodate large subjects and is relatively simple in terms of materials

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3 required. The primary component consists of a point light source. The resulting shadow effect produced by the refractive-index field can be observed on an imaging surface. In terms of cost, this method is probably the le ast expensive technique to set up and operate. However, this system is not very sensitive. Also it is not a method suitable for quantitative measurements of the fluid dens ity. However, it is a convenient method for obtaining a quick survey of flow fields with varying density, particularly shock waves. Another commonly used flow visu alization technique is the inte rferometric technique [1]. It is highly sensitive and can provide quantitative informati on. However, such systems are expensive, complex to set up, and can onl y deal with relatively small subjects. A shadowgraph system can be converted in to a schlieren system with a slight modification in its optical arrangement. Schlieren systems are intermediate in terms of sensitivity, system complexity, and cost. The German word “schliere” means “streaks,” since the variation in the refractive index show up as streaks. It was used in Germany for detection of an inhomogeneous medium in optic al glass, which is of ten manifested in the form of streaks. In principle, the ligh t passing through a medium with relatively small refractive differences bends light to directions other than th e direction of propagation of light. The simple schlieren set-up consists of optics to produce a point-light source, two lenses, a knife-edge and a screen The point light s ource is placed at the focus of Lens 1. The lens produces parallel light beams, which pass through the test section and are made convergent by the second lens called the sc hlieren head (Lens 2). An image of the light source is formed at the focal point of the schlieren head.

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4 LENS 1 KNIFE-EDGE TEST AREA POINT LIGHT SOURCE LENS 2 SCREEN 1 f 2 f Figure 1-1: Simple schlieren setup. At this position, a knife-edge (oriente d perpendicular to the desired density gradient component) cuts off a certain porti on of the light-source image and reduces the intensity of the recorded image plane. The edge is adjusted so that, if an optical disturbance is introduced such that a portion of the image of the source is displaced, the illumination of the corresponding part of th e image on the screen will decrease or increase according to whether the deflection is toward or away from the opaque side of the knife edge. Building on this model an extended light source is considered in Chapter 2. Schlieren systems can be configured to su it many different applications and sensitivity requirements. They can be used for obser ving sound waves, shock waves, and flaws in glass. Its principal limitations are field si ze (limited by the diam eter of the optical components) and optical aberrations.

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5 1.2 Optical Deflectometer Optical deflectometry involves the direct measurements of the density gradient in a flow using the schlieren technique. The conventional schlieren methods measure the total angular deflection experi enced by a light ray while cros sing the working section. This deflection can be directly related to the illumination of the image on the screen. The quantitative version of the schlieren technique involves the determination of the density gradient component from the meas ured deflection. E xperimental calibration and theory is used to characterize the relatio nship between light inte nsity fluctuation and density gradient. An adaptor is flush mount ed onto the screen, which is connected to a photo-sensor module using a fiber optic cable. The photosensor detects the instantaneous fluctuations in light intensity in the schlieren image and c onverts the optical signal to an electrical signal. 1.3 Review of an Optical Deflectometer The first important contribution to the de velopment of the deflectometer was made by Foucault [4] by using an explicit cutoff in th e form of a knife edge for the schlieren measurements. At the same time, the measurement technique was re-invented by Toepler [4] who named it “schlieren.” Though the knife edge was not developed by Toepler, he has been given credit historically for de veloping the schlieren imaging technique. The method was recognized to be a very valuable tool and used by many eminent scientists (Wood, Prandtl, etc.) [4]. Prin ciples and experimental setup of schlieren techniques [3] were later explained by Holder and North in 1963 as part of the Notes on Applied Science and was published by the Natio nal Physical Laborat ory. Fisher and Krause measured the light sca ttered [6] from two optical be ams crossed in the region of

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6 interest in a turbulent air jet. By cross-co rrelating the signal from each beam, information on the behavior of scatterer number-density n ear the intersection point was determined. It was observed that this method was suffi ciently general and independent of the method employed to obtain the desired fluctuat ion of light intensity. Taking advantage of this fact, Wilson and Damkevala [7] adapte d a cross-correlation technique to obtain statistical properties of scalar density fluctuat ion. In their method, two schlieren systems are used, each of which gives signals in the flow directi on integrated along the beam path. The optical beams are made to intersect in the turbulent field and, with the further assumption of locally isotropic conditions, the cross-correlati on of the two signals and the local mean–square density fluctuation in th e beam intersection point are determined. Davis [8] used a single-beam schlieren syst em and made a series of measurements to investigate the density fluctuations pres ent in the initial region of a supersonic axisymmetric turbulent jet. The difference in distribution of density fluctuation due to preheating was observed using th is method. Later, a quantit ative schlieren technique was used by Davis [9] to determine the local scales and intensity of turbulent density fluctuations. Recently, McIntyre et al. [4] developed a technique called “optical deflectometry” that was well suited to the study of coherent structures in compre ssible turbulent shear flows. A fiber-optic sensor was embedde d in a schlieren image to determine the convective velocities of large-s cale structures in a supersonic jet shear layer. The relative simplicity, low cost, and excellent frequency re sponse of the optical deflectometer makes it an ideal instrument for turbulence measur ements, especially in high-Reynolds number

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7 flows. But one of its drawbacks is that it provides the information integrated along the beam path. This limitation can be removed using a sharp focusing schlieren [4]. Weinstein [10] has recently provided the analysis and performance of a highbrightness large-field focusing schlieren sy stem. The system was used to examine complex twoand three-dimensional flows. Diffuse screen holography was used for three-dimensional photography, multiple colors were used in a time multiplexing technique, and focusing schlieren was obt ained through distorti ng optical elements. Also, Weinstein [11] described techniques that allow the focusing schlieren system to be used through slightly di storting optical elements. It was also mentioned that the system could be used to examine comple x twoand three-dimensional flows. Alvi and Settles [12] refined the quan titative schlieren system combining the focusing schlieren system with an optical deflectometer. This instrument is capable of making turbulence measurements and was verified by measurements of KelvinHelmholtz instabilities produced in a low-speed axisymmetric mixing layer. Garg and Settles [13] used the technique for the measurements of density gradient fluctuations confined to a thin slice of the fl ow field. The optical deflectometer was used to investigate the structure of a two-dime nsional, adiabatic, boundary layer at a free stream Mach number of 3. The results obtai ned were found to be in good agreement with that obtained using a hot wire anemometer. This result helped validate the new measurement technique. Further, Garg et al. [14,15] used the light-i ntensity fluctuations in a real-time schlieren image to obtain th e quantitative flow-field data in a twodimensional shear layer spanning an open cavit y. Instantaneous density gradient and density fields were obtained from the data collected. With the help of a reference

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8 microphone, phaselocked movies were created Surveys were carried out for a Mach 0.25 cavity shear layer using the schlieren inst rument as well as hot-wire anemometer. The results showed that the growth rates of in stability waves in the initial “linear” region of the shear layer could accurately be measured using this technique. A similar procedure was adopted by Kege rise et al. [16] using the optical deflectometer with various higher-order corr ections to experimentally study the modal components of the oscillations in a cavity flow field. Shear layer a nd acoustic near field measurements were performed at free stream Mach numbers of 0.4 and 0.6. Standing wave patterns were identified in the cavity using this method. These experiments have considerably improved the understanding of th e cavity physics that cause and maintain self-sustaining oscillations. In the most recent development, Cattafest a et al. [17] verified using primary instantaneous schlieren images that the multiple peaks of comparable strength in unsteady pressure spectra, which char acterize compressible flow-induced cavity oscillations, are the results of mode-switching phenomenon. 1.4 Research Objectives The objective of the current project is to develop a system, which can measure the density gradient in a normal impedance tube. This represents an intermediate step towards a focused schlieren system for nor mal acoustic impedance tube measurements for applications described earlier in the chapte r. It can be seen that all the measurement techniques used till date like the microphone method are flow intrusive. The main advantage of this technique is that there is no flow intrusion. Also two dimensional flow fields and effects of scattering can be determ ined using this technique. The aim of the

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9 experiment ultimately is to verify the results obtained from the deflectometer using the microphone measurements, thereby deve loping a more efficient technique. 1.5 Thesis Outline The thesis is organized into six chapters. This chapter presents the introduction, background, and application of the optical defl ectometer. Chapter 2 presents the theory of the schlieren system as well as that of the normal impedance tube. Theoretical formulations for the sensitivity are also developed in this chapter. Chapter 3 discusses the steps involved in setting up the system and the plane wave tube including the data acquisition system. Chapter 4 presents th e data analysis using the cross spectral technique. It also describes the static a nd dynamic calibration of the system. Chapter 5 presents the numerical and experi mental results obtained during the course of this project. Chapter 6 presents concluding remarks, propos ed design modifications, and future work for the optical deflectometer.

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10 CHAPTER 2 OPTICAL DEFLECTOMETER As discussed in Chapter 1, the goal of this project is to design and test an optical system capable of measuring the density gradie nt of plane waves in an impedance tube. Therefore, this chapter discusses the theory behind an optical deflectometer and the sensitivity of the system. First, an expre ssion for the pressure fluctuation in a normal impedance tube for plane waves is derived. Th e pressure fluctuation is then related to the density and density gradient, which causes the refraction of the light rays. Second, the relationship between the light intensity fluctu ation and the density gradient in the flowfield is presented. The sensitivity of the system with regard to the various optical parameters is then derived. 2.1 Normal Impedance Tube The flow visualized by the deflectometer is generated in a rectangular normal impedance tube. In the next section, theore tical derivations of th e acoustic field within the tube are given for plane wave. Single Impedance Termination. The acoustic flow fiel d is thus comprised of waves that have uniform pressure in all planes perpendicular to the direction of propagation and are termed as plane waves. The plane wave assumption is valid below the cut-on frequency [18] of a square tube, given by 2 c s where c is the isentropic sound speed and s is the width of the normal impedan ce tube. As can be seen in Figure

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11 2-1, an acoustic driver ultimately generate s plane waves on one end of the impedance tube of length l and the specimen is placed d l pi (x, t) pr(x, t) x ACOUSTIC DRIVER SPECIMEN Figure 2-1: Plane wave tube. at the other end. The pressure p inside the tube at a position d is given by ()ee,ikdikd irpdpp (2-1) where i p and r p are the incident and reflected pressure respectively. Here eikl ipA and eikl r p B are phasors. The time-harmonic dependence, ite is implicit in p The complex reflection coefficient R is defined as .r i p R p (2-2) Substituting Equation 2-2 in Equation 2-1, the expression for () p d becomes ()(ee).ikdikd ipdpR (2-3)

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12 In a plane wave, the velocity can be re lated to pressure through the equation p u c and, the velocity in the plane wave tube can be written as ()(eRe).ikdikd ip ud c (2-4) The specific acoustic impedance is defined as the ratio of the pressure to the velocity in the tube (ee) (). (ee)ikdikd ikdikdpR Zdc uR (2-5) When 0 d the impedance must equal the terminating impedance, which is the impedance of the specimen (1) (1)n R Zc R (2-6) Solving for R gives .n n Z c R Z c (2-7) When R is not equal to zero a standing wa ve pattern results and is often characterized by the [18] pressu re standing wave ratio (SWR) max min1 1 p R SWR p R (2-8) which is the ratio of maximum to minimum pr essure in the tube. The complex reflection factor R can be represented in polar notation as .rj R Re (2-8) The reflection factor R can also be split into re al and imaginary components as realimg R RjR (2-9)

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13 where sinimgrRR and cosrealrRR The reflection coefficient R can be measured [19] [20] using the two-microphone method. '2().Riks i jkls jksHe RRee eH (2-10) where s is the center-to-center spaci ng between the two microphones, 'l is the distance from the test sample to the nearest microphone and H is the geometric mean of the measured transfer function between the microphones 12,HHH (2-11) and 1H and 2H are the transfer functions for th e microphones in the standard and switched configurations. The normalized impedance can be obtained by substituting Equations 2-9 in Equation 2-7 as 22 221 (1)realimg real realimgRR Z RR (2-12) and 222 (1)img img realimgR Z RR (2-13) 2.2 Theory of a Schlieren System This section summarizes the theory of opera tion of a simple [3] schlieren system. Details of the quantitative part of the experimental set-up are discussed in Chapter 4.

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14 2.2.1 Deflection of Light by a Density Gradient In the experiments being conducted the refr active index of the test section varies along the direction of propagation. This is caused by the variation in the density of the medium. For an isentropic pr ocess, pressure and density fluctuations are related by 2. p c (2-13) The pressure gradient in the x -direction is 2. p c x x (2-14) From the above equation, it can be s een how the density varies in the x direction in an impedance tube. Substituting Equation 2-3 in Equation 2-14 and taking the derivative w.r.t d, we obtain the density gradient as 2() .jkdjkd ipjkeRe dc (2-15) The aim of the experimental program is to determine this density gradient using the schlieren system. In the case of gaseous subs tances the refractive index, n is related to density by the Gladstone-Dale equation [3] (Equation 2-16). 1. nk (2-16) where 432.25910 kmkgis the Gladstone-Dale constant. If there exists a gradient of refractive index normal to the light rays in a working section, the ra ys will be deflected since light travels more slowly in a nonvacuum media, with the velocity given by */, ccn (2-17)

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15 where *c is the velocity of light in vacuum. Th e basis of the schlie ren technique relies on the fact that the deflection of the light rays is a measure of the first derivative of the density with respect to distance (i.e., the density gradient). The derivation [1] is as follows: The curvature of the ray is proportional to the refractiv e-index gradient in the direction normal to the ray. If we take the z axis as the direction of the undisturbed ray, the curvatures in the x y and yz planes respectively are given by (see Appendix B) 2 21 x n znx (2.18) 2 21 yn zny (2.19) The total angular deflection in the x z and yz planes are taken as x' and y' respectively 1 '= xn dz nx (2-20) 1 '= yn dz ny (2-21) If the optical disturbance is in the work ing section of a wind tunnel, the light ray will be refracted on leaving the tunnel so that 0sin'sin, nn (2-22) where on is the refractive index of th e air surrounding the tunnel, and n is the refractive index in the working section. Thus, assumi ng small angular deflections, the final angular deflections measured beyond the tunnel are

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16 01 ,xn dz nx (2-23) 01 .yn dz ny (2-24) Since the refractive index of air is approxima tely equal to 1, the expressions can be written as .xn dz x (2-25) .yn dz y (2-26) In the case of two-dimensional flow in a tunnel of width W, the expressions become ,xn W x (2-27) .yn W y (2-28) where the deflection is toward the re gion of highest density. 2.2.2 The Toepler Method In the Toepler schlieren system [4] a rect angular slit source is used, with the long dimension of the slit parallel to the knife-e dge. As shown in Figure 2-2 an image of the slit source produced at the knife-edge, of which only a part of height a is allowed to pass over the knife-edge. With a homogeneous test field, the recording plane is evenly illuminated with an intensity ) (y x I= const, which is propor tional to the value of a. Light rays deflected by an angle x due to a disturbance in the test field cause a (vertical) shift of the light source image by an amount a

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17 2tan.xaf For small values of deflection since tan()x is approximately x so 2,xaf (2-29) where2 f is the focal length of Lens 2. Let P be the power of the light beam. The power can be defined as I A P where I is the intensity (Power per unit area) and A is the illumination area. Assuming the power to be conserved between th e knife-edge and the screen K EKEscreenscreenIAIA (2-30) or K EKE screen s creen I A I A (2-31) where K E I is the intensity of light passing the knife -edge. In the case of half knife-edge cut-off, the image on the rear side of the knife-edge has a height b and breadth 2 a and hence the area '. 2KEa Ab (2-32) Equation 2-31 can be expressed as 2KE screen s creen I ab I A (2-33) When the light source image shifts by an amount a 2KE screen screena I ab II A (2-34) And the intensity change on the screen can be expressed as

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18 KE s creen I ab I A (2-35) UNDISTURBED IMAGE OF THE SOURCE IMAGE WITH OPTICAL DISTURBANCE KNIFE-EDGE a abb Figure 2-2: Light source in the plane of the knife-edge The recording plane receives an intensity changed by I in the corresponding image point; the relative intensity change also known as contrast C is obtained from Equation 2-33 and Equation 2-35 as 22 2 .x screenf Ia C Iaa (2-36)

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19 The contrast sensitivity is given by 22c f dC S da (2-37) This sensitivity [14] in terms of the contrast is important in the traditional qualitative method since the photographic films are se nsitive relative in tensity changes. But in the case of quantitative schliere n technique, the photo-sensor can detect absolute light intensity change. Hence th e quantitative sensitivity is defined as 22 ()qscreenf dI SI da (2-38) Substituting the expression for s creenI we obtain 2()KE q s creen f Ib dI S dA (2-39) Using the value of angular deflec tion from Equation 2-27 we obtain 22screenfW I n I ax (2-40) Using the Gladstone-Dale Equation 2-16 we get 22 .screenkfW I I ax (2-41) Alternatively, the relative intensity change in theydirection can be obtained by turning the knife-edge by 90o, and the relative intensity change is 22 .screenkfW I I ay (2-42) The schlieren system “sensitivity” is often equated to the relative change in illumination intensity, which is a measure of th e refractive index or density gradient in the direction normal to the knife-edge.

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20 In the case of a normal impedance tube with plane waves, since the density gradient exists only in the x direction, the measured intens ity change is therefore given by Equation 2-41. 2.3 Sensitivity Analysis A detailed sensitivity analysis of the optical system is performed in this section. Since a point source, discussed in Chapter 1, is not realistic, an extended light source is considered in the following derivation. The extended light source is in the form of a rectangular aperture, and a c ondenser lens is used to focus the light beams from the source at the rectangural aperture. The experime ntal set-up will be discussed in detail in Chapter 3. Figure 2-3 shows the ray diagram of the sch lieren setup. It can be seen that the light beams from the source are focused by the co ndenser lens at the re ctangular aperture. From this point onwards, the rectangular aperture/slit with height h and width d is treated as the source. Lens 1 is placed at a distance equivalent to it s focal length from the slit and collimates the light beam. Since the light source is no-longer a point source, Lens 1 no longer produces rays parallel to the optical axis. It can be taken [4] as an array of the source distributed along the height h. This can be proved by taking four rays from the top of the slit into consideration. Ray (1) passes th rough the center of the lens and is undeviated. Ray (2) which initially travels parallel to the optical axis is deviated so that it passes through the focal point. It can be seen from the figur e that these two rays comes out of the lens parallel to each other. Also any ray originating from the top of the slit emerges out of the lens parallel to Ray (1) and Ray (2). Two such rays, Ray (3) and Ray (4) are then taken into cons ideration. Ray (3) passes th rough the focal point of

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21 f 1 f1f2d1d2x h f2 1 2 3 4 cd LENS 1 LENS 2 CONDENSER LENS RECTANGULAR APERTURE SCREEN KNIFE-EDGE 5 6 Figure 2-3: Ray diagram of the schlieren setup. Lens 2 and is deviated to travel parallel to the optical axis. Ra y (4) passes through the center of the lens and is undevi ated. These two beams are fo cused at the focal point of Lens 2, where the knife-edge is placed. The image is further magnified and viewed on the screen placed at a distance x from the knife-edge. Rays (5) and (6) are the extreme rays originating from the top of the slit. The power of the light beam emitted by the slit is ().slitslitslitslitPIAIdh (2-43) It is assumed that the power is conserved th rough out the optical se tup for a case without the knife-edge, therefore I A P is constant. /, s litslitscreenwoscreenIAIA or /.slit screenwo s creen I dh I A (2-44) where / s creenwoI is the intensity of the image on the screen with out knife-edge. Also, as the light passes through any lens both source dimensions are magnified such that the image area at any point will be ()() mdmh or 2mdh where m is the magnification ratio defined as the ratio of im age dimension to object dimension. For a

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22 lens combination, as in Figure 2-3, the to tal magnification is th e product of each lens magnification such that 123screenmmmm And, area of the image on the screen is 2,screenscreen A mdh (2-45) where s creenm is the magnification on the screen. In the case of a bi convex lens, the magnifi cation is also equal to the ratio of the image distance from the lens to the object distance from the lens. Therefore the magnification due to the Lens 1 is 1 1, m f (2-46) since the slit is placed at the foal point of Lens 1 and the image is formed at infinity. The magnification due to the Lens 2 is 2 2, f m (2-47) since parallel rays falls on the lens, and the im age is formed at the focal point of Lens 2. The magnification between the focal point of Lens 2 and the screen (see Appendix A) using similar triangles is 22 1 3 2 2 12 1 df h f mx f h f (2-48) where x is the distance between the knife-edge and the traverse. Hence the total transverse magnification on the screen is

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23 22 21 2 2 1 12 1,screendf h ff mx f f h f 22 21 2 2 1 12 1,screendf h ff mx f f h f (2-49) Substituting Equation 2-49 into Equation 2-45 the area of the screen can be expressed as 2 22 2 21 2 2 1 12 1.screendf h ff Axdh f f h f (2-50) The resulting intensity on the image plane is / 2 22 2 21 2 2 1 12 1slit screenwoI I df h ff x f f h f (2-51) In practice however, a knife-edge is placed at the focal point of Lens 2. There is no power loss prior to the knife-edge location. Hence K EKEsourcesourceIAIA (2-52) where K E I is the light intensity, and KEA is the area illuminated on the front side of the knife-edge. 2.KEKEAmdh (2-53)

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24 Also, the magnification at the knife-edge can be calculated as 1 2 2 1*f f m m mKE Substituting in Equation 2-52, the light intensit y at the knife-edge can be expressed as 2 2 1. s ource KEI I f f (2-54) The knife-edge blocks a portion of the light as shown in Figure 2-2. In the case of half-cutoff, the area of the image on the rear side of the knife-edge K EA (where the prime superscript denotes a position after th e knife-edge) has already been derived in Equation 2-32. As in section 2.2.2, the power can be assumed to be conserved from the knife-edge to the screen and Equation 2-30 can be applied. Also, the constants s creenA and K E I can be obtained from Equation 250 and Equation 2-54 respectivel y. The expression for the intensity of the image on the screen (Equation 2-33) can be modified as 2 22 4 21 2 2 1 12 21source screenIab I df h ff x dh f f h f (2-55) and the quantitative sensi tivity (Equation 2-39) of the instrument in the x axis can be expressed as 2 2 22 4 21 2 2 1 12 1source qIfb S df h ff x dh f f h f (2-56)

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25 Since 2 12 1() f bmmhh f f2d2x f2 LENS 2 S TEST AREA Figure 2-4: Schlieren head with the conjugate plane. Equation 2-56 can be written as 2 2 22 3 21 2 2 1 12 1source qIf S df h ff x d f f h f (2-57) If the test area is placed at a distance s from the focal point of Lens 2, the real image is formed at the conjugate plane. Using the thin lens formula, we obtain

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26 2 2 f x s (2-58) Hence if the screen is placed at the conjugate plane the sens itivity can be expressed as 2 2 22 3 21 1 12 1source qIf S df h ff d hs f f (2-59) The equation can be non-dimentionalized as 2 2 22 3 21 1 12 1q sourceS f I df h ff d hs f f (2-60) It can thus be seen that the quantitative sensitivity is a functi on of various optical parameters. Once the field of view 2d and the conjugate plane distance x are fixed, an optimal value for 2 1 f f (ratio of the focal length of Lens 2 and Lens 1) and h d (Ratio of the height and width of the slit), can be obtained for maximum sensitivity (taking diffraction effects into consideration).

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27 CHAPTER 3 EXPERIMENTAL SET UP Flow visualization and data acquisiti on using the optical deflectometer was performed at the Interdisciplin ary Microsystems Laboratory at the University of Florida. This chapter discusses the experimental setup in detail. The chapter is divided into three sections. The first section describes the op tical system used for detecting the density gradient field. The second section describe s the normal impedance tube that generates the acoustic field. The last section deals with the data acquisition system used in detecting the light intensity fluctuations. 3.1 Basic Schlieren Setup The schlieren setup used in the optical de flectometer is shown in figure 3-1. As discussed earlier, the point light source mentioned in Chapter 1 is not realistic. Hence an extended light source is created using a 22 mmmm rectangular aperture. A 100 W tungstenhalogen lamp with a custom alumin um housing is used as the light source. A combination of compressed air and a fan is us ed to cool the housing. A DC power supply (Twinfly model110012 P ) is used to supply 12 V to the lamp. The light passes through an 8-inch long tube for minimal loss. A condenser lens (Oriel model 39235) of diameter 50.8 mm and focal length 100 mm is placed at the end of this tube. The lens is achromatic in nature and prevents chromatic aberrations. The beam that passes through the condenser lens is focused onto a rectangular aperture (Coherent model 611137 ), which acts as the point source.

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28 dcd1d2 fcfc fc
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29 DSA A/D CONVERTER MIC POWER SUPPLY PULSE SIGNAL GENERATOR POWER AMPLIFIER COMPRESSION DRIVER MIC 1 MIC 2 RIGID BACK PLATE TEST SECTION 1d2d 25.4 mm OPTICAL GLASS HIGH PASS FILTER SPECIMEN d Figure 3-2: Normal impedance tube the test section. The metal used for the cons truction of the plane-wave tube is aluminum It has a length of 0.724 m with a cut-off frequency of 6.7 kHz for the plane wave mode. The walls are 22 mm thick so that incident sound pr oduces no appreciable vibration and validates the rigid wall approximation. The test -section is placed on one end of the tube. A membrane loudspeaker/compr ession driver (JBL model2426 H) is placed at the termination of the impedance tube at the end opposite to the sample holder. The loudspeaker is contained in a sound-insu lating box in order to minimize the sound produced by the speaker. Sinusoidal osci llations are genera ted using a signal generator, which is the PULSE (B&K Type 2827-002) system in our case. 3.2.2 Fabrication of the Test Section The test section was fabricated such that it can be attached to one end of the impedance tube. The transition from impedan ce tube to the test section is smooth and care is taken to minimize leakage at the joints. Optical glass is inse rted at the front and rear side of the test section for visualiz ing the flow. The length of the window is

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30 0.17 mand height 50.8 mm The sample holder is fixed at the termination of the testsection. It is a LIGHT SOURCE RECTANGULAR APERTURE LENS 1LENS 2 2-D TRAVERSE KNIFE-EDGE TEST-SECTION PLANE-WAVE TUBE Figure 3-3: Setup for normal impedance tube. separate unit and it is large enough to install test objects leaving air spaces of a required depth behind them. Since experiments were being conducted for a reflection coefficient of unity, the specimen used was made of aluminium and 22mm thick in order to provide sound hard boundary condition. 3.3 Data Acquisition System The microphone signal and both the mean a nd the fluctuating signal of the photo detector were measured. The following equipment was used.

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31 3.3.1 Photosensor Module Data was acquired using two photodiode modules (Hamamatsu Model 578420 H ). One detector was used to detect a reference signal (explained in Chapter 4) and the other measured the fluctuating light intensity on the scree n. The sensitivity of the module is 625510/ Vlm and the output of the detector varies from 15 to 15 V The output-offset voltage was 8.0 mV and the control voltage of the photo-detector module was set at 0.35 V The light is passed on to the photodiode through a fiber optic cable, which terminates in a SMA adaptor (Newport model 3FPSMA ). 3.3.2 Positioning System The screen with the photodiode is placed on a two-dimensional traverse (Velmex Model MB4012P40J-S4). The reso lution of the traverse is 1.6m and is controlled using a controller (Velmex Model VXM 1). The positioning system and data acquisition system was computer controlled using LabV IEW. For calibration of the system, the knife-edge was placed on a one-dimen sional traverse (Newport model 100ESP), which has a resolution of 1 m 3.3.3 Signal-Processing Equipment The output signal of the photo-diode and the microphone are filtered using a computer optimized filter (Kemo Model 35 VBF Multi-Channel Filter/Amplifier System) with a flat pass band and linear passband phase, that operates as a high pass filter with a cut-in frequency of 850 Hz The filtered signal passes to a 16-bit A-D converter (National Instrument Model 4552NI) to remove the low frequency components. The time invariant component of the signal is simultaneously sampled using a multimeter (Keithley model 2400). The entire process is computer controlled using LabVIEW.

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32 CHAPTER 4 DATA ANALYSIS This chapter discusses the quantitative exte nsion of the schliere n technique, namely optical deflectometry. As discussed in Chapte r 1, this involves the measurement of lightintensity fluctuations at a poi nt on the image plane using a fiber optic sensor. The first part of this chapter deals with the static calibration, which relates light intensity fluctuations to the knife-e dge deflection. Subsequently, the cross-spectral correlation technique used to determine the magnitude and phase of the density gradient relative to a reference microphone is described. In the last part of the chapter, dynamic calibration of the instrument using a laser impulse is discussed. 4.1 Calibration of the Optical Deflectometer The objective of the calibration is to es tablish a relationship between the light intensity fluctuation and the density gradient. We take advantage of both theoretical and experimental methods to determine this relationship. The relationship between the angular defl ection and the varia tion in refractive index has already been derived in Chapter 2 in Equation 2-27 and Equation 2-28. Using the Gladstone-Dale relationshi p in Equation 2-16, assuming tw o-dimensional flow field, and 01 n for the surrounding air gives .xkW x (4-1) Prior to the experiments, theoretical calcu lations were done in order to obtain an estimate of the range of the light intensity fluctuation due to the flow. The maximum

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33 density gradient in the plane wave tube fo r a typical value of sound pressure level was compared to the maximum density gradient th at can be detected by the system. This provides an estimate of the fraction of the lin ear operating range that is occupied by the light-intensity fluctuations. It has already been derived (Equation 2-30) that the light intensity fluctuation is related to the angular deflecti on of the light rays. The image of the source at the knifeedge has been shown in Figure 2-2, and the pa rameters are described in Chapter 2. For a knife-edge at half cutoff, abkKNIFE-EDGE UNDISTURBED IMAGE DEFLECTED IMAGE a Figure 4-1: Light source in the plane of the knife-edge. 2 a k (4-2) The maximum deflection in the light ray, which can be detected by the system, occurs at 2 a a (4-3)

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34 This case has been shown in Figure 4-1. S ubstituting Equation 4-3 in Equation 2-29, 2 xaf gives 2. 2xa f (4-4) To determine the maximum density gradient that can be detected by the system, combining Equation 4-4 and Equation 4-1 gives max 2. 2a x fkW (4-5) Also, the magnification of the s ource at the knife-edge is 2 1 KE f m f (4-6) and ,KEamd (4-7) where dis the width of the rectangular apertu re. Using Equation 4-7, the density gradient can be rewritten as max 1. 2 d x fkW (4-8) Substituting the value of the focal length 1100 f mm the width of the test-section 0.0254 Wm and the value of 2 dmm the maximum density gradient that can be detected by the schlieren system is 331.7410/ kgm The expression for the density gradient in the normal impedance tube is given by Equation 2-15. Differentia ting with respect to d, and equating it to zero we obtain the value ofd at which maxima occur for a reflection coefficient of unity as (21), 4 dn (4-9)

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35 where n is a positive integer and is the wavelength of the acoustic wave. For an SPL (re 20 Pa ) value of 120 dB and a unity reflection coeffi cient, which corresponds to a maximum pressure of 2i p occurring in the normal impedance tube, and for a frequency of5 kHz the density gradient at 4 is 30.016/ kgm The ratio of this density gradient fluctu ation in the impedance tube to maximum detectable gradient of the system is 510 Hence, it can be concluded that the density gradient fluctuation occupies only a minute fraction of the dynamic range of the device. Static Calibration. In the absence of a knife-edge, the light intensity at the image plane is represented as max I and, when fully blocked by the knife-edge, the intensity is 0, assuming that diffraction effects are negligible Thus the light intensity varies from max0 I When the knife-edge blocks a part of the light, given by ak as seen in Figure 4.1, the intensity on the image plane is given by max.screenk I I a (4-10) When light is refracted due to the density gr adient in the test s ection the expression for intensity is modified as max, ka I I a (4-11) where a is the knife-edge deflecti on as described in Chapter 2 and is given by Equation 2-29. From the above expression, the static se nsitivity with respect to the deflection of the image at the knife-edge is given by max. I I K aa (4-12)

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36 Since the light intensity is linearly relate d to the knife-edge deflection via Equation 4-11, a direct calibration can be done in an undisturbed (no-flow) cas e, by recording the light-intensity for several knife-edge positions ranging from no cutoff to full cutoff. The calibration curve which gives the voltage va riation (directly propor tional to the light intensity) vs knife-edge position can be used to determine the angular deflection of light rays that pass through a flow with a density gr adient for a fixed knife-edge. In order to ensure that all the measurements are taken in the linear range, the schlieren system was operated at half cut-off, in which case the kni fe-edge blocks half the image of the source. 0 0.5 1 1.5 2 2.5 3 -1.25-0.75-0.250.250.751.25Knife-edge position (mm)Vpd (V) Figure 4-2: Knife-edge calib ration of photodiode sensor. A typical knife-edge calibration at the cen ter of the image plane is shown in Figure 4-2. It shows the output voltage of the phot o-detector as the kn ife-edge location is varied. The x -axis is rescaled so that the y axis passes through the operating point at half cutoff. The source intensity variations with time are accounted for using a second reference photo-detector that measures the source intensity direc tly. Figure 4-3 shows the calibration curve after the photo-detecto r signal has been normalized using the reference detector signal. It can be seen that the temporal non-uniformity of the light

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37 source shown in Figure 4-2 has been corrected by the reference photo detector in Figure 4-3. 0 0.05 0.1 0.15 0.2 -1.25-0.75-0.250.250.751.25Knife-edge position (mm)Vpd/Vref_pd Figure 4-3: Photodiode knife-edge calibration. The experiments also account for any zer o offsets in the photo-detectors. A dimensionless parameter is defined which takes into c onsideration the two corrections mentioned above. ___,pdpddark refpdrefpddarkVV VV (4-13) where pdV is the DC voltage of the photo-detector, _refpdV is the voltage of the reference photo-detector and pddarkV and __ refpddarkV are the “no-light” voltage offset of the photo-detector and reference photo-detector respectively. But, the equation does not take into consideration the spatial variation of light intensity in the image plane. Calibration curves were obtained for three lo cations in the image plane as shown in Figure 4-4. The plots show the effect of nonuniform cut-off in the image plane. This leads to non-uniform illumination of the sc reen, which causes the slope (or static sensitivity) to vary from point to point in the image plane as can be seen in Figure 4-4. Non-uniform illumination also causes the maximum intensity to vary from one position

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38 dNORMAL IMPEDANCE TUBE SCHLIEREN IMAGE 6.0 dcm 7.6 dcm 9.1 dcm 0 0.5 1 1.5 2 2.5 3 00.511.52Knife-edge position (mm) d=6.0 cm d=7.6 cm d=9.1 cm Figure 4-4: Photodiode knife-edge calibrations at three locations. to another as can be seen in Figure 4-4. Th e second effect is not of major concern to us since the operating point is at the center of th e curve and the intensit y fluctuation is very small when compared to the linear range of the curve as shown in the beginning of this chapter. The data were normalized using the equation min maxmin,norm (4-14) where min and max are the minimum and maximum value of respectively, at a particular location. This nor malization allows us to compare slopes at the three locations, which otherwise have different linear ranges. Linear curve fits were obtained for the

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39 linear portion of the curv es using Excel Regression Tool as shown in Figure 4-5. The variation in the slope is clear ly visible in the figure and was not found to fall within the 95% confidence interval of each other. 0 0.2 0.4 0.6 0.8 1 1.2 00.511.52Knife-edge position (mm)norm d=6.0 cm d=7.6 cm d=9.1 cm Figure 4-5: Linear re gion of the calibration curves at three locations after regression analysis. It was determined that the non-uniform cut-o ff is mainly due to the finite filament size of the tungsten-halogen lamp. The c ondenser lens-slit combination could not produce uniform illumination over the entire area slit. This effect was mitigated by placing ground glass behind th e rectangular slit. Figure 4-6 shows the calibration curves at the three locations after the ground glass was inserted. It can be clearly seen from the plots that the variation in the slope of the curve (static sensitivity) is reduced considerably. This was verified using linear regression analysis. The slope of the calibration curve is summari zed in Table 4-1. It can be seen that there still exists a small vari ation in the slope. In the e xperiments conducted, since data was being taken only at twenty locations al ong the tube length, a calibration curve was found at each of the twenty loca tions and the local slope was used for data reduction.

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40 0 0.2 0.4 0.6 0.8 1 1 2 0.10.150.20.250.3Knife-edge position (mm) d=6.0 cm d=7.6 cm d=9.1 cm Figure 4-6: Calibration curves af ter the ground glass was inserted. Table 4-1: Slope of the calibration curve at three different locations. Location Slope (1mm ) Right 0.1580.001 Center 0.1680.002 Left 0.1700.001 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 5.756.757.758.75Distance from specimen (cm)Calibration Curve Slope (1/mm) Figure 4-7: Slope of th e calibration curve plotted along the test section.

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41 The slope of the calibration curve at the twenty points taken prior to an experiment has been shown in Figure 4-7 with the 95% uncertainty estimates obtained from the linear regression analysis in Excel. The x axis gives the distance from the end of the normal impedance tube that contains the specimen. The data reduction procedure for one location is now summarized. The dimensionless parameter, for a no-flow case is defined in Equation 4-13. For a case where there is light intensity fluctuation, this parameter is modified as ___'pdpddark refpdrefpddarkVVV VV (4-15) where 'V is the fluctuating term in the phot o-detector signal caused by the acoustic density gradient in the impe dance tube. The corresponding fluctuation element can be obtained by subtracting the undi sturbed intensity (DC opera ting light intensity) from Equation 4-15 as ' ___ refpdrefpddarkV VV (4-16) 4.2 Data Reduction Procedure Optical deflectometry is based on the prin ciple of a cross correlation between two points. This section briefly discusses th e procedure followed to obtain the density gradient fluctuations Calculation of Density Fluctuation. The fluctuation in light intensity is measured using a photo-detector mounted on a traverse in the image plane that moves along the length of the impedance tube. It has been show n in Chapter 2 that the fluctuation in light intensity is a measure of the refractive inde x or instantaneous dens ity gradient in the direction normal to the knife-e dge. This optical signal is conv erted to an electrical signal

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42 using a photo-detector. The signal is then f iltered and amplified us ing a filter/amplifier and digitized at high speed by a 16-bit AD converter (Nationa l Instrument Model 4552NI). The digitized data is transferred to a computer fo r subsequent analysis. A signal from a fixed microphone located at 6.4 dcm is also sampled simultaneously. Using cross-spectral analysis, the coherent power in the photo-detector signal and the relative phase difference betw een the photo-detector signal and the microphone signal are determined. The density fluctuation [12] can be represented as '(,)Re().it x x txe x (4-17) The spatially dependent term in the above equation is a complex quantity, which can be expressed as ()()().ixxxe xx (4-18) The magnitude and the phase of the a bove equation are determined by a crossspectral analysis described below. The frequency response function between the input microphone signal x and output photo detector signal y is defined as [19] ,xy xy ii xxxxG G HeHe GG (4-19) where xyG is the cross spectrum between the input and the output signal, and xxG is the auto spectrum of the input signal. The voltage fluctuation detected by the photo-detector at a ny location can be represented as

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43 ()'(,)Re'(),ixitVxVxee (4-20) and the reference microphone signal at a distance 1d in the plane-wave tube is given by 1() 11(,)Re().id itPdPdee (4-21) In Equation 4-19, the magnitude H is the ratio of the out put to the input signal, and the phase 1()()() x xd is the phase difference be tween the output and input signals. The amplitude of 1(,) Pd is obtained from the power spectrum, 2 1(,)rmsPd of the microphone signal and the phase at 1d with respect to 0d using two-microphone method. These values are then used to co mpute the amplitude and phase distribution of the voltage fluctuations. The calculated amplitude of '(,)Vx is then substituted in Equation 4-16 to obtain the corresponding values of The slope of the calibration curve can be written as slope a (4-22) where is related to the fluctuating el ement of the light intensity and a is the shift in the light rays in the plane of the knife-edge, wh ich causes the light intensity fluctuation. Using the slope of the calibration curve at the corresponding locat ion, the shift in light ray a is obtained using Equation 4-22. Subsequently, the angular deflection x corresponding to the magnitude of light in tensity fluctuation can be obtained using Equation 2-29. Finally, the density gradient is obtained by substituting x in Equation 4-1.

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44 The data reduction procedure describe d above was automated using a MATLAB code for all the locations, and the density gr adient distribution (ma gnitude and phase) are determined along the length of the tube. A sc hematic of the data reduction procedure is shown in Figure 4-8. CALIBRATION CURVE SLOPE REFERENCE MICROPHONE POWER SPECTRUM FREQUENCY RESPONSE FUNCTION input output Experimental Photodetector Reference microphone a (1) Pd '() () (1) Vx Hx Pd '() Vx refDCrefdarkV VV refDCV refdarkV a 2 xa f 2 f x x kW W k Figure 4-8: Data reduction procedure. Further, frequency domain correlation tools, such as coherence and coherent power spectrum are used. The ordinary coherence function for x as input signal and y as output signal is defined as 2 2.xy xy xxyyG GG (4-23)

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45 The coherence function is related to the portion of y that is linearly correlated to the input signal x The coherence power spectral density is defined as 2 ,.yycohyyGG (4-24) It is a direct measure of the power-spectra l density that is linearly coherent with the reference pressure. These two parameters help determine the qual ity of the frequency response measurements. 4.3 Dynamic Calibration The dynamic system sensitivity of the system was determined after the completion of the impedance tube experiments. The refl ected output from a laser pulse was directed towards the photo-detector at the operational gain of the system. The pulse input duration was found to be much shorter (20sec) n than the photo-detector system time constant. Hence, the input was treated as an impulse. Figure 4-9 shows the measured impulse response of the photo-detector. Figure 4-9: Impulse response of the experimental photo-detector.

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46 Figure 4-10: Frequency response of the experi mental photo-detector at the experimental gain setting.

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47 The impulse response shows that the syst em behaves like an over damped secondorder system. A curve fit was performed using the equation 22(1)(1) 2(), 21nntt nK htee (4-25) where is the damping ratio, n the undamped natural frequency and K is related to the strength of the pulse. The damping ratio was found to be 1.0018 and n = 51.9310sec rad Since it is an over-damped second-order system, it can be concluded that there is a time delay and an amplitude attenuation. The frequency-response characteristics obtained from the damping ratio and the undamped natural frequency are shown in Figur e 4-10. The gain variation was found to be 0.2dB and the phase lag 018.5 at 5kHz. From the response, it can be seen that while the gain is negligible, the phase lag has a finite valu e. This was accounted for while reducing the data in Chapter 5. 4.4 Experimental Procedure Two experimental methods were employe d simultaneously to investigate the density gradient fluctuations in the normal impedance tube: the optical deflectometer and the two-microphone method [25]. The entire experimental procedure was au tomated using a LabVIEW program. The program simultaneously acquired data from two photo-detectors and two microphones, controlled a 2-D traverse and partially proce ssed the data. Data was acquired at twenty different locations with a spacing of 1.7mm across the schlieren image. At each location, the filtered fluctuating signal of the experimental photo-detector and the pressure fluctuation signals from two microphones placed at a distance

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48 16.4m dc and 24.5m dc from the specimen, respectively, were acquired. The dc component of the experimental and the re ference photodector were acquired using a digital multimeter (Keithley model 2400). For the deflectometer, the frequency re sponse function with the first microphone signal (reference microphone) as input and expe rimental photo detector signal as output is calculated simultaneously by the LabV IEW program. The DC component of the photo-detector signals is used to correct the temporal non-un iformity in light source as discussed in section 4.1. For the two-microphone method, signals from the two microphones were used to constructing the standing wave pattern in the impedance tube. The reflection coefficient is calculated using Equation 2-10 by th e program as the data is acquired. The analysis was carried out entirely in the frequency domain and ensemble averaging was performed over 10,000 blocks of data with 1024 data points in each block. Uncorrelated noise is reduced by ensemble averaging. Thus the random noise variations in the spectrum are smoothed out. The sampling frequency of the data acquisition system was set at 102400secsamples and hence f was 100Hz. Since the signals were periodic, a rect angular widow was applied and spectral leakage was avoided [19]. Once the entire procedure was completed, the static calibration was repeated. It was observed that the variation in the slope of the calibration curve at a particular location was negligible during th e course of the experimental procedure. The traverse was moved to the adjacent location. Static calibrations were performed at the next location, and the measurement procedure wa s repeated for all twenty locations.

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49 CHAPTER 5 RESULTS AND DISCUSSION This chapter presents the theoretical, computational, and experimental results obtained during the course of the research. The results obtained from two experimental methods with the uncertainty estimates are co mpared in the latter pa rt of the chapter. 5.1 Theoretical Results Prior to the experiments, the variations in pressure, density, and density gradient were calculated along the lengt h of the impedance tube. Since the experiments were being conducted for a sound hard termination, the value of R should ideally be unity and was thus used for these calculations. Fr om Equation 2-3 pressure distribution for 1 R is ()(ee).ikdikd ipdp (5-1) From Equation 2-13, the density distribution can be obtained as 2()(ee).ikdikd ip d c (5-2) And using Equation 2-15, the density gradient for 1 R is derived as 2() .jkdjkd ipjkee dc (5-3) Figure 5-1 provides a comparis on of pressure, density and density gradient fields. It can be seen that pressure doubling occurs at 0,,,... 2d and pressure nodes occur at

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50 2 i p c 2 id ipkc i p p Figure 5-1: Comparison of pr essure, density and density gradient distributions. 35 ,,,... 444d Also, from the figure it can be obs erved that the nodes of the pressure wave correspond to the maxima of the de nsity gradient and vice versa. The variation in the phase of pressure and density gradient with respect to kd for 1 R is shown in Figure 5-2. A 180degree shift phase occurs at 35 ,,... 444d in the case of the pressure distribution and, at 3 0,,,... 22d in density distribution. In practice, however, leaks/losses exist in the impedance tube and, hence, the magnitude and phase of the reflection co efficient were calculated using the twomicrophone method as discussed in Chapter 2.

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51 Figure 5-2: Phase variation for a rigid termination. The density gradient distri bution in the impedance tube can be obtained using the pressure signals at two locations, 1d and 2d (see Figure 2-1), using the following procedure. The amplitude of the pressure signal (Appendix D) is derived from Equation 2-3 as 22()(cos(())cos())(sin(())sin()),rr p dAkdlRkdklkdlRkdkl (5-4) where l is the length of the impedance tube, d is the distance from the test specimen, R and r are the magnitude and phase of the re flection coefficient, respectively. Substituting the pressure magnitude at a location 1d in Equation 5-4, the constant A is determined. This value can then be verified by using the pressure amplitude at location 2d. The density gradient amplitude is derived from Equation 2-15 as

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52 22 2()(cos(())cos())(sin(())sin()).rrAk dkdlRkdklkdlRkdkl xc (5-5) The density gradient amplitude for a known value of A and reflection coefficient is thus obtained vs. position along the le ngth of the tube using the above expression. The phase of the density gradient is found from Equation 2-15 1(cos(())cos()) tan. (sin(())sin())r rkdlRkdkl kdlRkdkl (5-6) The magnitude and phase of the density gradient will be used to verify the experimental results obtained from the deflectometer in Section 5.3.2. 5.2 Numerical Results Data analysis similar to the test conditi ons was performed for simulated pressure waves. The pressure across the plane wave t ube at a particular pos ition at a particular instant of time is given by ReeeRee,iklikdikdiwtpA (5-7) where k is the wave number, l is the length of the tube, R is the complex reflection coefficient and is the frequency of the signal. A reference signal was taken at the origin, 0kd. Cross-spectral analysis of the signa l at various locations with respect to this reference signal was then performed. Pr essure signals were reconstructed using the amplitude and phase information. The resulting animation consists of 72 frames (i.e., the phase is advanced by 5 deg. between frames).

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53 0009002700180 Figure 5-3: Pressure waves at various phases for R = 1. 0180 0180090000270 Figure 5-4: Pressure waves at various phases for R = 0.

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54 0270018009000 incident wave Figure 5-5: Pressure waves at various phases for R = 0.5. Phase locked movies for various values of R were obtained. Except for 0R standing wave patterns exist. Figure 53 shows sample snapshots of the movie for 1 R case. This represents a rigid termination or sound-hard boundary condition. Pressure doubling occurs at the interface 0kd and the pressure reduces to zero at the nodes. The termination acoustic impedance defined by Equation 2-6 is infinite. Also, the standing wave ratio is infinite [18]. A purely progressive pressure wave in a tube with a c termination is shown in Figure 5-4. It can be seen that a standing wave does not exist for 0R. The pressure waves propagate towards the open end of the tube as the phasor is increased from 0 to 270

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55 Simulations were also obtained for a general resistive termination (see Figure 5-5). The value of reflection coefficient was taken to be 1 2 The standing wave ratio is 3 and 03n Z Z 5.3 Experimental Results As mentioned in Chapter 4, two e xperimental methods were employed simultaneously to investigate the density gradient fluctuations in the plane-wave tube: the optical deflectometer and the two-microphone me thod. This section describes the results obtained from the experiments conducted. In particular, the results obtained using these two methods are compared later in this section. 5.3.1 Measurement of Density Gradient Using the Optica l Deflectometer An experimental set-up as de scribed in Chapter 2 was constructed. Light intensity fluctuations were measured along the length of the test se ction using a photo detector mounted on a traverse. Pressure waves were generated at 5 kHz in the plane-wave tube. The noise floor of the detector for a no-lig ht case was determined and expressed as power spectral density in Figure 5-6. The power spectra of the detector for light-on cases, with various values of the gain are pl otted in Figure 5-7. The power spectrum of the detector was seen to increase as the gain was varied from 0 to 0.4. Beyond this gain, the power spectrum reduced steeply and remained closer to the 0 gain spectrum. This is likely due to the saturation of the photo-detector. Hence the operating amplifier gain was set at 0.35. Figure 5-8 and Figure 5-9 show exampl es of photodiode an d microphone spectra, respectively, at 145.4 dB These spectra were measured for illustrative purpose using Virtual bench. The dominance of power at 5 kHz is clearly visible from both the plots.

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56 Figure 5-6: Noise floor of the experimental photo-detect or at an operational gain of 0.35. Figure 5-7: Noise floor of the experimental photo-detector (light-on) for various gains.

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57 Figure 5-10 shows the coherence between the photo-detector and the reference microphone signal. Although the coherence value is low (0.19) at the operating frequency, the coherent power of the photo-detector at5 kHz is approximately three orders-of-magnitude above the noise floor (Figure 5-11). Figure 5-8: Example of photo-detector pow er spectrum at 145.4 dB Figure 5-9: Example of referen ce microphone power spectrum at 145.4 dB

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58 Figure 5-10: Example of coherent spectrum at 145.4 dB Figure 5-11: Example of photo-de tector coherence power at 145.4dB.

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59 Figure 5-12: Magnitude of the fr equency response function at 145.4 dB Figure 5-13: Density gradient amplitude along the length of the tube at 145.4 dB

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60 Light intensity fluctuations were measured at 20equally spaced locations. The frequency response function gain has been pl otted in Figure 5-12. Data was reduced according to the procedure described in Chapte r 4, and the density gradient distribution was obtained in the impedance tube and is s hown in Figure 5-13. The density gradient distribution reveals a standing wave pattern and follows the trend discussed in Section 5.1. Data was acquired for Sound Pressure Levels ranging from123155 dB From Figure 5-14, for example, the power of th e photo-detector signal at the operating frequency is close to the noise floor at 126.4 dB Also, at low SPL, the coherence of the signal reduced considerably, the coherence at 126.4 dB was 0.0009. Nevertheless, the coherent power at 5 kHz was still dominant (Figure 5-16). The density gradient distribution at 126.38 dB is shown in Figure 5-18. Figure 5-14: Example of photo-de tector power spectrum at 126.4 dB

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61 Figure 5-15: Example of referen ce microphone power spectrum at 126.4 dB Figure 5-16: Coherent power of the photo-detector at 126.4 dB

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62 Figure 5-17: Example of coherence spectrum at 126.4dB. Figure 5-18: Density gradient amplit ude along the length of the tube at 126.4dB.

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63 The phase difference between the photodetector signal and the microphone signal is obtained from the frequency response func tion (Figure 5-19). The phase is found to remain constant with a 0180 shift at the density-gradien t node. To obtain the absolute value of the phase of the de nsity gradient, we require the phase information of the reference signal, which is obtained from the two-microphone method in Section 5.3.2. Figure 5-19: Phase difference between the phot o-detector signal and the reference microphone signal at 145.4 dB 5.3.2 Measurement of Density Gradient Using microphones The density gradient along the normal im pedance tube length was calculated as described in section 5.1. The amplitude of the pressure signal at a distance 4.5cm and 6.4cm is obtained from the power spect rum of the microphone signals.

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64 Figure 5-20: Density gradient amplitude along the length of the tube using the microphone method at 145.4dB. Figure 5-21: Density gradient phase along the length of th e tube using the microphone method at 145.4 dB The reflection coefficient is obtaine d from the two-microphone method as discussed in Chapter 2. The magnitude and phase of the density gradient at various

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65 locations of the impedance tube are calcul ated using Equation 5-5 and Equation 5-6 and are plotted in Figure 5-20 and Figure 5-21, respectively. Fr om Figure 5-21, the phase at the reference microphone location was found to be 089.6 and used to obtain the phase angle from the deflectometer method re lative to the spec imen location (0) d 5.3.3 Comparison of Results Obtained by the Two Methods The amplitude of the density gradients obt ained using the two methods is plotted with the uncertainty estimates (see uncertain ty analysis in Appe ndix-C), in Figure 5-22 and Figure 5-23 at 145.4 dB and 126.4 dB respectively. Figure 5-22: Magnitude of the density gr adient using the two methods at 145.4 dB

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66 Figure 5-23: Magnitude of the density gr adient using the two methods at 126.4 dB It can be seen that at higher SPL, the density gradient fluctuations obtained from the two methods are very similar. The defl ectometer result deviated from the microphone method near the maxima of the density gradie nt. For the higher SPL, the estimates were close but did not overlap near the maxima. The most dominant term in the uncertainty estimate was the error due to the frequency re sponse function and this error is dependent on the coherence. Interestingly, the agreement at the node is better than at the anti-node. In the case of the lower SPL, the results obtained with the schlieren method had similar results and agreement as at higher SPL The error in the density gradient field was larger and this is caused due to the lo wer coherence between the signals. The error bars were found to fall within the range of each other.

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67 Figure 5-24: Phase of the density grad ient using the two methods at 145.4 dB Figure 5-25: Phase of the density gr adient using the two methods at 145.4dB after the phase correction from the photo detector.

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68 Figure 5-26: Phase of the density grad ient using the two methods at 126.4 dB after the phase correction from the photo detector. The phase distribution obtained from the two experimental tec hniques is compared in Figure 5-24. The reference micr ophone phase was calculated at 6.4 cm and added to the phase of the FRF to obtain the phase of the density gradient. The result obtained from the optical method differed from the microphone method by a finite value. This shift was caused by the phase lag in the photo detector and was corrected via the dynamic calibration describe d in Section 4.3. Figure 5-25 and Figure 526 shows the corrected values of the phase w ith the error bars. They are found to fall well within the range of each other.

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69 CHAPTER 6 CONCLUSION AND FUTURE WORK This chapter summarizes the work done duri ng the course of this project. Future work required to improve the overall performanc e of the instrument is discussed in the latter part of this chapter. 6.1 Conclusions The ultimate objective of the research activity was to device a technique to visualize and measure the acoustic field in a normal acoustic impedance tube. A schlieren technique used for visualizing co mpressible flow was extended to measure the one-dimensional acoustic field with a much smaller density gradient. The instrument successfully detected density gradient fl uctuations and data was obtained for sound pressure levels ranging from123157 dB The amplitude and phase distributions of the density gradient were measured using the deflectometer. A second method based on the two-microphone method was used to verify the results. The results were in fairly good agreement except at the antinodes of the density gradient dist ribution. The discrepancy in the results can partially be explaine d using the uncertainty estimation. 6.2 Future Work The study conducted provides rudimentary resu lts, which can be used as a basis to study the behavior of an acoustic field in a normal acoustic impedance tube. The optical deflectometer system can be improved to obtain better results. The results of the detailed sensitivity analysis were obtai ned after the experiments. Th ese results can be used to design a more sensitive instrument.

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70 The system field of view is limited by the diameter of the lens/mirror, such that only a limited field of view c ould be obtained. Hence a larger diameter lens/mirror is required to visualize flow at lower frequencies. Various types of spherical and achromatic aberrations were encountered using the lens-based system. Hence, a mirror-based system is highly recommended. Though the f number of the condenser lens and that of th e first schlieren lens or mirror are shown to be equal in the ray diagram, it is advisable to have the f number of the condenser lens to be 1.5 to 2 times smaller [4]. This avoi ds non-paraxial effects caused by the condenser lens and also the effects of the reduced illu mination of the light beam in the periphery. Amplitude and phase mismatches were not corrected using the microphone switching method [25] for different values of SPL while the experiments were being conducted. This can be avoided by implemen ting the switching tec hnique prior or during the test. The photo-detectors were characterized as over damped second order system after the completion of the experiments. A more sensitive photo-detector with lower noise and increased bandwidth, such as a c ooled photo-multiplier tube should be used in future work. Finally, using suitable boundary conditions, th e pressure field can be obtained from the density gradient fluctuations by developi ng a suitable procedure. The technique can thus be extended to determine twodimensi onal fields and can be used in the study of scattering effects. Also, the technique can now be extended to focused schlieren for the characterization of various specimens using the normal impedance tube.

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71 APPENDIX A KNIFE-EDGE GEOMETRY The magnification between the focus of Lens 2 and the screen can be derived as follows. k 22 d k 2 f o A B C D x y Figure A-1: Magnification of the source on the screen. The Figure shows the region between the sch lieren head and the screen (Figure 23) in the setup with the extreme rays origin ating from the top of the object. The image has a height k at the focus of Lens 2 and yk on the screen. It can be seen from Figure A-1 that AOBCOD. Hence

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72 2 22 d k yx f ( 1) Magnification 31. yky m kk ( 2) Substituting Equation A1 in Equation A3 2 3 22 1 d k mx kf ( 3) The total magnification at focus of Lens 2 is equal to m1*m2 is 22 11. f f f f ( 4) Therefore we have h f f k1 2. Substituting in the Equation A3 22 1 3 2 2 12 1. df h f mx f h f ( 5)

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73 APPENDIX B LIGHT RAYS IN AN INHOMOGENEOUS FLUID The derivation is done under the assu mption that physical phenomena like diffraction or disper sion does not exist. The refractive index is assumed to vary as a function of the three spatial coordinates. (,,). nnxyz (B1) The incident ray is initially parallel to th e direction. According to the Ferment’s principle, the variation of opti cal path length along a light ra y in the refractive field must vanish. P Q x y x y 2 z1z Image Plane l Figure B-1: Deflection of a light ray in inhomogeneous test object. (,,)0, nxyzds (B2)

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74 where s denotes the arc le ngth along the ray and 2222. dsdxdydz (B3) Equation B2 is equivalent to tw o sets of differential equations 22 2 211 1, dxdxdyndxn dzdzdznxdznz (B4) 22 2 211 1. dydxdyndyn dzdzdznydznz (B5) Assuming that the slopes of the ray dx dz and dy dz are very small as compared to unity and assuming n x and n y are of same order of magnitude Equation B4 and Equation B5 simplifies to 2 21 ; dxn dznx (B6) 2 21 dyn dzny (B7)

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75 APPENDIX C UNCERTAINTY ANALYSIS This section estimates the error in bo th magnitude and phase of the density gradients measured using the deflectome ter and the two-microphone method. The analysis uses a technique that employs a first order uncertainty estimate. For the uncertainty analysis of the microphone met hod, the technique is extended to complex variables [23]. The uncertainti es in the individual variab les are propagated through the data reduction equation into the result. For a general case [24] if is a function of measur ed variables, 12(,,.....).J R RXXX (C1) The uncertainty in the result is given by 122222 12....,JrXXX JRRR UUUU XXX (C2) where ixU is the uncertainty in variable i X The partial derivative i R X is defined as the sensitivity coefficient. C.1 Uncertainty in Amplitude C.1.1 Deflectometer Method The data reduction procedure explained in section (5.3.2) is followed and a final expression for density gradient is derived as shown below.

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76 The equations for (Equation 4-16) and the slop e of the calibration curve (Equation 4-17) were derived in section (4.1.1). Substituting, Equation 4-16 in Equation 4-17 we obtain the expression for a as __'() (). ()(())refDCrefdarkVd ad slopedVdV (C3) Substituting Equation C3 into Equation 2-29, we obtain the angular deflection as 2__'() (). ()(())x refDCrefdarkVd d slopedfVdV (C4) Subsequently, the density gradient is obtai ned by substituting Equation C4 in Equation 4-1 as 11 2__2 (), ()(())refDCrefdarkHG d x slopedfVdVKW (C5) where 11G is the power spectrum of the input microphone signal at the operating frequency. The numerator essentially gives the voltage amplitude as a product of the magnitude of the frequency response function and the reference pressure amplitude. The density gradient can be written in the form of Equation C1 as a function of various parameters as shown 112__()((),,(),,(),,).refDCrefdarkdHdGslopedfVdVW xx (C6) Representing x as x various sensitivity coefficients can be written as follows. 11 2__2 ; ()x refDCrefdarkG HslopefVVKW (C7)

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77 11 112__; 2()x refDCrefdarkH G slopeGfVVKW (C8) 11 2 112__2 ; 2()x refDCrefdarkHG slope slopeGfVVKW (C9) 11 2 22__2 ; ()x refDCrefdarkHG f slopefVVKW (C10) 11 2 _2__2 ; ()x refDCrefDCrefdarkHG VslopefVVKW (C11) 11 2 _2__2 ; ()x refdarkrefDCrefdarkHG VslopefVVKW (C12) Figure C-1: Error bar in the amplitude of the density gradient using the deflectometer at 145.4 dB

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78 11 2 2__2 ; ()x refDCrefdarkHG WslopefVVKW (C13) The uncertainties [19] 1XUin H and 11G were 21 2 n and 1 n respectively, where is the coherence and n is the number of averages. The remaining parameters were determined from the instrument specifica tions. Finally the total uncertainty in the density gradient was computed at each of th e twenty locations and plotted in Figure C-1 for a SPL of 145.4 dB The uncertainties in various parameters have been tabulated in C-1. Table C-1: Uncertainties in various parameters for a deflectometer. Parameter ixU H 21 2 n 11G 1 n slope 95% error from the calibration curve 2 f 8mm _refDCV 0.6mV _refdarkV 0.6 mV W 0.0254mm

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79 C.1.1 Two Microphone Method A similar approach was also followed for the two-microphone method. The expression for the absolute pressure in the no rmal acoustic impedance tube is derived in Equation 5-4. The constant A can be expressed as 11 22(2) (cos((1))cos(1))(sin((1))sin(1))rrG A kdlRkdklkdlRkdkl (C14) The expression for density gradient has al ready been derived in Equation 2-15 as 2() ().ri iklikdikdAikeeRee d xc It can be written as a function of four parameters as shown 11()((1,,),,,).rdAdlGlR xx (C15) Various sensitivity coefficients are calculated as follows 2() ;ri iklikdikd xikeeRee Ac (C16) 2;ri iklikd xAikeRee Rc (C17) 2 2() ;ri iklikdikd xAkeeRee lc (C18) 2;ri iklikd x rAkeRee c (C19) It can be seen from Equation C14 that the constant A itself is a function of 1d 11(1,,) AAdlG (C20)

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80 And partial derivatives are given as 22 11 111 ; (2)(cos((1))cos(1))(sin((1))sin(1))rrA G GkdlRkdklkdlRkdkl (C21 ) 0; A l (C22) 3 11sin(21) ; 12rARkd A dG (C23) Table C-2: Uncertainties in various parameters for the microphone method. Parameter ixU R 0.03 11G 1 n l 0.025 mm 1d 0.025mm r 0.015 The uncertainty in A is 112222 1 11. 1rlGdAAA UUUU lGd (C24)

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81 Figure C-2: Error bar in the amp litude using the microphone method 145.4 dB Figure C-3: Comparison of microphone and deflectometer method at 145.4dB.

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82 The uncertainty in R and r were calculated using the method of Schultz et al. [23]. The total uncertainty in the magnitude of the density gradient is obtained [23] using the equation '' '22 '' '1 Re()Re()Im()Im()xx xxx xUUU (C25) and plotted in Figure C-2. In Figure C-3, the error bars obtained from the microphone method and the deflectometer method are compared. The uncertainties in various parameters are shown in Table C-2. C.2 Uncertainty in Phase C.2.1 Deflectometer Method Figure C-4: Error bar in the phase using the deflectometer at 145.4 dB

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83 The phase of the density gradient using the deflectometer method is essentially the sum of the phase difference obtained fr om the frequency response function and the phase of the reference microphone at location 1d as shown in Equation C26 1sin(())sin() tan. cos(())cos()r rkdlRkdkl kdlRkdkl (C26) The uncertainty in is given by [19] the expression 21 2 n and the uncertainty from the reference signal at 1d is obtained from the two-microphone method. Figure C-6: Error bar in the amplitu de using the deflectometer at 145.4 dB

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84 Figure C-7: Comparison of microphone and deflectometer method at 145.4dB. C.2.2 Microphone method Procedure similar to Section C.1.2 was followed and the uncertainty in the phase is obtained [23] from the expression '' '22 '' 2 '1 Im()Re()Re()Im().xx xxx xUUU (C27) The error bar has been plotte d in Figure C-6 and the comparison in Figure C-7.

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85 APPENDIX D DENSITY GRADIENT As discussed in Chapter 2 the pressure di stribution in the impedance tube is given by ()(),iklikdiklikdpdAeeBee (D1) where ikl i p Ae and ikl r p Ae and R defined as r i p R p Equation D-2 can be expressed as ()().ri iklikdikdpdAeeRee (D2) Using Equation 2-8, the pressure distributi on can be con be conve rted into real and imaginary terms as ()(cos(())sin(())cos()sin()).rrPdAkdlikdlRkdklRikdkl(D3) The amplitude is given by 22()(cos(())cos())(sin(())sin()).rr p dAkdlRkdklkdlRkdkl (D4) And the phase is given by 1sin(())sin() ()tan. cos(())cos()r rkdlRkdkl d kdlRkdkl The density gradient can be obt ained from Equation 2-15 as

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86 2() ().ri iklikdikdAikeeRee d xc It can be expressed as real and imaginary terms as 2()(cos(())sin(())cos()sin()).rrAk dikdlkdliRkdklRkdkl xc (D4) The amplitude of Equation D-5 is given by 22 2()(cos(())cos())(sin(())sin()).rrAk dkdlRkdklkdlRkdkl xc (D 6) And the phase is obtained from 1(cos(())cos()) tan. (sin(())sin())r rkdlRkdkl kdlRkdkl (D5)

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87 LIST OF REFERENCES [1] Merzkirch W (1987) Flow visualization. 2nd edition, Academic Press, Inc., Orlando, FL. [2] Dyke V Album of Fluid Motion, Parabolic Press, Incorporated, Stanford California [3] Holder DW; North RJ (1963) Schlieren Methods. Na tional Physical Laboratory, Teddington, Middlesex. [4] Settles GS (2001) Schlieren and Shadowgr aph Techniques.ISBN 3-540-66155-7 Springer-Verlag Berlin Heidelberg NewYork. [5] Coleman HW; Steele WG (1999) Experimentation and Uncertainity Analysis for Engineers, 2nd ed. New York: John Wiley & Sons, Inc.. [6] Davis MR (1971) Measurements in a subsonic turbulent jet usi ng a quantitative schlieren technique. J.Fluid Mech (1971), vol. 46, part4, pp. 631-656. [7] Wilson LN; Damkevala RJ (1969) Statistical Propert ies of Turbulent Density Fluctuations. J.Fluid Mech,vol. 43, part 2, pp. 291-303. [8] Davis MR (1972) Quantitative schlieren measur ement in a supersonic turbulent jet. J.Fluid Mech. (1972), vol. 51, part 3, pp. 435-447. [9] Davis MR (1974) Intensity, Scale and Conv ection of Turbulent Density Fluctuations. J.Fluid Mech (1975), vol. 70, part 3, pp. 463-479. [10] Weinstein LM (1991) An improved large-field focusing schlieren system. AIAA Paper 91-0567. [11] Alvi FS; Settles GS (1993) A Sharp-Focusing Schlieren Optical Deflectometer. AIAA Paper 93-0629. [12] Weinstein LM (1993) Schlieren system and met hod for moving objects. NASA CASE NO. LAR 15053-1. [13] Garg S; Settles GS (1998) Measurements of a supersonic turbulent boundary layer by Focusing schlieren deflectomet ry. Experiments in Fluids 25 254-264.

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88 [14] Garg S; Cattafesta LN 111; Kegerise MA; Jones GS (1998) Quantitative schlieren measurements of coherent struct ures in planar turbulent shear flows. Proc. 8th Int. Symp.Flow Vis, Sorrento, Italy. [15] Garg S; Cattafesta LN 111 (2000) Quantitative schlieren measurements of coherent structures in a cavity shear la yer. Experiments in Fluids 30 pp. 123-124. [16] Michael A. Kegerise; Eric F. Spina; Louis N. Cattafesta 111 (1999) An Experimental Investigation of Flow-i nduced Cavity Oscillation. AIAA-99-3705. [17] Cattafesta LN; Kegerise ma; Jones GS (1998) Experiments on Compressible Flow-Induced Cavity Oscillations. AIAA-98-2912. [18] Blackstock DT (2000) Fundamentals of Physical Acoustics. New York: John Wiley & Sons, Inc. [19] ASTM-E1050-90, "Impedance and Absorption of Acoustical Materials Using a Tube, Two Microphones, and a Digita l Frequency Analysis System." [20] Ebert AF and Ross DF(1977) “Experimental determination of acoustic properties using a two-microphone random-excitation technique” Journal of the Acoustical Society of America 61(5). [21] Bendat JS and Piersol AG Random Data, 3rd ed. New York: John Wiley & Sons, Inc. [22] Taghavi R; Raman G(1996) Visualization of supe rsonic jets using a phase conditioned focusing schlieren system Experiments in Fluids 20 472-475 [23] Settles GS (2001) Schlieren and Shadowgr aph Techniques. Visualizing Phenomena in Transparent Media. [24] Leonard M. Weinstein (1991) Large-Field High-B rightness Focusing Schlieren System.AIAA Paper 91-0567 [25] Schultz T; Louis N. Cattafest a 111; Nishida T; Sheplak M (1998) Uncertainty Analysis of the Two-Microphone Met hod for Acoustic Impedence Testiong [26] Horowitz S; Nishida T; Cattafesta L N; Sheplak M (2001) CompliantBackplate Helmholtz Resonators for Active Noise Control Application, 39 th Aerospace Science Meeting & Exhibit, 2001

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89 BIOGRAPHICAL SKETCH Priya Narayanan was born in 1979 in Malappuram, Kerala, India. She moved to the State of Kuwait in 1981 and graduated from The Indian School in Salmiya, Kuwait in 1997. She went back to India for her Bachel ors degree and obtained her Bachelor’s of Technology degree in Aerospace Engineering from Indian Institute Of Technology, Madras, India in May 2001. She is currently pursuing her Master of Science degree in the Department of Mechanical and Aerospace E ngineering at the University of Florida.


Permanent Link: http://ufdc.ufl.edu/UFE0001445/00001

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Title: Quantitative Measurement of the Density Gradient Field in a Normal Impedance Tube Using an Optical Deflectometer
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Copyright Date: 2008

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Source Institution: University of Florida
Holding Location: University of Florida
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Permanent Link: http://ufdc.ufl.edu/UFE0001445/00001

Material Information

Title: Quantitative Measurement of the Density Gradient Field in a Normal Impedance Tube Using an Optical Deflectometer
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0001445:00001


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QUANTITATIVE MEASUREMENT OF THE DENSITY GRADIENT FIELD IN A
NORMAL IMPEDANCE TUBE USING AN OPTICAL DEFLECTOMETER
















By

PRIYA NARAYANAN


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2003

































Copyright 2003

by

Priya Narayanan















ACKNOWLEDGMENTS

I would like to thank foremost my advisor, Dr. Louis N Cattafesta, for his

guidance, support and patience. His continual guidance and motivation made this work

possible. I would also like to express my heartfelt gratitude to my co-advisor Dr. Mark

Sheplak for his support.

I owe special thanks to Dr. Bruce Caroll and Dr. Paul Hubner for their help during

the design of the experimental set-up. I thank all of the students in the Interdisciplinary

Microsystems Group, particularly Ryan Holman, for his help during data acquisition. I

would also like to express my gratitude to my colleagues Todd Schultz, Steve Horowitz,

Anurag Kasyap, Karthik Kadirvel and David Martin for their help during the course of

this project.

I would like to thank my undergraduate advisor Dr. Job Kurian for his motivation

and encouragement. I would also like to thank my roommates and friends for making my

stay in Gainesville a memorable one. Finally, I want to thank my parents and my sister

Poornima for their endless support.
















TABLE OF CONTENTS
page

A C K N O W L E D G M E N T S ......... .................................................................................... iii

LIST OF TABLES ........... ................... .............. ....... ....... vi

L IST O F F IG U R E S .... ...... ................................................ .. .. ..... .............. vii

A B STR A C T ................................................. ..................................... .. x

1 IN TR OD U CTION ............................................... .. ......................... ..

1.1 Basic Schlieren M ethod .................................. ......................................2
1.2 O ptical D eflectom eter ............................................. ................... ...............
1.3 Review of an Optical Deflectometer ............ .............................................5
1.4 R research Objectives ........................................................... ..................8
1.5 T hesis O outline .............................................. ........................ 9

2 OPTICAL DEFLECTOM ETER ........................................ .......................... 10

2.1 N orm al Im pedance Tube........................................................ ............... 10
2.2 Theory of a Schlieren System ..................................................................... 13
2.2.1 Deflection of Light by a Density Gradient.......................... .........14
2.2.2 The Toepler Method.. .. ......................... ...................16
2.3 Sensitivity Analysis.............. .... ......... ...... .... ..... ............... 20

3 EXPERIM EN TAL SET U P ............................................... ............................ 27

3.1 B asic Schlieren Setup............ ................................................ ............... 27
3.2 N orm al Im pedance Tube........................................................ ............... 28
3.2.1 C om ponents ............................................. ........ .... ........ ......28
3.2.2 Fabrication of the Test Section ..... ............................29
3.3 Data Acquisition System ............ ..... ........ ................... 30
3.3.1 Photosensor M odule ..................................... .......... 31
3.3.2 P ositioning System ........................................... .......... ............... 31
3.3.3 Signal-Processing Equipm ent ..................................... ......... ......... 31

4 D A T A A N A L Y SIS .......................................................................... ....................32

4.1 Calibration of the Optical Deflectometer ................ .................................. 32
4.2 D ata R education Procedure........................................... .......................... 41









4.3 D ynam ic Calibration ............................................................ ............... 45
4.4 Experim ental Procedure ........................................................ ............... 47

5 RESULTS AND DISCU SSION ........................................... .......................... 49

5.1 Theoretical Results .................. ............................. .......... ........ .. 49
5.2 N um erical Results .................. ......................... .. .. .. ..... .......... .... 52
5.3 E xperim ental R results ........................ ....... ............. .. ... ..................... ... 55
5.3.1 Measurement of Density Gradient Using the Optical Deflectometer...55
5.3.2 Measurement of Density Gradient Using microphones.....................63
5.3.3 Comparison of Results Obtained by the Two Methods ...................65

6 CONCLUSION AND FUTURE WORK ....................................... ............... 69

6 .1 C o n c lu sio n s ................................................................................................. 6 9
6.2 Future Work ............................................................................. ......... ........................69

APPENDIX

A KNIFE-EDGE GEOMETRY .......................................................................71

B LIGHT RAYS IN AN INHOMOGENEOUS FLUID.................. ...................73

C UNCERTAINTY ANALYSIS ............................................................................75

C 1 U uncertainty in A m plitude............................................ ........... ............... 75
C 1.1 D eflectom eter M ethod ........................................ ....... ............... 75
C.1.1 Tw o M microphone M ethod .......................................... ............... 79
C .2 U uncertainty in P hase .............................................................. .....................82
C .2.1 D eflectom eter M ethod ........................................ ....... ............... 82
C .2.2 M icrophone m ethod ........................................ ......................... 84

D D E N SIT Y G R A D IEN T ................................................. ......................... ..............85

L IST O F R EFE R EN C E S ................................... ............................... ...........................87

B IO G R A PH IC A L SK E TCH ..................................................................... ..................89
















LIST OF TABLES
Table p

4-1 Slope of the calibration curve at three different locations................... ............40

C-l Uncertainties in various parameters for a deflectometer............... ...................78

C-2 Uncertainties in various parameters for the microphone method............................. 80
















LIST OF FIGURES
Figure page

1-1 S im p le sch lieren setu p .......................................................................... .......... .. ..... 4

2-1 Plane w ave tube....................................... ................ .... .. ............ 11

2-2 Light source in the plane of the knife-edge......................... ................18

2-3 Ray diagram of the schlieren setup. .............................................. ............... 21

2-4 Schlieren head with the conjugate plane. ...................................... ...............25

3-1 D eflectom eter set-up. ...................................................................... ...................28

3-2 N orm al im pedance tube ........................................ .............................................29

3-3 Setup for normal impedance tube ................................................ 30

4-1 Light source in the plane of the knife-edge.......................... ............... 33

4-2 Knife-edge calibration of photodiode sensor. ............. .................... ........ ... .... 36

4-3 Photodiode knife-edge calibration. ........................................ ....... ............... 37

4-4 Photodiode knife-edge calibrations at three locations................ .............. 38

4-5 Linear region of the calibration curves at three locations after regression analysis.39

4-6 Calibration curves after the ground glass was inserted..................... ..............40

4-7 Slope of the calibration curve plotted along the test section.............................. 40

4-8 D ata reduction procedure. ............................................................. .....................44

4-9 Impulse response of the experimental photo-detector................... ................45

4-10 Frequency response of the experimental photo-detector at the experimental gain
setting .............................................................................. 4 6

5-1 Comparison of pressure, density and density gradient distributions......................50

5-2 Phase variation for a rigid termination. ........................................ ............... 51









5-3 Pressure waves at various phases for R = 1. ................................. .................53

5-4 Pressure waves at various phases for R = 0. ................................. .................53

5-5 Pressure waves at various phases for R = 0.5. ................................. ............... 54

5-6 Noise floor of the experimental photo-detector at an operational gain of 0.35 .......56

5-7 Noise floor of the experimental photo-detector (light-on) for various gains. ..........56

5-8 Example of photo-detector power spectrum at 145.4dB ................ .................57

5-9 Example of reference microphone power spectrum at 145.4dB ...........................57

5-10 Example of coherent spectrum at 145.4dB ....................................... ................. 58

5-11 Example of photo-detector coherence power at 145.4dB ......................................58

5-12 Magnitude of the frequency response function at 145.4dB ................ ................59

5-13 Density gradient amplitude along the length of the tube at 145.4dB ....................59

5-14 Example of photo-detector power spectrum at 126.4dB ...................................60

5-15 Example of reference microphone power spectrum at 126.4dB ...........................61

5-16 Coherent power of the photo-detector at 126.4dB ...........................................61

5-17 Example of coherence spectrum at 126.4dB ................... ......................... 62

5-18 Density gradient amplitude along the length of the tube at 126.4dB ....................62

5-19 Phase difference between the photo-detector signal and the reference microphone
sig n al at 14 5 .4 d B ....................................................................................6 3

5-20 Density gradient amplitude along the length of the tube using the microphone
m ethod at 145.4dB ........................... .......... .. .. ... ............ 64

5-21 Density gradient phase along the length of the tube using the microphone method
at 14 5 .4 dB ..............................................................................6 4

5-22 Magnitude of the density gradient using the two methods at 145.4dB ...................65

5-23 Magnitude of the density gradient using the two methods at 126.4dB ...................66

5-24 Phase of the density gradient using the two methods at 145.4dB .........................67









5-25 Phase of the density gradient using the two methods at 145.4 dB after the phase
correction from the photo detector. ........................................ ....................... 67

5-26 Phase of the density gradient using the two methods at 126.4dB after the phase
correction from the photo detector. ........................................ ....................... 68

A-1 M agnification of the source on the screen. ................................... ............... 71

B-l Deflection of a light ray in inhomogeneous test object...............................73

C-1 Error bar in the amplitude of the density gradient using the deflectometer at
145.4 dB .................... ........ .... ......... .. .. ...... ............ 77

C-2 Error bar in the amplitude using the microphone method 145.4 dB .....................81

C-3 Comparison of microphone and deflectometer method at 145.4 dB ......................81

C-4 Error bar in the phase using the deflectometer at 145.4 dB ....................................82

C-6 Error bar in the amplitude using the deflectometer at 145.4 dB .............................83

C-7 Comparison of microphone and deflectometer method at 145.4 dB .......................84















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

QUANTITATIVE MEASUREMENT OF DENSITY GRADIENT FIELD IN A
NORMAL IMPEDANCE TUBE USING AN OPTICAL DEFLECTOMETER

By

Priya Narayanan

December 2003

Chair: Louis N. Cattafesta
Cochair: Mark Sheplak
Department: Mechanical and Aerospace Engineering

Interest is growing in optical flow-visualization techniques because they are

inherently nonintrusive. A commonly used optical flow visualization technique is the

schlieren method. This technique normally provides a qualitative measure of the density

gradient by visualizing changes in refractive index that accompany the density changes in

a flowfield. The "optical deflectometer" instrument extends the schlieren technique to

quantitatively measure the density gradient. The optical deflectometer has been

successfully used to characterize highly compressible flows. In this thesis, an optical

deflectometer is studied that can provide quantitative measurements of the acoustic field

in a normal impedance tube.

Results of the static calibration performed on the instrument are presented. The

frequency response of the instrument is inferred using a laser impulse response test.

Two-point cross-spectral analysis between the light intensity fluctuations in a schlieren









image and a reference microphone signal are used to determine the density gradient field

in the normal impedance tube. Numerical simulations were obtained for test cases to

validate the data-reduction method. In addition, the two-microphone method is used to

verify the results obtained from the deflectometer.

Results of the experiments performed for normal sound pressure levels of

145.4 dB and 126.4 dB for a plane wave at 5 kHz are presented. A detailed uncertainty

analysis is also performed. The results were in good agreement with each other except at

the density gradient maxima (pressure minima) at the higher sound pressure level.














CHAPTER 1
INTRODUCTION

Many modern-day research activities involve studies of substances that are

colorless and transparent. The flow and temperature distribution [1] of many of these

substances are of significant importance (for example, mixing of gases and liquids,

convective heat transfer, plasma flow).

As suggested by the proverb "Seeing is believing," suggests, visualization is one of

the best ways to understand the physics of any flow. Visualization also aids flow

modeling. Hence, over the years numerous techniques have been developed to visualize

the motion of fluids. These flow visualization techniques [2] have been used extensively

in the field of engineering, physics, medical science, and oceanography .

In aerospace engineering, flow visualization has been an important tool in the field

of fluid dynamics. Several flow visualization techniques [3] have been used in the study

of flow past an airfoil, jet mixing in supersonic flows and acoustic oscillations.

Flow visualization can be classified as non-optical and optical techniques. In the

former, seed particles are usually added and their motion observed. This indirectly gives

information about the motion of the fluid itself. However, in the case of unsteady flow,

these methods [2] are prone to error because of the finite size of the seed particles.

Purely optical techniques, on the other hand, are based on the interaction of light rays

with fluid flow in the absence of macroscopic seed particles. The information recorded is

dependent on the change in optical properties of the fluid. One of the commonly studied

properties is the variation in refractive index with fluid flow.









Several methods are commonly used to visualize refractive-index variation in

fluids. Common methods [3] include shadowgraph, schlieren and interferometric

techniques. Since these techniques are non-intrusive, the flow is not disturbed by the

measurement technique.

Classical optical flow visualization tools (like schlieren and shadowgraphy) were

used in the study of compressible or high-density gradient flows [2] (shock waves in

wind tunnels, turbulent flow, convection patterns in liquids, etc.) Many of these

techniques have been extended for the quantitative study [4] of the fluctuating properties

of the flow under consideration.

The objective of this thesis is to use one of the classical techniques to detect

acoustic waves in a normal impedance tube. These waves, unlike the flow fields in the

previous studies, produce very small density gradients, and there is not a mean flow. If

successful, this technique could be used to study various acoustic fields. One immediate

application is the characterization of compliant back plate Helmholtz resonators [5] at the

Interdisciplinary Microsystems Laboratory at the University of Florida using the normal

impedance tube. These resonators will later serve as fundamental components of the

electromechanical acoustic liner used for jet noise suppression. A quantitative optical

flow visualization technique can be used to study phenomena (such as scattering effects

in the acoustic field) that cannot otherwise be determined quantitatively using only

microphone measurements.

1.1 Basic Schlieren Method

Of all the flow-visualization techniques mentioned earlier, shadowgraph is perhaps

the simplest. Shadowgraphy is often used when the density gradients are large. This

technique can accommodate large subjects and is relatively simple in terms of materials









required. The primary component consists of a point light source. The resulting shadow

effect produced by the refractive-index field can be observed on an imaging surface. In

terms of cost, this method is probably the least expensive technique to set up and operate.

However, this system is not very sensitive. Also it is not a method suitable for

quantitative measurements of the fluid density. However, it is a convenient method for

obtaining a quick survey of flow fields with varying density, particularly shock waves.

Another commonly used flow visualization technique is the interferometric technique [1].

It is highly sensitive and can provide quantitative information. However, such systems

are expensive, complex to set up, and can only deal with relatively small subjects.

A shadowgraph system can be converted into a schlieren system with a slight

modification in its optical arrangement. Schlieren systems are intermediate in terms of

sensitivity, system complexity, and cost. The German word "schliere" means "streaks,"

since the variation in the refractive index show up as streaks. It was used in Germany for

detection of an inhomogeneous medium in optical glass, which is often manifested in the

form of streaks. In principle, the light passing through a medium with relatively small

refractive differences bends light to directions other than the direction of propagation of

light.

The simple schlieren set-up consists of optics to produce a point-light source, two

lenses, a knife-edge and a screen. The point light source is placed at the focus of Lens 1.

The lens produces parallel light beams, which pass through the test section and are

made convergent by the second lens called the schlieren head (Lens 2). An image of the

light source is formed at the focal point of the schlieren head.










f, f2


KNIFE-EDGE








POINT LIGHT
SOURCE
TEST AREA


LENS 1 LENS 2 SCREEN

Figure 1-1: Simple schlieren setup.

At this position, a knife-edge (oriented perpendicular to the desired density

gradient component) cuts off a certain portion of the light-source image and reduces the

intensity of the recorded image plane. The edge is adjusted so that, if an optical

disturbance is introduced such that a portion of the image of the source is displaced, the

illumination of the corresponding part of the image on the screen will decrease or

increase according to whether the deflection is toward or away from the opaque side of

the knife edge. Building on this model an extended light source is considered in Chapter

2. Schlieren systems can be configured to suit many different applications and sensitivity

requirements. They can be used for observing sound waves, shock waves, and flaws in

glass. Its principal limitations are field size (limited by the diameter of the optical

components) and optical aberrations.









1.2 Optical Deflectometer

Optical deflectometry involves the direct measurements of the density gradient in a

flow using the schlieren technique. The conventional schlieren methods measure the

total angular deflection experienced by a light ray while crossing the working section.

This deflection can be directly related to the illumination of the image on the screen.

The quantitative version of the schlieren technique involves the determination of

the density gradient component from the measured deflection. Experimental calibration

and theory is used to characterize the relationship between light intensity fluctuation and

density gradient. An adaptor is flush mounted onto the screen, which is connected to a

photo-sensor module using a fiber optic cable. The photosensor detects the instantaneous

fluctuations in light intensity in the schlieren image and converts the optical signal to an

electrical signal.

1.3 Review of an Optical Deflectometer

The first important contribution to the development of the deflectometer was made

by Foucault [4] by using an explicit cutoff in the form of a knife edge for the schlieren

measurements. At the same time, the measurement technique was re-invented by Toepler

[4] who named it "schlieren." Though the knife edge was not developed by Toepler, he

has been given credit historically for developing the schlieren imaging technique.

The method was recognized to be a very valuable tool and used by many eminent

scientists (Wood, Prandtl, etc.) [4]. Principles and experimental setup of schlieren

techniques [3] were later explained by Holder and North in 1963 as part of the Notes on

Applied Science and was published by the National Physical Laboratory. Fisher and

Krause measured the light scattered [6] from two optical beams crossed in the region of









interest in a turbulent air jet. By cross-correlating the signal from each beam, information

on the behavior of scatterer number-density near the intersection point was determined.

It was observed that this method was sufficiently general and independent of the

method employed to obtain the desired fluctuation of light intensity. Taking advantage of

this fact, Wilson and Damkevala [7] adapted a cross-correlation technique to obtain

statistical properties of scalar density fluctuation. In their method, two schlieren systems

are used, each of which gives signals in the flow direction integrated along the beam

path. The optical beams are made to intersect in the turbulent field and, with the further

assumption of locally isotropic conditions, the cross-correlation of the two signals and the

local mean-square density fluctuation in the beam intersection point are determined.

Davis [8] used a single-beam schlieren system and made a series of measurements

to investigate the density fluctuations present in the initial region of a supersonic axi-

symmetric turbulent jet. The difference in distribution of density fluctuation due to

preheating was observed using this method. Later, a quantitative schlieren technique was

used by Davis [9] to determine the local scales and intensity of turbulent density

fluctuations.

Recently, McIntyre et al. [4] developed a technique called "optical deflectometry"

that was well suited to the study of coherent structures in compressible turbulent shear

flows. A fiber-optic sensor was embedded in a schlieren image to determine the

convective velocities of large-scale structures in a supersonic jet shear layer. The relative

simplicity, low cost, and excellent frequency response of the optical deflectometer makes

it an ideal instrument for turbulence measurements, especially in high-Reynolds number









flows. But one of its drawbacks is that it provides the information integrated along the

beam path. This limitation can be removed using a sharp focusing schlieren [4].

Weinstein [10] has recently provided the analysis and performance of a high-

brightness large-field focusing schlieren system. The system was used to examine

complex two- and three-dimensional flows. Diffuse screen holography was used for

three-dimensional photography, multiple colors were used in a time multiplexing

technique, and focusing schlieren was obtained through distorting optical elements.

Also, Weinstein [11] described techniques that allow the focusing schlieren system

to be used through slightly distorting optical elements. It was also mentioned that the

system could be used to examine complex two- and three-dimensional flows.

Alvi and Settles [12] refined the quantitative schlieren system combining the

focusing schlieren system with an optical deflectometer. This instrument is capable of

making turbulence measurements and was verified by measurements of Kelvin-

Helmholtz instabilities produced in a low-speed axisymmetric mixing layer.

Garg and Settles [13] used the technique for the measurements of density gradient

fluctuations confined to a thin slice of the flow field. The optical deflectometer was used

to investigate the structure of a two-dimensional, adiabatic, boundary layer at a free

stream Mach number of 3. The results obtained were found to be in good agreement with

that obtained using a hot wire anemometer. This result helped validate the new

measurement technique. Further, Garg et al. [14,15] used the light-intensity fluctuations

in a real-time schlieren image to obtain the quantitative flow-field data in a two-

dimensional shear layer spanning an open cavity. Instantaneous density gradient and

density fields were obtained from the data collected. With the help of a reference









microphone, phase- locked movies were created. Surveys were carried out for a Mach

0.25 cavity shear layer using the schlieren instrument as well as hot-wire anemometer.

The results showed that the growth rates of instability waves in the initial "linear" region

of the shear layer could accurately be measured using this technique.

A similar procedure was adopted by Kegerise et al. [16] using the optical

deflectometer with various higher-order corrections to experimentally study the modal

components of the oscillations in a cavity flow field. Shear layer and acoustic near field

measurements were performed at free stream Mach numbers of 0.4 and 0.6. Standing

wave patterns were identified in the cavity using this method. These experiments have

considerably improved the understanding of the cavity physics that cause and maintain

self-sustaining oscillations.

In the most recent development, Cattafesta et al. [17] verified using primary

instantaneous schlieren images that the multiple peaks of comparable strength in

unsteady pressure spectra, which characterize compressible flow-induced cavity

oscillations, are the results of mode-switching phenomenon.

1.4 Research Objectives

The objective of the current project is to develop a system, which can measure the

density gradient in a normal impedance tube. This represents an intermediate step

towards a focused schlieren system for normal acoustic impedance tube measurements

for applications described earlier in the chapter. It can be seen that all the measurement

techniques used till date like the microphone method are flow intrusive. The main

advantage of this technique is that there is no flow intrusion. Also two dimensional flow

fields and effects of scattering can be determined using this technique. The aim of the









experiment ultimately is to verify the results obtained from the deflectometer using the

microphone measurements, thereby developing a more efficient technique.

1.5 Thesis Outline

The thesis is organized into six chapters. This chapter presents the introduction,

background, and application of the optical deflectometer. Chapter 2 presents the theory

of the schlieren system as well as that of the normal impedance tube. Theoretical

formulations for the sensitivity are also developed in this chapter. Chapter 3 discusses

the steps involved in setting up the system and the plane wave tube including the data

acquisition system. Chapter 4 presents the data analysis using the cross spectral

technique. It also describes the static and dynamic calibration of the system. Chapter 5

presents the numerical and experimental results obtained during the course of this project.

Chapter 6 presents concluding remarks, proposed design modifications, and future work

for the optical deflectometer.














CHAPTER 2
OPTICAL DEFLECTOMETER

As discussed in Chapter 1, the goal of this project is to design and test an optical

system capable of measuring the density gradient of plane waves in an impedance tube.

Therefore, this chapter discusses the theory behind an optical deflectometer and the

sensitivity of the system. First, an expression for the pressure fluctuation in a normal

impedance tube for plane waves is derived. The pressure fluctuation is then related to the

density and density gradient, which causes the refraction of the light rays. Second, the

relationship between the light intensity fluctuation and the density gradient in the flow-

field is presented. The sensitivity of the system with regard to the various optical

parameters is then derived.

2.1 Normal Impedance Tube

The flow visualized by the deflectometer is generated in a rectangular normal

impedance tube. In the next section, theoretical derivations of the acoustic field within

the tube are given for plane wave.

Single Impedance Termination. The acoustic flow field is thus comprised of

waves that have uniform pressure in all planes perpendicular to the direction of

propagation and are termed as plane waves. The plane wave assumption is valid below


the cut-on frequency [18] of a square tube, given by -c, where c. is the isentropic
2s

sound speed and s is the width of the normal impedance tube. As can be seen in Figure









2-1, an acoustic driver ultimately generates plane waves on one end of the impedance

tube of length / and the specimen is placed

ACOUSTIC SPECIMEN
SPECIMEN
DRIVER




Pi (x, t)



Pr(X, t)


x d




Figure 2-1: Plane wave tube.

at the other end. The pressure p inside the tube at a position d is given by


p(d) = p, e"kd + p, ekd


(2-1)


where p, and p, are the incident and reflected pressure respectively. Here p, = Ae ek

and p = B ekl are phasors. The time-harmonic dependence, e'", is implicit in p The

complex reflection coefficient R is defined as

R=p. (2-2)
P,

Substituting Equation 2-2 in Equation 2-1, the expression for p(d) becomes


p(d)= p,(ekd + Re -kd).


(2-3)










In a plane wave, the velocity can be related to pressure through the equation u = and,
pc

the velocity in the plane wave tube can be written as


u(d)= (ekd- R- e kd). (2-4)
PC.

The specific acoustic impedance is defined as the ratio of the pressure to the

velocity in the tube

p (e'"k + R- e kd)
Z(d)= = pc (ekd d) (2-5)
u (e' Re d)

When d = 0, the impedance must equal the terminating impedance, which is the

impedance of the specimen

(1+ R)
(1 R)

Solving for R gives

Z pc0
R = Z- (2-7)
Zn + pcm

When R is not equal to zero a standing wave pattern results and is often

characterized by the [18] pressure standing wave ratio (SWR)


SWR=P max +R (2-8)
Pmn 1- R

which is the ratio of maximum to minimum pressure in the tube. The complex reflection

factor R can be represented in polar notation as

R = Rej. (2-8)

The reflection factor R can also be split into real and imaginary components as

R = Rea + jR-mg (2-9)









where RmgR = R sin and Re, = R cos ,

The reflection coefficient R can be measured [19] [20] using the two-microphone

method.

H- eks
R = R e = e2k(l+s) (2-10)
e jks -H

where s is the center-to-center spacing between the two microphones, I is the distance

from the test sample to the nearest microphone and H is the geometric mean of the

measured transfer function between the microphones

H = HIH, (2-11)

and H1 and H2 are the transfer functions for the microphones in the standard and

switched configurations.

The normalized impedance can be obtained by substituting Equations 2-9 in

Equation 2-7 as

1-R 2 2
Zreal real ,mg2 (2-12)
(1 -reail )2 + Rmg

and

2R
Z'g =( R 2 +mg (2-13)
(1-Rel) + R^,

2.2 Theory of a Schlieren System

This section summarizes the theory of operation of a simple [3] schlieren system.

Details of the quantitative part of the experimental set-up are discussed in Chapter 4.









2.2.1 Deflection of Light by a Density Gradient

In the experiments being conducted the refractive index of the test section varies

along the direction of propagation. This is caused by the variation in the density of the

medium. For an isentropic process, pressure and density fluctuations are related by

P = c (2-13)

The pressure gradient in the x -direction is

--=c2 (2-14)
ax ax

From the above equation, it can be seen how the density varies in the x direction in an

impedance tube. Substituting Equation 2-3 in Equation 2-14 and taking the derivative

w.r.t d, we obtain the density gradient as

ap pjk(ekd -Re-kd)
O-2(2-15)
ad c

The aim of the experimental program is to determine this density gradient using the

schlieren system.

In the case of gaseous substances the refractive index, n is related to density by the

Gladstone-Dale equation [3] (Equation 2-16).

n-1= kp. (2-16)

where k = 2.259 x 104 m3/kg is the Gladstone-Dale constant. If there exists a gradient of

refractive index normal to the light rays in a working section, the rays will be deflected

since light travels more slowly in a non-vacuum media, with the velocity given by

c = c n, (2-17)









where c' is the velocity of light in vacuum. The basis of the schlieren technique relies on

the fact that the deflection of the light rays is a measure of the first derivative of the

density with respect to distance (i.e., the density gradient). The derivation [1] is as

follows:

The curvature of the ray is proportional to the refractive-index gradient in the

direction normal to the ray. If we take the z axis as the direction of the undisturbed ray,

the curvatures in the x -y and y z planes respectively are given by (see Appendix B)


X (2.18)
d2z n a'

az2 y lan (2.19)
z2 n oy

The total angular deflection in the x- z and y z planes are taken as e'x and E'y

respectively

1 n
S -Adz (2-20)
x n ox


e' =- z (2-21)
Sn oy

If the optical disturbance is in the working section of a wind tunnel, the light ray

will be refracted on leaving the tunnel so that

n sin E' = no sin E, (2-22)

where no is the refractive index of the air surrounding the tunnel, and n is the refractive

index in the working section. Thus, assuming small angular deflections, the final angular

deflections E measured beyond the tunnel are










S= -dz, (2-23)
Hno x

1 an
=- n dJz. (2-24)
no y

Since the refractive index of air is approximately equal to 1, the expressions can be

written as


=1-az. (2-25)


On
EY = dz. (2-26)


In the case of two-dimensional flow in a tunnel of width W, the expressions

become


= Wn (2-27)
ax'


S=Wn. (2-28)
ay

where the deflection is toward the region of highest density.

2.2.2 The Toepler Method

In the Toepler schlieren system [4] a rectangular slit source is used, with the long

dimension of the slit parallel to the knife-edge. As shown in Figure 2-2 an image of the

slit source produced at the knife-edge, of which only a part of height a is allowed to pass

over the knife-edge. With a homogeneous test field, the recording plane is evenly

illuminated with an intensity I(x, y)= const, which is proportional to the value of a.

Light rays deflected by an angle Ex due to a disturbance in the test field cause a (vertical)

shift of the light source image by an amount Aa











Aa = f tan. E,

For small values of deflection since tan (E,) is approximately E, so

Aa = f/E,, (2-29)

where f2 is the focal length of Lens 2.

Let P be the power of the light beam. The power can be defined as P = IA where

I is the intensity (Power per unit area) and A is the illumination area. Assuming the

power to be conserved between the knife-edge and the screen

IKEAKE I ... sAscreen (2-30)


or Ien= IKEAKE (2-31)
screen


where 'KE is the intensity of light passing the knife-edge. In the case of half knife-edge

a
cut-off, the image on the rear side of the knife-edge has a height b and breadth and
2

hence the area

a
AKE = b. (2-32)
2

Equation 2-31 can be expressed as

IKab
Ie IKEab (2-33)
S 2ACee 2A


When the light source image shifts by an amount Aa


IKE + Aa b
screen + AI = (2-34)
screen

And the intensity change on the screen can be expressed as










A I Aab
A
screen


(2-35)


KNIFE-EDGE


IMAGE WITH
OPTICAL
DISTURBANCE


Ab







b


UNDISTURBED
IMAGE OF THE
SOURCE


Aa


Figure 2-2: Light source in the plane of the knife-edge

The recording plane receives an intensity changed by AI in the corresponding

image point; the relative intensity change also known as contrast C is obtained from

Equation 2-33 and Equation 2-35 as


Al
C _
I
screen


2Aa 2f2
a x.
a a


(2-36)


r









The contrast sensitivity is given by

S dC 2 (2-37)
Sd a

This sensitivity [14] in terms of the contrast is important in the traditional qualitative

method since the photographic films are sensitive relative intensity changes.

But in the case of quantitative schlieren technique, the photo-sensor can detect

absolute light intensity change. Hence the quantitative sensitivity is defined as

d(A) 2f
screen (2-38)
dE a

Substituting the expression for Iscren we obtain

d(AI) f2IKEb
S d (2-39)
dE A

Using the value of angular deflection from Equation 2-27 we obtain

/ 2 f2W cn
f n (2-40)
I a ~c
screen a 9

Using the Gladstone-Dale Equation 2-16 we get

AI 2kf2W Op
2k p (2-41)
Screen a (2

Alternatively, the relative intensity change in they direction can be obtained by

turning the knife-edge by 90, and the relative intensity change is

Al 2kf2W (p
(2-42)
Screen a (

The schlieren system "sensitivity" is often equated to the relative change in

illumination intensity, which is a measure of the refractive index or density gradient in

the direction normal to the knife-edge.









In the case of a normal impedance tube with plane waves, since the density gradient

exists only in the x direction, the measured intensity change is therefore given by

Equation 2-41.

2.3 Sensitivity Analysis

A detailed sensitivity analysis of the optical system is performed in this section.

Since a point source, discussed in Chapter 1, is not realistic, an extended light source is

considered in the following derivation. The extended light source is in the form of a

rectangular aperture, and a condenser lens is used to focus the light beams from the

source at the rectangural aperture. The experimental set-up will be discussed in detail in

Chapter 3.

Figure 2-3 shows the ray diagram of the schlieren setup. It can be seen that the

light beams from the source are focused by the condenser lens at the rectangular aperture.

From this point onwards, the rectangular aperture/slit with height h and width d is

treated as the source. Lens 1 is placed at a distance equivalent to its focal length from the

slit and collimates the light beam. Since the light source is no-longer a point source, Lens

1 no longer produces rays parallel to the optical axis. It can be taken [4] as an array of

the source distributed along the height h. This can be proved by taking four rays from

the top of the slit into consideration. Ray (1) passes through the center of the lens and is

undeviated. Ray (2) which initially travels parallel to the optical axis is deviated so that it

passes through the focal point. It can be seen from the figure that these two rays comes

out of the lens parallel to each other. Also any ray originating from the top of the slit

emerges out of the lens parallel to Ray (1) and Ray (2). Two such rays, Ray (3) and Ray

(4) are then taken into consideration. Ray (3) passes through the focal point of









CONDENSER
LENS LENS 1 LENS 2

SCREEN
.. .... ....KNIFE-EDGE





RECTANGULAR : (1
APERTURE



f' f1 f2 f,
1 1 2 2 "

Figure 2-3: Ray diagram of the schlieren setup.

Lens 2 and is deviated to travel parallel to the optical axis. Ray (4) passes through the

center of the lens and is undeviated. These two beams are focused at the focal point of

Lens 2, where the knife-edge is placed. The image is further magnified and viewed on

the screen placed at a distance x from the knife-edge. Rays (5) and (6) are the extreme

rays originating from the top of the slit.

The power of the light beam emitted by the slit is

PsI I = IsbhAslh = I'sl (dh). (2-43)

It is assumed that the power is conserved through out the optical setup for a case without

the knife-edge, therefore P = IA is constant.


IsiAslit = Iscreen woAscreen

Iso ,dh
or Isceen wo (2-44)
screen

where Iscreen wo is the intensity of the image on the screen with out knife-edge.

Also, as the light passes through any lens, both source dimensions are magnified

such that the image area at any point will be (md)(mh) or m2dh where m is the

magnification ratio defined as the ratio of image dimension to object dimension. For a









lens combination, as in Figure 2-3, the total magnification is the product of each lens

magnification such that

screen = M2 M3

And, area of the image on the screen is

Ascreen = seenn )2 dh, (2-45)

where mscreen is the magnification on the screen.

In the case of a bi convex lens, the magnification is also equal to the ratio of the

image distance from the lens to the object distance from the lens. Therefore the

magnification due to the Lens 1 is

m =- (2-46)


since the slit is placed at the foal point of Lens 1 and the image is formed at infinity.

The magnification due to the Lens 2 is


m2 = ,2 (2-47)


since parallel rays falls on the lens, and the image is formed at the focal point of Lens 2.

The magnification between the focal point of Lens 2 and the screen (see Appendix

A) using similar triangles is

d2 f2h
2 f
m3 2 1 x+1 (2-48)
f22h


where x is the distance between the knife-edge and the traverse. Hence the total

transverse magnification on the screen is














.screen ]x+1} (2-49)


d2 2f
'sce eni w [f i (2-49)



Substitituting Equation 2-49 into Equation 2-45 the area of the screen can be expressed as



=. [ d2 fh +l (2-
2] 2 f,
IKeAn -dh. (2-50)



The resulting intensity on the image plane is

kn e.... wo I'ht (2-51)
1 screen/wo (2-51)
^-A-2

A2 2 / f i

fi


In practice however, a knife-edge is placed at the focal point of Lens 2. There is no

power loss prior to the knife-edge location. Hence

IKEAKE = source .... Source (2-52)

where IKE is the light intensity, and AKE is the area illuminated on the front side of the

knife-edge.


AKE = (mKE2 dh.


(2-53)









Also, the magnification at the knife-edge can be calculated as m, = m1 m2
f,

Substituting in Equation 2-52, the light intensity at the knife-edge can be expressed as

IKE ='s-.. (2-54)



The knife-edge blocks a portion of the light as shown in Figure 2-2. In the case of

half-cutoff, the area of the image on the rear side of the knife-edge AKE' (where the

prime superscript denotes a position after the knife-edge) has already been derived in

Equation 2-32.

As in section 2.2.2, the power can be assumed to be conserved from the knife-edge

to the screen and Equation 2-30 can be applied. Also, the constants As,,,. and IK can

be obtained from Equation 2-50 and Equation 2-54 respectively. The expression for the

intensity of the image on the screen (Equation 2-33) can be modified as


Screen Isourceab- (2-55)
Sd2 f J
222 f2 X +I dh

f -


and the quantitative sensitivity (Equation 2-39) of the instrument in the x axis can be

expressed as


S, Iource f2 -2 (2-56)

f dr2 f2h
f








Since


f
J\


LENS 2


TEST AREA


ID


S x2 2

Figure 2-4: Schlieren head with the conjugate plane.

Equation 2-56 can be written as


Sf
surce2_f2
-2


! f /


(2-57)


If the test area is placed at a distance s from the focal point of Lens 2, the real

image is formed at the conjugate plane. Using the thin lens formula, we obtain






26

f2
S= 2 (2-58)
s

Hence if the screen is placed at the conjugate plane the sensitivity can be expressed as


S, Is..u f2 (2-59)
]3L2 -2hJ
Shs



The equation can be non-dimentionalized as
Sq_ (2-60)
Source L]3 fhJ
f2 2 f +1 d
f hsI


It can thus be seen that the quantitative sensitivity is a function of various optical

parameters. Once the field of view d2 and the conjugate plane distance x are fixed, an

fi h
optimal value for (ratio of the focal length of Lens 2 and Lens 1) and (Ratio of the
fd d

height and width of the slit), can be obtained for maximum sensitivity (taking diffraction

effects into consideration).














CHAPTER 3
EXPERIMENTAL SET UP

Flow visualization and data acquisition using the optical deflectometer was

performed at the Interdisciplinary Microsystems Laboratory at the University of Florida.

This chapter discusses the experimental setup in detail. The chapter is divided into three

sections. The first section describes the optical system used for detecting the density

gradient field. The second section describes the normal impedance tube that generates

the acoustic field. The last section deals with the data acquisition system used in

detecting the light intensity fluctuations.

3.1 Basic Schlieren Setup

The schlieren setup used in the optical deflectometer is shown in figure 3-1. As

discussed earlier, the point light source mentioned in Chapter 1 is not realistic. Hence an

extended light source is created using a 2mm 2mm rectangular aperture. A 100 W

tungsten- halogen lamp with a custom aluminum housing is used as the light source. A

combination of compressed air and a fan is used to cool the housing. A DC power supply

(Twinfly model P -100 -12) is used to supply 12 V to the lamp. The light passes

through an 8-inch long tube for minimal loss. A condenser lens (Oriel model 39235) of

diameter 50.8 mm and focal length 100 mm is placed at the end of this tube. The lens is

achromatic in nature and prevents chromatic aberrations. The beam that passes through

the condenser lens is focused onto a rectangular aperture (Coherent model 61-1137),

which acts as the point source.




























LIGHT SOURCE E LENS 1 LENS 2 REEN
REFERENCE PHOTODIODE
PHOTODIODE
TRANSIMPEDANCE
AMPLIFIER

HIGH PASS
FILTER
MULTIMETER FILTER
16-BIT A/D


Figure 3-1: Deflectometer set-up.

The test-section is placed in between the two schlieren achromatic lens (Oriel


model 39235) of diameter 50.8 mm and focal length 100 mm, so that a collimated beam


passes through the flow-field. The knife-edge consists of a razor blade and is placed on a


X Y Z positioning movement (Edmund Scientific Model NT03 607 ) for fine


adjustments. The screen is made of translucent paper and is mounted on a traverse. All


the optical components are mounted using mounting posts and holders. The system is


placed on a rail (Edmund model NT54 402) so that the distance between various


optical components can be adjusted easily.


3.2 Normal Impedance Tube

3.2.1 Components

The impedance tube shown in Figure 3-2 is straight and is of a constant square


cross-section of width 25.4 mm rigid, smooth, non-porous walls without holes or slits in










TEST SECTION RIGID BACK
PLATE
COMPRESSION SPECIMEN
DRIVER OPTICAL GLASS
SMIC2 MIC1



254 mm





POWER PULSE SIGNAL DSA A/D HIGH PASS MIC POWER
AMPLIFIER GENERATOR CONVERTER FILTER SUPPLY


Figure 3-2: Normal impedance tube

the test section. The metal used for the construction of the plane-wave tube is aluminum

It has a length of 0.724 m with a cut-off frequency of 6.7 kHz for the plane wave mode.

The walls are 22 mm thick so that incident sound produces no appreciable vibration and

validates the rigid wall approximation. The test-section is placed on one end of the tube.

A membrane loudspeaker/compression driver (JBL model 2426H ) is placed at the

termination of the impedance tube at the end opposite to the sample holder. The

loudspeaker is contained in a sound-insulating box in order to minimize the sound

produced by the speaker. Sinusoidal oscillations are generated using a signal

generator, which is the PULSE (B&K Type 2827-002) system in our case.

3.2.2 Fabrication of the Test Section

The test section was fabricated such that it can be attached to one end of the

impedance tube. The transition from impedance tube to the test section is smooth and

care is taken to minimize leakage at the joints. Optical glass is inserted at the front and

rear side of the test section for visualizing the flow. The length of the window is










0.17 m and height 50.8 mm. The sample holder is fixed at the termination of the test-

section. It is a

RECTANGULAR
APERTURE LENS1 ENS2 2-D TRAVERSE KNIFE-EDGE







LIGHT
SOURC

















TEST-SECTION PLANE-WAVE
TUBE


Figure 3-3: Setup for normal impedance tube.

separate unit and it is large enough to install test objects leaving air spaces of a required

depth behind them. Since experiments were being conducted for a reflection coefficient

of unity, the specimen used was made of aluminium and 22 mm thick in order to provide

sound hard boundary condition.

3.3 Data Acquisition System

The microphone signal and both the mean and the fluctuating signal of the photo

detector were measured. The following equipment was used.









3.3.1 Photosensor Module

Data was acquired using two photodiode modules (Hamamatsu Model

H5784 20). One detector was used to detect a reference signal (explained in Chapter

4) and the other measured the fluctuating light intensity on the screen. The sensitivity of

the module is 255 x106 V /Im and the output of the detector varies from -15 to +15 V .

The output-offset voltage was -8.0 mV and the control voltage of the photo-detector

module was set at 0.35 V. The light is passed on to the photodiode through a fiber optic

cable, which terminates in a SMA adaptor (Newport model FP3 SMA).

3.3.2 Positioning System

The screen with the photodiode is placed on a two-dimensional traverse (Velmex

Model MB4012P40J-S4). The resolution of the traverse is 1.6 pUm and is controlled

using a controller (Velmex Model VXM 1). The positioning system and data acquisition

system was computer controlled using LabVIEW. For calibration of the system, the

knife-edge was placed on a one-dimensional traverse (Newport model ESP100), which

has a resolution of 1 umn.

3.3.3 Signal-Processing Equipment

The output signal of the photo-diode and the microphone are filtered using a

computer optimized filter (Kemo Model VBF35 Multi-Channel Filter/Amplifier System)

with a flat pass band and linear passband phase, that operates as a high pass filter with a

cut-in frequency of 850 Hz. The filtered signal passes to a 16-bit A-D converter

(National Instrument Model N14552) to remove the low frequency components. The

time invariant component of the signal is simultaneously sampled using a multimeter

(Keithley model 2400). The entire process is computer controlled using LabVIEW.














CHAPTER 4
DATA ANALYSIS
This chapter discusses the quantitative extension of the schlieren technique, namely

optical deflectometry. As discussed in Chapter 1, this involves the measurement of light-

intensity fluctuations at a point on the image plane using a fiber optic sensor. The first

part of this chapter deals with the static calibration, which relates light intensity

fluctuations to the knife-edge deflection. Subsequently, the cross-spectral correlation

technique used to determine the magnitude and phase of the density gradient relative to a

reference microphone is described. In the last part of the chapter, dynamic calibration of

the instrument using a laser impulse is discussed.

4.1 Calibration of the Optical Deflectometer

The objective of the calibration is to establish a relationship between the light

intensity fluctuation and the density gradient. We take advantage of both theoretical and

experimental methods to determine this relationship.

The relationship between the angular deflection and the variation in refractive

index has already been derived in Chapter 2 in Equation 2-27 and Equation 2-28. Using

the Gladstone-Dale relationship in Equation 2-16, assuming two-dimensional flow field,

and no & 1 for the surrounding air gives


E = kW (4-1)


Prior to the experiments, theoretical calculations were done in order to obtain an

estimate of the range of the light intensity fluctuation due to the flow. The maximum









density gradient in the plane wave tube for a typical value of sound pressure level was

compared to the maximum density gradient that can be detected by the system. This

provides an estimate of the fraction of the linear operating range that is occupied by the

light-intensity fluctuations.

It has already been derived (Equation 2-30) that the light intensity fluctuation is

related to the angular deflection of the light rays. The image of the source at the knife-

edge has been shown in Figure 2-2, and the parameters are described in Chapter 2. For a

knife-edge at half cutoff,



DEFLECTED
I IMAGE
ak A
A ------ ^


b






UNDISTURBED
IMAGE
KNIFE-EDGE

Figure 4-1: Light source in the plane of the knife-edge.

a
k = (4-2)
2

The maximum deflection in the light ray, which can be detected by the system,

occurs at

a
Aa = (4-3)









This case has been shown in Figure 4-1. Substituting Equation 4-3 in Equation

2-29, Aa = f2, gives

a
E (4-4)
2f2,

To determine the maximum density gradient that can be detected by the system,

combining Equation 4-4 and Equation 4-1 gives

P- a (4-5)
max 2f2kW

Also, the magnification of the source at the knife-edge is


mKEf2 (4-6)


and a = mKEd, (4-7)

where d is the width of the rectangular aperture. Using Equation 4-7, the density

gradient can be rewritten as

Op) d (4-8)
max 2fkW


Substituting the value of the focal length f = 100 mm, the width of the test-section

W = 0.0254 m, and the value of d = 2 mm, the maximum density gradient that can be

detected by the schlieren system is 1.74 x 103 kg /m3.

The expression for the density gradient in the normal impedance tube is given by

Equation 2-15. Differentiating with respect to d, and equating it to zero we obtain the

value ofd at which maxima occur for a reflection coefficient of unity as


d=(2n+l) (4-9)
4









where n is a positive integer and A is the wavelength of the acoustic wave. For an SPL

(re 20 /Pa) value of 120 dB, and a unity reflection coefficient, which corresponds to a

maximum pressure of 2p, occurring in the normal impedance tube, and for a frequency


of5 kHz, the density gradient at is 0.016 kg/m3.
4

The ratio of this density gradient fluctuation in the impedance tube to maximum

detectable gradient of the system is 10 5. Hence, it can be concluded that the density

gradient fluctuation occupies only a minute fraction of the dynamic range of the device.

Static Calibration. In the absence of a knife-edge, the light intensity at the image

plane is represented as Imax and, when fully blocked by the knife-edge, the intensity is 0,

assuming that diffraction effects are negligible. Thus the light intensity varies from

0 Ima When the knife-edge blocks a part of the light, given by a k as seen in Figure

4.1, the intensity on the image plane is given by

k
seen = Imax (4-10)
a

When light is refracted due to the density gradient in the test section the expression for

intensity is modified as

k + Aa
Imax (4-11)
a

where Aa is the knife-edge deflection as described in Chapter 2 and is given by Equation

2-29. From the above expression, the static sensitivity with respect to the deflection of

the image at the knife-edge is given by


K = 0 -max (4-12)
8Aa a










Since the light intensity is linearly related to the knife-edge deflection via Equation

4-11, a direct calibration can be done in an undisturbed (no-flow) case, by recording the

light-intensity for several knife-edge positions ranging from no cutoff to full cutoff. The

calibration curve which gives the voltage variation (directly proportional to the light

intensity) vs knife-edge position can be used to determine the angular deflection of light

rays that pass through a flow with a density gradient for a fixed knife-edge. In order to

ensure that all the measurements are taken in the linear range, the schlieren system was

operated at half cut-off, in which case the knife-edge blocks half the image of the source.


-------- 4 -------1-2.5- -


1.5-T--- ---- ---- --
----------- --------- ------


-1.25 -0.75 -0.25 0.25 0.75 1.25
> **T


**, 0.5 +



-1.25 -0.75 -0.25 0.25 0.75 1.25

Knife-edge position (mm)


Figure 4-2: Knife-edge calibration of photodiode sensor.

A typical knife-edge calibration at the center of the image plane is shown in Figure

4-2. It shows the output voltage of the photo-detector as the knife-edge location is

varied. The x -axis is rescaled so that the y axis passes through the operating point at

half cutoff.

The source intensity variations with time are accounted for using a second

reference photo-detector that measures the source intensity directly. Figure 4-3 shows

the calibration curve after the photo-detector signal has been normalized using the

reference detector signal. It can be seen that the temporal non-uniformity of the light










source shown in Figure 4-2 has been corrected by the reference photo detector in Figure

4-3.

0.2

i----------9-- ------------
I *
A*
01.


> - - ^ - -~-- -^ '&- ---4 --- - -- -



-1.25 -0.75 -0.25 0.25 0.75 1.25
Knife-edge position (mm)

Figure 4-3: Photodiode knife-edge calibration.

The experiments also account for any zero offsets in the photo-detectors. A

dimensionless parameter D is defined which takes into consideration the two corrections

mentioned above.


S= d pd dark (4-13)
Vref pd ref pd dark

where Vpd is the DC voltage of the photo-detector, Vef pd is the voltage of the


reference photo-detector and Vd _dark and Vref pd dark are the "no-light" voltage offset of


the photo-detector and reference photo-detector, respectively. But, the equation does not

take into consideration the spatial variation of light intensity in the image plane.

Calibration curves were obtained for three locations in the image plane as shown in

Figure 4-4. The plots show the effect of non-uniform cut-off in the image plane. This

leads to non-uniform illumination of the screen, which causes the slope (or static

sensitivity) to vary from point to point in the image plane as can be seen in Figure 4-4.

Non-uniform illumination also causes the maximum intensity to vary from one position











SCHLIEREN
IMAGE



NORMAL IMPEDANCEt d
TUBE


d = 9.1 cm d =6.0 cm
d =7.6 cm

3
2. -----------------------------------d-6.0-c
2.5- d=6.0 cm
2.5 ,
Sd=7.6 cm
2 --------------------
d=9.1 cm
g 1.5 -----------
e1-

0.5

0 i-
0 0.5 1 1.5 2
Knife-edge position (mm)


Figure 4-4: Photodiode knife-edge calibrations at three locations.

to another as can be seen in Figure 4-4. The second effect is not of major concern to us

since the operating point is at the center of the curve and the intensity fluctuation is very

small when compared to the linear range of the curve as shown in the beginning of this

chapter.

The data were normalized using the equation


(0 =m mm n (4-14)
max mmin

where (mmn and max are the minimum and maximum value of 0, respectively, at a


particular location. This normalization allows us to compare slopes at the three locations,

which otherwise have different linear ranges. Linear curve fits were obtained for the










linear portion of the curves using Excel Regression Tool as shown in Figure 4-5. The

variation in the slope is clearly visible in the figure and was not found to fall within the

95 % confidence interval of each other.

1.2







0.2 -
0
0 0.5 1 1.5 2
Knife-edge position (mm)


Figure 4-5: Linear region of the calibration curves at three locations after regression
analysis.

It was determined that the non-uniform cut-off is mainly due to the finite filament

size of the tungsten-halogen lamp. The condenser lens-slit combination could not

produce uniform illumination over the entire area slit. This effect was mitigated by

placing ground glass behind the rectangular slit.

Figure 4-6 shows the calibration curves at the three locations after the ground glass

was inserted. It can be clearly seen from the plots that the variation in the slope of the

curve (static sensitivity) is reduced considerably. This was verified using linear

regression analysis.

The slope of the calibration curve is summarized in Table 4-1. It can be seen that

there still exists a small variation in the slope. In the experiments conducted, since data

was being taken only at twenty locations along the tube length, a calibration curve was

found at each of the twenty locations and the local slope was used for data reduction.

















0.8


e 0.6


0.4


0.2


0


d=6.0 cm
-s-. d=7.6 cm
!. d=9.1 cm
U.
U.


nnnn T i


0.1 0.15 0.2 0.25 0.3

Knife-edge position (mm)


Figure 4-6: Calibration curves after the ground glass was inserted.


Table 4-1: Slope of the calibration curve at three different locations.


Location Slope (mm 1)


Right 0.158+0.001


Center 0.168 0.002


Left 0.170+0.001


0.2
0.18-
0.16-
0.14
0.12-
0.1
0.08-
0.06-
0.04-
0.02-
n-


5.75 6.75 7.75 8.75

Distance from specimen (cm)



Figure 4-7: Slope of the calibration curve plotted along the test section.


- -
I *;e $
I I ~f~









The slope of the calibration curve at the twenty points taken prior to an

experiment has been shown in Figure 4-7 with the 95 % uncertainty estimates obtained

from the linear regression analysis in Excel. The x axis gives the distance from the end

of the normal impedance tube that contains the specimen.

The data reduction procedure for one location is now summarized. The

dimensionless parameter, for a no-flow case is defined in Equation 4-13. For a case

where there is light intensity fluctuation, this parameter is modified as

SVpd + V'-pd dark (4-15)
Vref pd ref _pd dark

where V' is the fluctuating term in the photo-detector signal caused by the acoustic

density gradient in the impedance tube. The corresponding fluctuation element can be

obtained by subtracting the undisturbed intensity (DC operating light intensity) from

Equation 4-15 as

V
SAO=O-O= (4-16)
ref _pd ref _pd _dark

4.2 Data Reduction Procedure

Optical deflectometry is based on the principle of a cross correlation between two

points. This section briefly discusses the procedure followed to obtain the density

gradient fluctuations

Calculation of Density Fluctuation. The fluctuation in light intensity is measured

using a photo-detector mounted on a traverse in the image plane that moves along the

length of the impedance tube. It has been shown in Chapter 2 that the fluctuation in light

intensity is a measure of the refractive index or instantaneous density gradient in the

direction normal to the knife-edge. This optical signal is converted to an electrical signal









using a photo-detector. The signal is then filtered and amplified using a filter/amplifier

and digitized at high speed by a 16-bit A-D converter (National Instrument Model

N14552). The digitized data is transferred to a computer for subsequent analysis. A

signal from a fixed microphone located at d = 6.4 cm is also sampled simultaneously.

Using cross-spectral analysis, the coherent power in the photo-detector signal and the

relative phase difference between the photo-detector signal and the microphone signal are

determined.

The density fluctuation [12] can be represented as


px (x,t)= Re (x)ect (4-17)


The spatially dependent term in the above equation is a complex quantity, which

can be expressed as

S(x) = (x) e). (4-18)
Ox x

The magnitude and the phase of the above equation are determined by a cross-

spectral analysis described below.

The frequency response function between the input microphone signal x and

output photo detector signal y is defined as [19]


H ee = H eee, (4-19)
G. G.

where G, is the cross spectrum between the input and the output signal, and G, is the

auto spectrum of the input signal.

The voltage fluctuation detected by the photo-detector at any location can be

represented as









V '(x, co) = Re [V '(x) el(x)e"'t ], (4-20)

and the reference microphone signal at a distance d, in the plane-wave tube is given by

P(d,co) = Re IP(d) e'"(le't ]. (4-21)

In Equation 4-19, the magnitude H is the ratio of the output to the input signal,

and the phase 0(x) = O(x) yi(d) is the phase difference between the output and input

signals.

The amplitude of P(d1,w) is obtained from the power spectrum, P<, (dj1,w), of the

microphone signal and the phase Vf at d, with respect to d = 0 using two-microphone

method. These values are then used to compute the amplitude and phase distribution of

the voltage fluctuations.

The calculated amplitude of V'(x, w) is then substituted in Equation 4-16 to obtain

the corresponding values of AO The slope of the calibration curve can be written as

AO
slope = (4-22)
Aa

where AO is related to the fluctuating element of the light intensity and Aa is the shift in

the light rays in the plane of the knife-edge, which causes the light intensity fluctuation.

Using the slope of the calibration curve at the corresponding location, the shift in

light ray Aa is obtained using Equation 4-22. Subsequently, the angular deflection E"

corresponding to the magnitude of light intensity fluctuation can be obtained using

Equation 2-29. Finally, the density gradient is obtained by substituting E, in Equation

4-1.










The data reduction procedure described above was automated using a MATLAB

code for all the locations, and the density gradient distribution (magnitude and phase) are

determined along the length of the tube. A schematic of the data reduction procedure is

shown in Figure 4-8.


detector


Ox kW


Figure 4-8: Data reduction procedure.

Further, frequency domain correlation tools, such as coherence and coherent power

spectrum are used. The ordinary coherence function for x as input signal and y as

output signal is defined as


2 G 2
G"
XYGG
xxy ~ yr'


(4-23)










The coherence function is related to the portion of y that is linearly correlated to

the input signal x. The coherence power spectral density is defined as


G, .= ;2G,. (4-24)
Gyy,coh Gyy. (4-24)


It is a direct measure of the power-spectral density that is linearly coherent with

the reference pressure. These two parameters help determine the quality of the frequency

response measurements.

4.3 Dynamic Calibration

The dynamic system sensitivity of the system was determined after the completion

of the impedance tube experiments. The reflected output from a laser pulse was directed

towards the photo-detector at the operational gain of the system.

The pulse input duration was found to be much shorter (D 20 nsec) than the

photo-detector system time constant. Hence, the input was treated as an impulse. Figure

4-9 shows the measured impulse response of the photo-detector.

1.2

1----------------
1 ------ ------- ------- -------------- ------ I------- I--------.------- ------


0.8 ---- -- -- --------------- ---------------------------- -----------

0.6 --









-0.2
-0.2 I I A
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time (s) x 105


Figure 4-9: Impulse response of the experimental photo-detector.














OPERATING POINT


0


-0.1


-0.2


-0.3


-0.4


-0.5


-0.6


-0.7


-0.8


-0.9
1


5000 10000


OPERATING POINT--_


0



-5



-10



-15



-20



-25



-30



-35



-A


10 100 1000 5000 10000
Frequency (Hz)



Figure 4-10: Frequency response of the experimental photo-detector at the experimental

gain setting.


100 1000
Frequency (Hz)


-------~---~-,- ~~~~------- -- ,-- r r r~---- ------- ~ ~r, r






-~------~.--- -.--ri r r. -.------.------- r r r.-


- - I r - -- I -



-- - - -I- -


0


.. . . . -. .. .-- -. ..- -,----- - - --- L















------ ---- --- -- -- ------ -- -- -- ---- ---- -
,
, , ,









- -- - - -,-- --r- n- ,- -,--' -,--r '--- -


' '"" '""' '""


i i . .


-


--









The impulse response shows that the system behaves like an over damped second-

order system. A curve fit was performed using the equation


h(t) K ne 4S--)te~ )t (4-25)


where is the damping ratio, o,, the undamped natural frequency and K is related

to the strength of the pulse. The damping ratio was found to be 1.0018 and

on, =1.93 x 05 rad/sec. Since it is an over-damped second-order system, it can be

concluded that there is a time delay and an amplitude attenuation.

The frequency-response characteristics obtained from the damping ratio and the

undamped natural frequency are shown in Figure 4-10. The gain variation was found to

be -0.2 dB and the phase lag -18.50 at 5 kHz. From the response, it can be seen that

while the gain is negligible, the phase lag has a finite value. This was accounted for

while reducing the data in Chapter 5.

4.4 Experimental Procedure

Two experimental methods were employed simultaneously to investigate the

density gradient fluctuations in the normal impedance tube: the optical deflectometer and

the two-microphone method [25].

The entire experimental procedure was automated using a LabVIEW program. The

program simultaneously acquired data from two photo-detectors and two microphones,

controlled a 2-D traverse and partially processed the data. Data was acquired at twenty

different locations with a spacing of 1.7 mm across the schlieren image.

At each location, the filtered fluctuating signal of the experimental photo-detector

and the pressure fluctuation signals from two microphones placed at a distance









d, = 6.4 cm and d = 4.5 cm from the specimen, respectively, were acquired. The dc

component of the experimental and the reference photodector were acquired using a

digital multimeter (Keithley model 2400).

For the deflectometer, the frequency response function with the first microphone

signal (reference microphone) as input and experimental photo detector signal as output

is calculated simultaneously by the LabVIEW program. The DC component of the

photo-detector signals is used to correct the temporal non-uniformity in light source as

discussed in section 4.1.

For the two-microphone method, signals from the two microphones were used to

constructing the standing wave pattern in the impedance tube. The reflection coefficient

is calculated using Equation 2-10 by the program as the data is acquired.

The analysis was carried out entirely in the frequency domain and ensemble

averaging was performed over 10, 000 blocks of data with 1024 data points in each

block. Uncorrelated noise is reduced by ensemble averaging. Thus the random noise

variations in the spectrum are smoothed out. The sampling frequency of the data

acquisition system was set at 102400 samples/sec and hence Af was 100Hz. Since the

signals were periodic, a rectangular widow was applied and spectral leakage was avoided

[19].

Once the entire procedure was completed, the static calibration was repeated. It

was observed that the variation in the slope of the calibration curve at a particular

location was negligible during the course of the experimental procedure. The traverse

was moved to the adjacent location. Static calibrations were performed at the next

location, and the measurement procedure was repeated for all twenty locations.














CHAPTER 5
RESULTS AND DISCUSSION

This chapter presents the theoretical, computational, and experimental results

obtained during the course of the research. The results obtained from two experimental

methods with the uncertainty estimates are compared in the latter part of the chapter.

5.1 Theoretical Results

Prior to the experiments, the variations in pressure, density, and density gradient

were calculated along the length of the impedance tube. Since the experiments were

being conducted for a sound hard termination, the value of R should ideally be unity and

was thus used for these calculations. From Equation 2-3 pressure distribution for R = 1

is

p(d) = p, (ekd e- kd). (5-1)

From Equation 2-13, the density distribution can be obtained as


p(d) = -(ekd +e-kd). (5-2)


And using Equation 2-15, the density gradient for R = 1 is derived as

Op p jk(eJkd -e kd)
Od c2

Figure 5-1 provides a comparison of pressure, density and density gradient fields.


It can be seen that pressure doubling occurs at d = 0, A,... and pressure nodes occur at
2






50




1.8 ---- -------------- ------- ----- -- ------ ----- .- -------


\ 2 c i c2
l^---i^-----\-.-- i{-';-------\----.-^---^ -
1.68 --- ---







10.4 -------- ------L------- --------.------------------------ ----L-_
1.2 6 -------- ---- -- ------ --------- --------- ---- --- ----












0 1- 2 3 4 5 G 7
kd


Figure 5-1: Comparison of pressure, density and density gradient distributions.
L4 1
0 .8 .... - - - - - I - I - I - - 1 - -


















wave correspond to the maxima of the density gradient and vice versa.

The variation in the phase of pressure and density gradient with respect to kd for

i i i
R = 1 is shown in Figure 5-2. A 180 degree shift phase occurs at d = ,-- in the



case of the pressure distribution and, at d = 0, ,, ... in density distribution.
2I I
0 1 2 3 4 5 6 7 6
kd

Figure 5-1: Comparison of pressure, density and density gradient distributions.

d A 3A 5A








In practice, however, from thleaks/losses exist n e observedancet ube and, hence,es of the pressure

magnitude correspond to phase maxima of the reflection coefficty gradient were calculated using the two-

microphone variation in the phase discuss of pressure and density gradient with respect to kd for
R = 1 is shown in Figure 5-2. A 180 degree shift phase occurs at d in the


case of the pressure distribution and, at d = 0, ,... in density distribution.


In practice, however, leaks/losses exist in the impedance tube and, hence, the

magnitude and phase of the reflection coefficient were calculated using the two-

microphone method as discussed in Chapter 2.












100

80 ---------
OU~~~~~ ~ ~ ---------r--------r------- r - - T - --- -----T-- -




40 ......... r ........ r ........ ..... ........ ,-



0 -------4- ---- -------------- ------ -- -------- -------- ------












043 as 2
I I I
; i







F --------i r-- A --- -2: Phs v ari o fo -------- --------a ------
-AQ ---------h ---- ---h -- -----h --------+ ----__ __ _ _ -




-20 --------- L ----. ---- --------- L -------- I ----- .-- I -------- --------. -------

-40 --------- : ----- --- : -------- .-------- .----- -- -------- -- ------ --------

-100-----Ir ------------------ ---T--------T--------T-------


0 1 2 3 4 7 8(5-4)













where is tThe density gradient distribution inof the impedance tube, d is the distance fromobtained using the test specimen,



Substituting the pressure signalsmagnitude at two locations, d and d (see Figure 2-1), using 5-4, the follconstant A is


determinedure. Thise amplitude fcan then be verified by using the pressure ampsignal (Appendix D) is derived from Equlocation
pd2. The density gradient amplitude is cos( kd k2 + (sin( + Equation 2-15 as kd















d2. The density gradient amplitude is derived from Equation 2-15 as
-8 0 .. . . . ... . -
i
r 0
S_0 1 2 3 4 5 6 7 8~~~L~~~~~I~~~
I I I I kd
Figr 5-2 Phs vraio fo a riiemiain
The dest grdin ditibto in th ipdnetbcabeoaine snh
presur sinl at tw loaios d1 an 2(e iue21) snholwn





p -d) = A /(os-k--l) -r cos(-r- -kd ----kl) +(inkr-l))+ -r sin( r-kd-kl)2

I I I (5-4)
wher 1 is th legt of th meac ue stedsac rmtets pcmn
R~ an ar th mantd adpaeothrelcinoeiintrsetivey

Susttuin th prssr mantd at a loaind nEuto 54hosatAi
deerind Th~~L~~~~~~~~~~~~~is au a te evrfidb sngtepesreapiue tlcto
d2. Th dest grdin amltd is deie rmEuto -5a









-(d) = Ak J(cos(k(d- 1)) R cos( kd- k)) + (- sin(k(d 1)) + Rin kd- kl))2.

(5-5)

The density gradient amplitude for a known value of A and reflection coefficient is thus

obtained vs. position along the length of the tube using the above expression. The phase

of the density gradient is found from Equation 2-15


S= tan I (cos(k(d- )) R cos( kd- kl)) (5-6)
(-sin(k(d 1))+ R1 sin(o kd kl))

The magnitude and phase of the density gradient will be used to verify the

experimental results obtained from the deflectometer in Section 5.3.2.

5.2 Numerical Results

Data analysis similar to the test conditions was performed for simulated pressure

waves. The pressure across the plane wave tube at a particular position at a particular

instant of time is given by

p =Re [Ae ek (ekd + R e kd)et], (5-7)

where k is the wave number, I is the length of the tube, R is the complex reflection

coefficient and o) is the frequency of the signal. A reference signal was taken at the

origin, kd = 0. Cross-spectral analysis of the signal at various locations with respect to

this reference signal was then performed. Pressure signals were reconstructed using the

amplitude and phase information. The resulting animation consists of 72 frames (i.e., the

phase is advanced by 5 deg. between frames).







53


4 I I I I I
iricident wave:
0 Pressur
3 ---------------------- --0 I ------------
S janti-nod





|) 9(0 2700
0 ------------










-4 I I


- ~ ~~ ~ ~ - - - -- -- - - -
-2 --------------------------------------- ---P-rt


-3 ------------------ ---------- --------------
)800

0 2 4 6 8 10
kd


Figure 5-3: Pressure waves at various phases for R = 1.



4 I I
incide t










-3 ------------ ----------- -----------0 -----------O 00-- -----
-4 II, ,
2 - -,- .. ,-- -



S0 -- -- -------- -------^- -T------ ---T---- ^------ T




-2 -----------,--I----------,--I----------- ----------- ----------- -


-3 ------------ ------------ ------------ ----------- ----------- -

-4
12 10 8 6 4 2
kd

Figure 5-4: Pressure waves at various phases for R = 0.







54


4
-4 I-------------i--------inideint w I-

incident wave
3 ----------- ---------------------- -------------- ----- -:--------
o0 900 1800 2700

S-----------.-- ------------ ----------- ------ ---- ------ -------

















12 10 8 6 4 2 0
kd


Figure 5-5: Pressure waves at various phases for R = 0.5.

Phase locked movies for various values of R were obtained. Except for R = 0,

standing wave patterns exist. Figure 5-3 shows sample snapshots of the movie for R =1

case. This represents a rigid termination or sound-hard boundary condition. Pressure

doubling occurs at the interface kd = and the pressure reduces to zero at the nodes.

The termination acoustic impedance defined by Equation 2-6 is infinite. Also, the

standing wave ratio is infinite [18].

A purely progressive pressure wave in a tube with a pc termination is shown in


Figure 5-4. It can be seen that a standing wave does not exist for R = 0 The pressure

waves propagate towards the open end of the tube as the phasor is increased from 00 to

2700.
.. .. T




-- - - - --- --r- -- ---- ---- ---T---- r
-4 1 1
12 10 8 i
i i d





















2700. Ti ersnsargdtriaino on-adbudr odto.Pesr









Simulations were also obtained for a general resistive termination (see Figure 5-5).

1
The value of reflection coefficient was taken to be The standing wave ratio is 3 and
2

Z = 3Zo

5.3 Experimental Results

As mentioned in Chapter 4, two experimental methods were employed

simultaneously to investigate the density gradient fluctuations in the plane-wave tube: the

optical deflectometer and the two-microphone method. This section describes the results

obtained from the experiments conducted. In particular, the results obtained using these

two methods are compared later in this section.

5.3.1 Measurement of Density Gradient Using the Optical Deflectometer

An experimental set-up as described in Chapter 2 was constructed. Light intensity

fluctuations were measured along the length of the test section using a photo detector

mounted on a traverse. Pressure waves were generated at 5 kHz in the plane-wave tube.

The noise floor of the detector for a no-light case was determined and expressed as

power spectral density in Figure 5-6. The power spectra of the detector for light-on

cases, with various values of the gain are plotted in Figure 5-7. The power spectrum of

the detector was seen to increase, as the gain was varied from 0 to 0.4.

Beyond this gain, the power spectrum reduced steeply and remained closer to the 0

gain spectrum. This is likely due to the saturation of the photo-detector. Hence the

operating amplifier gain was set at 0.35.

Figure 5-8 and Figure 5-9 show examples of photodiode and microphone spectra,

respectively, at 145.4 dB. These spectra were measured for illustrative purpose using

Virtual bench. The dominance of power at 5 kHz is clearly visible from both the plots.





































frequency


Figure 5-6: Noise floor of the experimental photo-detector at an operational gain of 0.35.


10-10


0 0.5


1 1.5 2 2.5 3 3.5 4
Frequency (Hz) x 104


Figure 5-7: Noise floor of the experimental photo-detector (light-on) for various gains.











Figure 5-10 shows the coherence between the photo-detector and the reference


microphone signal. Although the coherence value is low (0.19) at the operating


frequency, the coherent power of the photo-detector at 5 kHz is approximately three


orders-of-magnitude above the noise floor (Figure 5-11).


107


10-8





m 10'9
101



o "_ 10 1

10-

10


10-14
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Frequency (Hz)


Figure 5-8: Example of photo-detector power spectrum at 145.4dB.

1IU


102
10
10


E 102

-4 /
10




10
10 -8-----


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Frequency (Hz)


Figure 5-9: Example of reference microphone power spectrum at 145.4dB.

































0 50 100 150 200 250 300 350 400 450
Frequency (Hz)

Figure 5-10: Example of coherent spectrum at 145.4dB.


0 1000 2000 3000 4000 5000 6000 7000
Frequency (Hz)


8000 9000 10000


Figure 5-11: Example of photo-detector coherence power at 145.4dB.














x 10-6
1 --------- -- ---------- ---------


0 .9 .5 ; .-. . . .- . : ..-
.








0 47 ---------------------- --------- --------- j----------- ---------- -------- -
0. ------------------------------ --------- --------
0.6










0.2


0 1 --------- ------------------- ---------
0
*,







6 6.5 7 7.5 8 8.5 9 9.5
Distance (cm)



Figure 5-12: Magnitude of the frequency response function at 145.4dB .


0.35



0.3


E
-r
SE 0.25

U
-n

0.2
E


| 0.15
-n


CD

0.05


0.05


6 6.5 7 7.5 8 8.5 9 9.5
Distance (cm)



Figure 5-13: Density gradient amplitude along the length of the tube at 145.4dB.


---------- -------------------- --------- I -------------------- I--------







-------- --------------------- ---------I-------------------- ------- --




---------- ----------- --------- --------- --
---------- ---------- ----------- --------- I ---------- ------*----- --*-------
I I







-- -- -I--------l-------- C---- -----------,------ - - -










Light intensity fluctuations were measured at 20 equally spaced locations. The

frequency response function gain has been plotted in Figure 5-12. Data was reduced

according to the procedure described in Chapter 4, and the density gradient distribution

was obtained in the impedance tube and is shown in Figure 5-13. The density gradient

distribution reveals a standing wave pattern and follows the trend discussed in Section

5.1.

Data was acquired for Sound Pressure Levels ranging from 123 -155 dB. From

Figure 5-14, for example, the power of the photo-detector signal at the operating

frequency is close to the noise floor at 126.4 dB. Also, at low SPL, the coherence of the

signal reduced considerably, the coherence at 126.4 dB was 0.0009.

Nevertheless, the coherent power at 5 kHz was still dominant (Figure 5-16). The

density gradient distribution at 126.38 dB is shown in Figure 5-18.



10






10
10
a-




10-9
10 10



10-11

10'12
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Frequency (Hz)

Figure 5-14: Example of photo-detector power spectrum at 126.4dB.
































1000 2000 3000 4000 5000 6000 7000
Frequency(Hz)


8000 9000 10000


Figure 5-15: Example of reference microphone power spectrum at 126.4dB.


10s6


10--





107


10-l






1011


10
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Frequency (Hz)

Figure 5-16: Coherent power of the photo-detector at 126.4dB.


10o


E
!5
S105


10-
Co


10-10


10-1 L
0








































0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Frequency (Hz)


Figure 5-17: Example of coherence spectrum at 126.4dB.


0.035
U.UJS



0.03



0.025



0.02



0.015



0.01



0.005


n


6 6.5 7 7.5 8 8.5 9
Distance (cm)


Figure 5-18: Density gradient amplitude along the length of the tube at


**.

---------- ---------- --------- -------- ------------------- ---------

jI. . ... - ..I I_ __







---------- -------------------- --------- --------------------- ------------ --- - -- -- -- -----





---------- -------- - - i - - -- - - r - --
---
-- --

- --_ -_ -_ -_ -_-_ _- - - - ----_-_-_-_- -_-_-_-_-_ _-_-_- -_-_-_- --__- --
.. .


9.5



126.4dB.


" ""'








63



The phase difference between the photo-detector signal and the microphone signal


is obtained from the frequency response function (Figure 5-19). The phase is found to


remain constant with a 1800 shift at the density-gradient node. To obtain the absolute


value of the phase of the density gradient, we require the phase information of the


reference signal, which is obtained from the two-microphone method in Section 5.3.2.



9 *



L *
2.6 ---------- ----------- --------L---------- ----------- -----------L-------- -



2 .---------. ---------- ---------- ---------. -------------------- .--------



1- I -I -

^ ---------- ------------------------------------------ --------------------_



0.1 ---------- -------.--- ---------- --------- ---------- ---------- -------- _
0~ 5 ----,----------,----,----------,---------;










-0.5 I I I
6 6.5 7 7.5 8 8.5 9 9.5
Distance (cm)


Figure 5-19: Phase difference between the photo-detector signal and the reference

microphone signal at 145.4dB


5.3.2 Measurement of Density Gradient Using microphones


The density gradient along the normal impedance tube length was calculated as


described in section 5.1. The amplitude of the pressure signal at a distance 4.5 cm and


6.4 cm is obtained from the power spectrum of the microphone signals.
I I I II
I I I II
I I I II
I I I II
I I I II
I I I II











cr,
-----------------------------
,-----,-

0.5
.. . 4 @ .


























6.4 cm is obtained from the power spectrum of the microphone signals.












0.35


0.3

E
0 0.25


S0.2
E

. 0.15
.-

0.1


0.05


0


Figure 5-20: Density


4 6 8 10 12
Distance (cm)


gradient amplitude along the length of the tube using the
microphone method at 145.4dB.


I I I I I
.- ... ... .---- ----. ---- ... -- --- .-. -- --- -----_ -- -_--.- ----- ----






,-__ -__- ------_ -- -_ -- -_-- --- --- ----- -_ -------- -
..........~~~~~~~~~~~ -.. -.. -.. -.. -.. .






-L---------- ----------- ----------- -------- -- _----- -- ---- -- --- --- -






L----- ---- L -----..- -- --- ---------


2 4 6 8 10 1
Distance (cm)


Figure 5-21: Density gradient phase along the length of the tube using the microphone
method at 145.4dB.


The reflection coefficient is obtained from the two-microphone method as


discussed in Chapter 2. The magnitude and phase of the density gradient at various










locations of the impedance tube are calculated using Equation 5-5 and Equation 5-6 and

are plotted in Figure 5-20 and Figure 5-21, respectively. From Figure 5-21, the phase at

the reference microphone location was found to be -89.60 and used to obtain the phase

angle from the deflectometer method relative to the specimen location (d = 0).

5.3.3 Comparison of Results Obtained by the Two Methods

The amplitude of the density gradients obtained using the two methods is plotted

with the uncertainty estimates (see uncertainty analysis in Appendix-C), in Figure 5-22

and Figure 5-23 at 145.4 dB and 126.4 dB respectively.

0.35 1
Microphone method Deflectpmeter
S: method







EI I
<_ 0.15 ---p---- --[------f--.--L-- -----L ----i- --- -- --l----- l---- -- -- _



S0.5 --------------- ------------ T---- T---------
E0.15 ------------ -- -------- ---- -------

o.("
0.1 -- --------- ---- -- ------------ --- .------T------ --- T ----------

-0.05 ;--------- ----- --- ----------- ------- ------ '-
03 -- - - -- I -- -- - -i- - - -



-0.05I I I
0 2 4 6 8 10 12
kd

Figure 5-22: Magnitude of the density gradient using the two methods at 145.4dB.







66


0.04 I I I
Microphone method Deflectome ter method
0.035 ------ -- --------- -





47 0.015 --_---- ----r- l-- j.----- -^---_ T -- - -- -- T- -
S0.035 -------------------- ------------ -------- ------

0.025---------- L. ---------

0.02

S-- ------ ---- ---------- ------------ -- ----------
0.015----------------------------------------


CD 0.005 .. -------
.0 - -- - -- --- ------- -- -- -- -- ---,






0 2 4 6 8 10 12
kd


Figure 5-23: Magnitude of the density gradient using the two methods at 126.4dB .

It can be seen that at higher SPL, the density gradient fluctuations obtained from

the two methods are very similar. The deflectometer result deviated from the microphone

method near the maxima of the density gradient. For the higher SPL, the estimates were

close but did not overlap near the maxima. The most dominant term in the uncertainty

estimate was the error due to the frequency response function and this error is dependent




In the case of the lower SPL, the results obtained with the schlieren method had

similar results and agreement as at higher SPL. The error in the density gradient field

was larger and this is caused due to the lower coherence between the signals. The error

bars were found to fall within the range of each other.
bars were found to fall within the range of each other.






































2 4 6 8 10


kd


Figure 5-24: Phase of the density gradient using the two methods at 145.4dB .


50


0o


-150 L
-2


8 10 12


Figure 5-25: Phase of the density gradient using the two methods at 145.4dB after the
phase correction from the photo detector.


-100




-150


,icrophone method




----------------------------









---------- ---------- ---------
. . .



----------.. -- ---------




















100------




50 --------




0 --------




-50 --------


-100 k----


-150 L
-2


6 8 10


kd


Figure 5-26: Phase of the density gradient using the two methods at 126.4dB after the

phase correction from the photo detector.


The phase distribution obtained from the two experimental techniques is compared


in Figure 5-24. The reference microphone phase was calculated at 6.4 cm and added to


the phase of the FRF to obtain the phase of the density gradient.


The result obtained from the optical method differed from the microphone method


by a finite value. This shift was caused by the phase lag in the photo detector and was


corrected via the dynamic calibration described in Section 4.3. Figure 5-25 and Figure 5-


26 shows the corrected values of the phase with the error bars. They are found to fall


well within the range of each other.


- ^ -------- -1---------- ^----- -

Microph ne method

-~---------- ---- 4 -- --- -----






I------------------ ------






: --------- ___,___-
I II
I I

Ii
I. .i. .







-- - - - - ,-- -


Deflectymeter method

I --------------









. -, -------~-------,-- -- --------_


I


----,--------


--














CHAPTER 6
CONCLUSION AND FUTURE WORK

This chapter summarizes the work done during the course of this project. Future

work required to improve the overall performance of the instrument is discussed in the

latter part of this chapter.

6.1 Conclusions

The ultimate objective of the research activity was to device a technique to

visualize and measure the acoustic field in a normal acoustic impedance tube. A

schlieren technique used for visualizing compressible flow was extended to measure the

one-dimensional acoustic field with a much smaller density gradient. The instrument

successfully detected density gradient fluctuations and data was obtained for sound

pressure levels ranging from 123-157 dB. The amplitude and phase distributions of the

density gradient were measured using the deflectometer. A second method based on the

two-microphone method was used to verify the results. The results were in fairly good

agreement except at the antinodes of the density gradient distribution. The discrepancy in

the results can partially be explained using the uncertainty estimation.

6.2 Future Work

The study conducted provides rudimentary results, which can be used as a basis to

study the behavior of an acoustic field in a normal acoustic impedance tube. The optical

deflectometer system can be improved to obtain better results. The results of the detailed

sensitivity analysis were obtained after the experiments. These results can be used to

design a more sensitive instrument.









The system field of view is limited by the diameter of the lens/mirror, such that

only a limited field of view could be obtained. Hence a larger diameter lens/mirror is

required to visualize flow at lower frequencies.

Various types of spherical and achromatic aberrations were encountered using the

lens-based system. Hence, a mirror-based system is highly recommended. Though the

f number of the condenser lens and that of the first schlieren lens or mirror are shown to

be equal in the ray diagram, it is advisable to have the f number of the condenser lens to

be 1.5 to 2 times smaller [4]. This avoids non-paraxial effects caused by the condenser

lens and also the effects of the reduced illumination of the light beam in the periphery.

Amplitude and phase mismatches were not corrected using the microphone

switching method [25] for different values of SPL while the experiments were being

conducted. This can be avoided by implementing the switching technique prior or during

the test. The photo-detectors were characterized as over damped second order system

after the completion of the experiments. A more sensitive photo-detector with lower

noise and increased bandwidth, such as a cooled photo-multiplier tube should be used in

future work.

Finally, using suitable boundary conditions, the pressure field can be obtained from

the density gradient fluctuations by developing a suitable procedure. The technique can

thus be extended to determine two- dimensional fields and can be used in the study of

scattering effects. Also, the technique can now be extended to focused schlieren for the

characterization of various specimens using the normal impedance tube.














APPENDIX A
KNIFE-EDGE GEOMETRY

The magnification between the focus of Lens 2 and the screen can be derived as

follows.


d
2 k
2


A-4


Figure A-i: Magnification of the source on the screen.

The Figure shows the region between the schlieren head and the screen (Figure 2-

3) in the setup with the extreme rays originating from the top of the object. The image

has a height k at the focus of Lens 2 and y + k on the screen. It can be seen from Figure

A-i that ZAOB = ZCOD. Hence











y= x,


Magnification


y+k
m3-k
k


Substituting Equation Al in Equation A3


m3 = x2+l (3)
kf2

The total magnification at focus of Lens 2 is equal to mi*m2 is

2 (4)


Therefore we have k


f2h. Substituting in the Equation A3
f,


d2 f2h
2 (5)
23 = x+l. (5)


y+
k















APPENDIX B
LIGHT RAYS IN AN INHOMOGENEOUS FLUID


The derivation is done under the assumption that physical phenomena like

diffraction or dispersion does not exist.

The refractive index is assumed to vary as a function of the three spatial

coordinates.


n = n(x, y, z). (Bl)

The incident ray is initially parallel to the direction. According to the Ferment's

principle, the variation of optical path length along a light ray in the refractive field must

vanish.

Ax

M. Image Plane
P











z1
z 2







Figure B-l: Deflection of a light ray in inhomogeneous test object.

f n(x, y, z)ds = 0, (B2)










where s denotes the arc length along the ray and


ds2 = dx2 +dy2 +dz2.


Equation B2 is equivalent to two sets of differential equations

dx 2 (dy2{ n d 1 anz
dz2 dz) dz) nn)x dz n azi


d2y
dz2


{1Kdx (dyl 1 an
I) +1d1) ,n
1+d -+
dz dz n O


dzy an
dz n Oz


Assuming that the slopes of the ray


an
unity and assuming
Dx


n
and -
ay


are of same order of magnitude Equation B4 and


Equation B5 simplifies to


d2x 1 an
dz2 n C


d2y 1 an
dz2 n 2 y


dx dy
- and are very small as compared to
dz dz


(B7)














APPENDIX C
UNCERTAINTY ANALYSIS


This section estimates the error in both magnitude and phase of the density

gradients measured using the deflectometer and the two-microphone method. The

analysis uses a technique that employs a first order uncertainty estimate. For the

uncertainty analysis of the microphone method, the technique is extended to complex

variables [23]. The uncertainties in the individual variables are propagated through the

data reduction equation into the result.

For a general case [24] if is a function of measured variables,

R = R(X,,X2,.....X). (Cl)


The uncertainty in the result is given by



U2 = T2 + U2 + ....+ U (C2)
OaXxC )a axJ X2

aR
where U, is the uncertainty in variable X The partial derivative is defined as the


sensitivity coefficient.


C.1 Uncertainty in Amplitude

C.1.1 Deflectometer Method

The data reduction procedure explained in section (5.3.2) is followed and a final

expression for density gradient is derived as shown below.









The equations for A' (Equation 4-16) and the slope of the calibration curve

(Equation 4-17) were derived in section (4.1.1). Substituting, Equation 4-16 in Equation

4-17 we obtain the expression for Aa as



Aa(d) = V'd(C3)
slope(d) x (Vef D (d) V, dak)


Substituting Equation C3 into Equation 2-29, we obtain the angular deflection as

V '(d)
Ex(d) = V'(d) (C4)
slope(d) x f x(Vef DC (d) Vef dak)

Subsequently, the density gradient is obtained by substituting Equation C4 in

Equation 4-1 as


P(d)=--, (C5)
x slope(d)xf, x(Vrf _D(d) VTf _,k) x K xW'

where G1, is the power spectrum of the input microphone signal at the operating

frequency. The numerator essentially gives the voltage amplitude as a product of the

magnitude of the frequency response function and the reference pressure amplitude. The

density gradient can be written in the form of Equation Cl as a function of various

parameters as shown


S(d) ( H(d) G, slope(d), f, Vf DC(d),Vf dark,W). (C6)


Representing as p various sensitivity coefficients can be written as follows.
LxI

p 2G,(C7)
aH slope xfx(Vref _DC Vref dark) x KW










p' H
9G, slope x x f2x e x_, 7 DC Vr ) x K x W'









ap" HVG
Oslope slope2 x f, X (V ,f _DC ,d k)K W'











OefDC slope x f2 x (Vref DC ref dark x W '





f dark slope x f2 x (Vref _DC 2 x K x W


0.3

0.25

0.2




0.1

0.05


0 .
-0.05


6 65 7 75 8 85 9


(C8)





(C9)





(C10)




(Cll)





(C12)


kd


Figure C-1: Error bar in the amplitude of the density gradient using the deflectometer at
145.4 dB.


SI







I I I I ---------
------- ---------------------r -----

------ -- ---- ------ I-------- I --------------- ------


-









SH 2G(C13)
8W slope x f x (Vf DC- Vrf dar) x K x W2


l-y2 1
The uncertainties [19] Ux in H and G,, were and respectively,


where y is the coherence and n is the number of averages. The remaining parameters

were determined from the instrument specifications. Finally the total uncertainty in the

density gradient was computed at each of the twenty locations and plotted in Figure C-l

for a SPL of 145.4 dB. The uncertainties in various parameters have been tabulated in

C-1.


Table C-l: Uncertainties in various parameters for a deflectometer.

Ux
Parameter


1--


GI


slope 95% error from the calibration curve


f2 8mm


Vref DC 0.6mV


Vref dark 0.6mV


W 0.0254mm









C.1.1 Two Microphone Method

A similar approach was also followed for the two-microphone method. The

expression for the absolute pressure in the normal acoustic impedance tube is derived in

Equation 5-4. The constant A can be expressed as


A = 11 (C14)
J(cos(k(dl- 1)) + IR cos( kdl- kl))2 + (sin(k(dl- )) + RI sin(, kdl- kl))2



The expression for density gradient has already been derived in Equation 2-15 as

dp Aikekl (ekd R e'e -kd)
(d) = 2
)C C2

It can be written as a function of four parameters as shown


P(d) = (A(d,l,G,),1, G R,). (C15)
Ox ox

Various sensitivity coefficients are calculated as follows

op' ike-zki (ezkd Re ekd)
-A = (C16)
8A c2


ap -Aike-1k R e -ked
= 2 (C17)
8iR c2

apx Ak2e 'ki(e'kd- R eOe-'kd)
-- (C18)
81 c


ap' Ake -'I R eekd
(C19)
8, c

It can be seen from Equation C14 that the constant A itself is a function of d,

A =A(dl,l,G,1) (C20)









And partial derivatives are given as


1)) + R cos(o -kdl kl))2 + (sin(k(dl


1))+ R sin(o, -kdl-kl))2
(C21)


A0
= 0;


cA _-A3 R sin(, 2kdl)
Odl 2G,,


Table C-2: Uncertainties in various parameters for the microphone method.

Ux
Parameter
R 0.03


1



/ 0.025mm


d, 0.025mm


01 0.015


The uncertainty in A is


U, A
'f a,)


SU +


UdA
9dl) dl


(C22)


(C23)


(C24)


sGn l (2Gii)/(cos(k(dl


U, + OA
OGw








81




0 35


0.3 -----
0235 ----- --- -------------I---- ------ ---------------------------

E






0.2 --- --------- ----- -- -------------- -- --T---- ------ -- -- -

0 2 4 8 10 12








kd



Figure C-2: Error bar in the amplitude using the microphone method 145.4 dB .





U.Jb I-I
Microphone method Deflectimeter

0 3 ------- ---------- -- ------- -- ---- -
0.2
E

<' 0.15





































02 ----- ------ L ------------------- ----- --- T----------- -------- _
E2 1 -- ---------- ---- -- ---------- ------ T----- T----- -


-0.05 --------- I------- ---- ----------- -------------
----
-0.105
0 05






















0 2 4 6 8 10 12
kd



Figure C-: Error bar in the amplitude using the microphone and deflectometer method 145.4 dB.
Microphone method Deflectpmeter
031-method:



0.25 -


02-

< 0.15 -


c3


2 0.05





-0.05
0 2 4 6 8 10 12
kd


Figure C-3: Comparison of microphone and deflectometer method at 145.4 dB.







82


The uncertainty in R and r, were calculated using the method of Schultz et al.


[23]. The total uncertainty in the magnitude of the density gradient is obtained [23] using

the equation


U -, I[Re(p,)Re(U )]2 +Im(p))Im(U )]
Ad 1x


(C25)


and plotted in Figure C-2.

In Figure C-3, the error bars obtained from the microphone method and the deflectometer

method are compared. The uncertainties in various parameters are shown in Table C-2.

C.2 Uncertainty in Phase


C.2.1 Deflec

80

60

40

20

l 0

-20

40

-60

-80

-100

-120
5.


tometer Method





-- -- ----------II---------------- I------ __ _ _ __--- __ -- -
----------- --------------------- .I---------------------------- -- ........





-- --- -- -- --- __-- __-- __ ,_----,_- _-- __-- __--,_-- __ --_ -- --
--
-- - - ,-, -
---
---------- ---------- ---------- --------- ---------- ----------- ---------

---- __------_ __ __ __ __ ---------- I--------- ---------- --------- -- -


-. -- ----------------- ------- ------- ---------- ---------


5 63


6.5 7 7.5


Figure C-4: Error bar in the phase using the deflectometer at 145.4 dB.








83



The phase of the density gradient using the deflectometer method is essentially


the sum of the phase difference obtained from the frequency response function and the


phase of the reference microphone at location d, as shown in Equation C26




Stan- sin(k(d- ))+ R sin(Q -kd- k) (
Scos(k(d- 1)) + R cos(, kd kl)




The uncertainty in 0 is given by [19] the expression -2 and the uncertainty from




the reference signal at d, is obtained from the two-microphone method.


0 2 4


8 10 12


Figure C-6: Error bar in the amplitude using the deflectometer at 145.4 dB.


-100




-150
-2


--------- ---------- ---- ---------









:r-- -- -- -- --- -- -- -- -- -


----- ---r --------- ------------







------- --------- --- ----- -- ---------------


I-- -------_












--L---------
: r __ _


---------,----------r---------r---------


----------.-









----------





----------J




- --


---,---------


-













100 --------


50 --------


100 --------


-150 L
-2


2 4


8 10 12


Figure C-7: Comparison of microphone and deflectometer method at 145.4 dB.

C.2.2 Microphone method

Procedure similar to Section C.1.2 was followed and the uncertainty in the phase is

obtained [23] from the expression


U = ;J m(p,)Re(U ) + [Re(p )Im(U ) .


(C27)


The error bar has been plotted in Figure C-6 and the comparison in Figure C-7.














APPENDIX D
DENSITY GRADIENT
As discussed in Chapter 2 the pressure distribution in the impedance tube is given


where p, = Ae -kl and p,


(Dl)


p(d)= (Ae- kle'kd +Be'kle kd),


Ae +ki and R defined as R =P


Equation D-2 can be expressed as


p(d) = Ae-ki(ekd + R e -e"kd).


(D2)


Using Equation 2-8, the pressure distribution can be con be converted into real and

imaginary terms as


P(d) = A(cos(k(d- 1)) + i sin(k(d


)) + Rcos(o -kd


kl)+ Rlisin(4


kd kl)). (D3)


The amplitude is given by


p(d) = AJ(cos(k(d


1)) + R cos(, kd kl))2 + (sin(k(d 1)) + R sin(o kd


And the phase is given by


y(d) tan ,sin(k(d -)) + R sin(, -kd- kl)
v/(d) = tan----------
S cos(k(d 1)) + R cos( kd kl)

The density gradient can be obtained from Equation 2-15 as


kl))2.
(D4)









(ap Aike kl (e kd R e' e -kd)
ax c2


It can be expressed as real and imaginary terms as





ap Ak
P(d) =- (i cos(k(d ))- sin(k(d ))- R\ cos(o kd kl) + R sin( kd kl)). (D4)
Cx c


The amplitude of Equation D-5 is given by


(d) = j (cos(k(d -)) R cos(- kd k))2 + (- sin(k(d 1)) + R sin( kd kl))2.

(D6)

And the phase is obtained from


S(cos(k(d- ))- R cos(( kd- k))
6 = tan(k(d 1)) + ( -kd ))
(-sin(k(d -1))+ R1 sin(A kd kl))


(D5)















LIST OF REFERENCES


[1] Merzkirch W (1987) Flow visualization. 2nd edition, Academic Press, Inc.,
Orlando, FL.

[2] Dyke V Album of Fluid Motion, Parabolic Press, Incorporated, Stanford,
California

[3] Holder DW; North RJ (1963) Schlieren Methods. National Physical Laboratory,
Teddington, Middlesex.

[4] Settles GS (2001) Schlieren and Shadowgraph Techniques.ISBN 3-540-66155-7
Springer-Verlag Berlin Heidelberg NewYork.

[5] Coleman HW; Steele WG (1999) Experimentation and Uncertainity Analysis for
Engineers, 2nd ed. New York: John Wiley & Sons, Inc..

[6] Davis MR (1971) Measurements in a subsonic turbulent jet using a quantitative
schlieren technique. J.Fluid Mech (1971), vol. 46, part4, pp. 631-656.

[7] Wilson LN; Damkevala RJ (1969) Statistical Properties of Turbulent Density
Fluctuations. J.Fluid Mech,vol. 43, part 2, pp. 291-303.

[8] Davis MR (1972) Quantitative schlieren measurement in a supersonic turbulent
jet. J.Fluid Mech. (1972), vol. 51, part 3, pp. 435-447.

[9] Davis MR (1974) Intensity, Scale and Convection of Turbulent Density
Fluctuations. J.Fluid Mech (1975), vol. 70, part 3, pp. 463-479.

[10] Weinstein LM (1991) An improved large-field focusing schlieren system. AIAA
Paper 91-0567.

[11] Alvi FS; Settles GS (1993) A Sharp-Focusing Schlieren Optical Deflectometer.
AIAA Paper 93-0629.

[12] Weinstein LM (1993) Schlieren system and method for moving objects. NASA
CASE NO. LAR 15053-1.

[13] Garg S; Settles GS (1998) Measurements of a supersonic turbulent boundary
layer by Focusing schlieren deflectometry. Experiments in Fluids 25 254-264.









[14] Garg S; Cattafesta LN 111; Kegerise MA; Jones GS (1998) Quantitative
schlieren measurements of coherent structures in planar turbulent shear flows.
Proc. 8th Int. Symp.Flow Vis, Sorrento, Italy.

[15] Garg S; Cattafesta LN 111 (2000) Quantitative schlieren measurements of
coherent structures in a cavity shear layer. Experiments in Fluids 30 pp. 123-124.

[16] Michael A. Kegerise; Eric F. Spina; Louis N. Cattafesta 111 (1999) An
Experimental Investigation of Flow-induced Cavity Oscillation. AIAA-99-3705.

[17] Cattafesta LN; Kegerise ma; Jones GS (1998) Experiments on Compressible
Flow-Induced Cavity Oscillations. AIAA-98-2912.

[18] Blackstock DT (2000) Fundamentals of Physical Acoustics. New York: John
Wiley & Sons, Inc.

[19] ASTM-E1050-90, "Impedance and Absorption of Acoustical Materials Using a
Tube, Two Microphones, and a Digital Frequency Analysis System."

[20] Ebert AF and Ross DF(1977) "Experimental determination of acoustic
properties using a two-microphone random-excitation technique" Journal of the
Acoustical Society of America 61(5).

[21] Bendat JS and Piersol AG Random Data, 3rd ed. New York: John Wiley & Sons,
Inc.

[22] Taghavi R; Raman G(1996) Visualization of supersonic jets using a phase
conditioned focusing schlieren system. Experiments in Fluids 20 472-475

[23] Settles GS (2001) Schlieren and Shadowgraph Techniques. Visualizing
Phenomena in Transparent Media.

[24] Leonard M. Weinstein (1991) Large-Field High-Brightness Focusing Schlieren
System.AIAA Paper 91-0567

[25] Schultz T; Louis N. Cattafesta 111; Nishida T; Sheplak M (1998) Uncertainty
Analysis of the Two-Microphone Method for Acoustic Impedence Testiong

[26] Horowitz S; Nishida T; Cattafesta L N; Sheplak M (2001) Compliant-
Backplate Helmholtz Resonators for Active Noise Control
Application, 39 th Aerospace Science Meeting & Exhibit, 2001















BIOGRAPHICAL SKETCH

Priya Narayanan was born in 1979 in Malappuram, Kerala, India. She moved to

the State of Kuwait in 1981 and graduated from The Indian School in Salmiya, Kuwait in

1997. She went back to India for her Bachelors degree and obtained her Bachelor's of

Technology degree in Aerospace Engineering from Indian Institute Of Technology,

Madras, India in May 2001. She is currently pursuing her Master of Science degree in

the Department of Mechanical and Aerospace Engineering at the University of Florida.