UFDC Home  myUFDC Home  Help 
PAGE 40
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 32: Normal impedance tube the test section. The metal used for the cons truction of the planewave tube is aluminum It has a length of 0.724 m with a cutoff 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 soundinsu 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 2827002) 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
PAGE 41
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 2D TRAVERSE KNIFEEDGE TESTSECTION PLANEWAVE TUBE Figure 33: 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.
PAGE 42
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 outputoffset voltage was 8.0 mV and the control voltage of the photodetector 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 twodimensional traverse (Velmex Model MB4012P40JS4). 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 knifeedge was placed on a onedimen sional traverse (Newport model 100ESP), which has a resolution of 1 m 3.3.3 SignalProcessing Equipment The output signal of the photodiode and the microphone are filtered using a computer optimized filter (Kemo Model 35 VBF MultiChannel Filter/Amplifier System) with a flat pass band and linear passband phase, that operates as a high pass filter with a cutin frequency of 850 Hz The filtered signal passes to a 16bit AD 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.
PAGE 43
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 knifee dge deflection. Subsequently, the crossspectral 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 227 and Equation 228. Using the GladstoneDale relationshi p in Equation 216, assuming tw odimensional flow field, and 01 n for the surrounding air gives .xkW x (41) 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
PAGE 44
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 lightintensity fluctuations. It has already been derived (Equation 230) 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 22, and the pa rameters are described in Chapter 2. For a knifeedge at half cutoff, abkKNIFEEDGE UNDISTURBED IMAGE DEFLECTED IMAGE a Figure 41: Light source in the plane of the knifeedge. 2 a k (42) The maximum deflection in the light ray, which can be detected by the system, occurs at 2 a a (43)
PAGE 45
34 This case has been shown in Figure 41. S ubstituting Equation 43 in Equation 229, 2 xaf gives 2. 2xa f (44) To determine the maximum density gradient that can be detected by the system, combining Equation 44 and Equation 41 gives max 2. 2a x fkW (45) Also, the magnification of the s ource at the knifeedge is 2 1 KE f m f (46) and ,KEamd (47) where dis the width of the rectangular apertu re. Using Equation 47, the density gradient can be rewritten as max 1. 2 d x fkW (48) Substituting the value of the focal length 1100 f mm the width of the testsection 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 215. 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 (49)
PAGE 46
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 knifeedge, the light intensity at the image plane is represented as max I and, when fully blocked by the knifeedge, the intensity is 0, assuming that diffraction effects are negligible Thus the light intensity varies from max0 I When the knifeedge 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 (410) 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 (411) where a is the knifeedge deflecti on as described in Chapter 2 and is given by Equation 229. From the above expression, the static se nsitivity with respect to the deflection of the image at the knifeedge is given by max. I I K aa (412)
PAGE 47
36 Since the light intensity is linearly relate d to the knifeedge deflection via Equation 411, a direct calibration can be done in an undisturbed (noflow) cas e, by recording the lightintensity for several knifeedge 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 knifeedge 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 knifeedge. In order to ensure that all the measurements are taken in the linear range, the schlieren system was operated at half cutoff, in which case the kni feedge blocks half the image of the source. 0 0.5 1 1.5 2 2.5 3 1.250.750.250.250.751.25Knifeedge position (mm)Vpd (V) Figure 42: Knifeedge calib ration of photodiode sensor. A typical knifeedge calibration at the cen ter of the image plane is shown in Figure 42. It shows the output voltage of the phot odetector as the kn ifeedge 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 photodetector that measures the source intensity direc tly. Figure 43 shows the calibration curve after the photodetecto r signal has been normalized using the reference detector signal. It can be seen that the temporal nonuniformity of the light
PAGE 48
37 source shown in Figure 42 has been corrected by the reference photo detector in Figure 43. 0 0.05 0.1 0.15 0.2 1.250.750.250.250.751.25Knifeedge position (mm)Vpd/Vref_pd Figure 43: Photodiode knifeedge calibration. The experiments also account for any zer o offsets in the photodetectors. A dimensionless parameter is defined which takes into c onsideration the two corrections mentioned above. ___,pdpddark refpdrefpddarkVV VV (413) where pdV is the DC voltage of the photodetector, _refpdV is the voltage of the reference photodetector and pddarkV and __ refpddarkV are the nolight voltage offset of the photodetector and reference photodetector 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 44. The plots show the effect of nonuniform cutoff in the image plane. This leads to nonuniform 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 44. Nonuniform illumination also causes the maximum intensity to vary from one position
PAGE 49
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 00.511.52Knifeedge position (mm) d=6.0 cm d=7.6 cm d=9.1 cm Figure 44: Photodiode knifeedge calibrations at three locations. to another as can be seen in Figure 44. 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 (414) 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
PAGE 50
39 linear portion of the curv es using Excel Regression Tool as shown in Figure 45. 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 00.511.52Knifeedge position (mm)norm d=6.0 cm d=7.6 cm d=9.1 cm Figure 45: Linear re gion of the calibration curves at three locations after regression analysis. It was determined that the nonuniform cuto ff is mainly due to the finite filament size of the tungstenhalogen lamp. The c ondenser lensslit 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 46 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 41. 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.
PAGE 51
40 0 0.2 0.4 0.6 0.8 1 1 2 0.10.150.20.250.3Knifeedge position (mm) d=6.0 cm d=7.6 cm d=9.1 cm Figure 46: Calibration curves af ter the ground glass was inserted. Table 41: 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 47: Slope of th e calibration curve plotted along the test section.
PAGE 52
41 The slope of the calibration curve at the twenty points taken prior to an experiment has been shown in Figure 47 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 noflow case is defined in Equation 413. For a case where there is light intensity fluctuation, this parameter is modified as ___'pdpddark refpdrefpddarkVVV VV (415) where 'V is the fluctuating term in the phot odetector 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 415 as ' ___ refpdrefpddarkV VV (416) 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 photodetector 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 knifee dge. This optical signal is conv erted to an electrical signal
PAGE 53
42 using a photodetector. The signal is then f iltered and amplified us ing a filter/amplifier and digitized at high speed by a 16bit 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 crossspectral analysis, the coherent power in the photodetector signal and the relative phase difference betw een the photodetector signal and the microphone signal are determined. The density fluctuation [12] can be represented as '(,)Re().it x x txe x (417) The spatially dependent term in the above equation is a complex quantity, which can be expressed as ()()().ixxxe xx (418) 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 (419) 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 photodetector at a ny location can be represented as
PAGE 54
43 ()'(,)Re'(),ixitVxVxee (420) and the reference microphone signal at a distance 1d in the planewave tube is given by 1() 11(,)Re().id itPdPdee (421) In Equation 419, 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 twomicrophone 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 416 to obtain the corresponding values of The slope of the calibration curve can be written as slope a (422) 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 knifeedge, 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 422. Subsequently, the angular deflection x corresponding to the magnitude of light in tensity fluctuation can be obtained using Equation 229. Finally, the density gradient is obtained by substituting x in Equation 41.
PAGE 55
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 48. 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 48: 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 (423)
PAGE 56
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 (424) It is a direct measure of the powerspectra 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 photodetector at the operational gain of the system. The pulse input duration was found to be much shorter (20sec) n than the photodetector system time constant. Hence, the input was treated as an impulse. Figure 49 shows the measured impulse response of the photodetector. Figure 49: Impulse response of the experimental photodetector.
PAGE 57
46 Figure 410: Frequency response of the experi mental photodetector at the experimental gain setting.
PAGE 58
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 (425) 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 overdamped secondorder system, it can be concluded that there is a time delay and an amplitude attenuation. The frequencyresponse characteristics obtained from the damping ratio and the undamped natural frequency are shown in Figur e 410. 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 twomicrophone method [25]. The entire experimental procedure was au tomated using a LabVIEW program. The program simultaneously acquired data from two photodetectors and two microphones, controlled a 2D 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 photodetector and the pressure fluctuation signals from two microphones placed at a distance
PAGE 59
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 photodetector signals is used to correct the temporal nonun iformity in light source as discussed in section 4.1. For the twomicrophone 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 210 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.
PAGE 60
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 23 pressure distribution for 1 R is ()(ee).ikdikd ipdp (51) From Equation 213, the density distribution can be obtained as 2()(ee).ikdikd ip d c (52) And using Equation 215, the density gradient for 1 R is derived as 2() .jkdjkd ipjkee dc (53) Figure 51 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
PAGE 61
50 2 i p c 2 id ipkc i p p Figure 51: 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 52. 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.
PAGE 62
51 Figure 52: 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 21), using the following procedure. The amplitude of the pressure signal (Appendix D) is derived from Equation 23 as 22()(cos(())cos())(sin(())sin()),rr p dAkdlRkdklkdlRkdkl (54) 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 54, 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 215 as
PAGE 63
52 22 2()(cos(())cos())(sin(())sin()).rrAk dkdlRkdklkdlRkdkl xc (55) 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 215 1(cos(())cos()) tan. (sin(())sin())r rkdlRkdkl kdlRkdkl (56) 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 (57) 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. Crossspectral 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).
PAGE 64
53 0009002700180 Figure 53: Pressure waves at various phases for R = 1. 0180 0180090000270 Figure 54: Pressure waves at various phases for R = 0.
PAGE 65
54 0270018009000 incident wave Figure 55: 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 soundhard 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 26 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 54. 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
PAGE 66
55 Simulations were also obtained for a general resistive termination (see Figure 55). 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 planewave tube: the optical deflectometer and the twomicrophone 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 setup 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 planewave tube. The noise floor of the detector for a nolig ht case was determined and expressed as power spectral density in Figure 56. The power spectra of the detector for lighton cases, with various values of the gain are pl otted in Figure 57. 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 photodetector. Hence the operating amplifier gain was set at 0.35. Figure 58 and Figure 59 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.
PAGE 67
56 Figure 56: Noise floor of the experimental photodetect or at an operational gain of 0.35. Figure 57: Noise floor of the experimental photodetector (lighton) for various gains.
PAGE 68
57 Figure 510 shows the coherence between the photodetector and the reference microphone signal. Although the coherence value is low (0.19) at the operating frequency, the coherent power of the photodetector at5 kHz is approximately three ordersofmagnitude above the noise floor (Figure 511). Figure 58: Example of photodetector pow er spectrum at 145.4 dB Figure 59: Example of referen ce microphone power spectrum at 145.4 dB
PAGE 69
58 Figure 510: Example of coherent spectrum at 145.4 dB Figure 511: Example of photode tector coherence power at 145.4dB.
PAGE 70
59 Figure 512: Magnitude of the fr equency response function at 145.4 dB Figure 513: Density gradient amplitude along the length of the tube at 145.4 dB
PAGE 71
60 Light intensity fluctuations were measured at 20equally spaced locations. The frequency response function gain has been pl otted in Figure 512. 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 513. 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 514, for example, the power of th e photodetector 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 516). The density gradient distribution at 126.38 dB is shown in Figure 518. Figure 514: Example of photode tector power spectrum at 126.4 dB
PAGE 72
61 Figure 515: Example of referen ce microphone power spectrum at 126.4 dB Figure 516: Coherent power of the photodetector at 126.4 dB
PAGE 73
62 Figure 517: Example of coherence spectrum at 126.4dB. Figure 518: Density gradient amplit ude along the length of the tube at 126.4dB.
PAGE 74
63 The phase difference between the photodetector signal and the microphone signal is obtained from the frequency response func tion (Figure 519). The phase is found to remain constant with a 0180 shift at the densitygradien 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 twomicrophone method in Section 5.3.2. Figure 519: Phase difference between the phot odetector 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.
PAGE 75
64 Figure 520: Density gradient amplitude along the length of the tube using the microphone method at 145.4dB. Figure 521: 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 twomicrophone method as discussed in Chapter 2. The magnitude and phase of the density gradient at various
PAGE 76
65 locations of the impedance tube are calcul ated using Equation 55 and Equation 56 and are plotted in Figure 520 and Figure 521, respectively. Fr om Figure 521, 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 ndixC), in Figure 522 and Figure 523 at 145.4 dB and 126.4 dB respectively. Figure 522: Magnitude of the density gr adient using the two methods at 145.4 dB
PAGE 77
66 Figure 523: 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 antinode. 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.
PAGE 78
67 Figure 524: Phase of the density grad ient using the two methods at 145.4 dB Figure 525: Phase of the density gr adient using the two methods at 145.4dB after the phase correction from the photo detector.
PAGE 79
68 Figure 526: 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 524. 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 525 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.
PAGE 80
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 onedimensional 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 twomicrophone 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.
PAGE 81
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 lensbased system. Hence, a mirrorbased 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 nonparaxial 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 photodetectors were characterized as over damped second order system after the completion of the experiments. A more sensitive photodetector with lower noise and increased bandwidth, such as a c ooled photomultiplier 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.
PAGE 82
71 APPENDIX A KNIFEEDGE 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 A1: 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 A1 that AOBCOD. Hence
PAGE 83
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)
PAGE 84
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 Ferments 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 B1: Deflection of a light ray in inhomogeneous test object. (,,)0, nxyzds (B2)
PAGE 85
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)
PAGE 86
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 twomicrophone 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.
PAGE 87
76 The equations for (Equation 416) and the slop e of the calibration curve (Equation 417) were derived in section (4.1.1). Substituting, Equation 416 in Equation 417 we obtain the expression for a as __'() (). ()(())refDCrefdarkVd ad slopedVdV (C3) Substituting Equation C3 into Equation 229, 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 41 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)
PAGE 88
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 C1: Error bar in the amplitude of the density gradient using the deflectometer at 145.4 dB
PAGE 89
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 C1 for a SPL of 145.4 dB The uncertainties in various parameters have been tabulated in C1. Table C1: 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
PAGE 90
79 C.1.1 Two Microphone Method A similar approach was also followed for the twomicrophone method. The expression for the absolute pressure in the no rmal acoustic impedance tube is derived in Equation 54. 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 215 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)
PAGE 91
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 C2: 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)
PAGE 92
81 Figure C2: Error bar in the amp litude using the microphone method 145.4 dB Figure C3: Comparison of microphone and deflectometer method at 145.4dB.
PAGE 93
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 C2. In Figure C3, the error bars obtained from the microphone method and the deflectometer method are compared. The uncertainties in various parameters are shown in Table C2. C.2 Uncertainty in Phase C.2.1 Deflectometer Method Figure C4: Error bar in the phase using the deflectometer at 145.4 dB
PAGE 94
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 twomicrophone method. Figure C6: Error bar in the amplitu de using the deflectometer at 145.4 dB
PAGE 95
84 Figure C7: 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 C6 and the comparison in Figure C7.
PAGE 96
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 D2 can be expressed as ()().ri iklikdikdpdAeeRee (D2) Using Equation 28, 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 215 as
PAGE 97
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 D5 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)
PAGE 98
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 3540661557 SpringerVerlag 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. 631656. [7] Wilson LN; Damkevala RJ (1969) Statistical Propert ies of Turbulent Density Fluctuations. J.Fluid Mech,vol. 43, part 2, pp. 291303. [8] Davis MR (1972) Quantitative schlieren measur ement in a supersonic turbulent jet. J.Fluid Mech. (1972), vol. 51, part 3, pp. 435447. [9] Davis MR (1974) Intensity, Scale and Conv ection of Turbulent Density Fluctuations. J.Fluid Mech (1975), vol. 70, part 3, pp. 463479. [10] Weinstein LM (1991) An improved largefield focusing schlieren system. AIAA Paper 910567. [11] Alvi FS; Settles GS (1993) A SharpFocusing Schlieren Optical Deflectometer. AIAA Paper 930629. [12] Weinstein LM (1993) Schlieren system and met hod for moving objects. NASA CASE NO. LAR 150531. [13] Garg S; Settles GS (1998) Measurements of a supersonic turbulent boundary layer by Focusing schlieren deflectomet ry. Experiments in Fluids 25 254264.
PAGE 99
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. 123124. [16] Michael A. Kegerise; Eric F. Spina; Louis N. Cattafesta 111 (1999) An Experimental Investigation of Flowi nduced Cavity Oscillation. AIAA993705. [17] Cattafesta LN; Kegerise ma; Jones GS (1998) Experiments on Compressible FlowInduced Cavity Oscillations. AIAA982912. [18] Blackstock DT (2000) Fundamentals of Physical Acoustics. New York: John Wiley & Sons, Inc. [19] ASTME105090, "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 twomicrophone randomexcitation 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 472475 [23] Settles GS (2001) Schlieren and Shadowgr aph Techniques. Visualizing Phenomena in Transparent Media. [24] Leonard M. Weinstein (1991) LargeField HighB rightness Focusing Schlieren System.AIAA Paper 910567 [25] Schultz T; Louis N. Cattafest a 111; Nishida T; Sheplak M (1998) Uncertainty Analysis of the TwoMicrophone 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
PAGE 100
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 Bachelors 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.



Full Text  
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 coadvisor 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 setup. 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 SignalProcessing 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 KNIFEEDGE 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 41 Slope of the calibration curve at three different locations................... ............40 Cl Uncertainties in various parameters for a deflectometer............... ...................78 C2 Uncertainties in various parameters for the microphone method............................. 80 LIST OF FIGURES Figure page 11 S im p le sch lieren setu p .......................................................................... .......... .. ..... 4 21 Plane w ave tube....................................... ................ .... .. ............ 11 22 Light source in the plane of the knifeedge......................... ................18 23 Ray diagram of the schlieren setup. .............................................. ............... 21 24 Schlieren head with the conjugate plane. ...................................... ...............25 31 D eflectom eter setup. ...................................................................... ...................28 32 N orm al im pedance tube ........................................ .............................................29 33 Setup for normal impedance tube ................................................ 30 41 Light source in the plane of the knifeedge.......................... ............... 33 42 Knifeedge calibration of photodiode sensor. ............. .................... ........ ... .... 36 43 Photodiode knifeedge calibration. ........................................ ....... ............... 37 44 Photodiode knifeedge calibrations at three locations................ .............. 38 45 Linear region of the calibration curves at three locations after regression analysis.39 46 Calibration curves after the ground glass was inserted..................... ..............40 47 Slope of the calibration curve plotted along the test section.............................. 40 48 D ata reduction procedure. ............................................................. .....................44 49 Impulse response of the experimental photodetector................... ................45 410 Frequency response of the experimental photodetector at the experimental gain setting .............................................................................. 4 6 51 Comparison of pressure, density and density gradient distributions......................50 52 Phase variation for a rigid termination. ........................................ ............... 51 53 Pressure waves at various phases for R = 1. ................................. .................53 54 Pressure waves at various phases for R = 0. ................................. .................53 55 Pressure waves at various phases for R = 0.5. ................................. ............... 54 56 Noise floor of the experimental photodetector at an operational gain of 0.35 .......56 57 Noise floor of the experimental photodetector (lighton) for various gains. ..........56 58 Example of photodetector power spectrum at 145.4dB ................ .................57 59 Example of reference microphone power spectrum at 145.4dB ...........................57 510 Example of coherent spectrum at 145.4dB ....................................... ................. 58 511 Example of photodetector coherence power at 145.4dB ......................................58 512 Magnitude of the frequency response function at 145.4dB ................ ................59 513 Density gradient amplitude along the length of the tube at 145.4dB ....................59 514 Example of photodetector power spectrum at 126.4dB ...................................60 515 Example of reference microphone power spectrum at 126.4dB ...........................61 516 Coherent power of the photodetector at 126.4dB ...........................................61 517 Example of coherence spectrum at 126.4dB ................... ......................... 62 518 Density gradient amplitude along the length of the tube at 126.4dB ....................62 519 Phase difference between the photodetector signal and the reference microphone sig n al at 14 5 .4 d B ....................................................................................6 3 520 Density gradient amplitude along the length of the tube using the microphone m ethod at 145.4dB ........................... .......... .. .. ... ............ 64 521 Density gradient phase along the length of the tube using the microphone method at 14 5 .4 dB ..............................................................................6 4 522 Magnitude of the density gradient using the two methods at 145.4dB ...................65 523 Magnitude of the density gradient using the two methods at 126.4dB ...................66 524 Phase of the density gradient using the two methods at 145.4dB .........................67 525 Phase of the density gradient using the two methods at 145.4 dB after the phase correction from the photo detector. ........................................ ....................... 67 526 Phase of the density gradient using the two methods at 126.4dB after the phase correction from the photo detector. ........................................ ....................... 68 A1 M agnification of the source on the screen. ................................... ............... 71 Bl Deflection of a light ray in inhomogeneous test object...............................73 C1 Error bar in the amplitude of the density gradient using the deflectometer at 145.4 dB .................... ........ .... ......... .. .. ...... ............ 77 C2 Error bar in the amplitude using the microphone method 145.4 dB .....................81 C3 Comparison of microphone and deflectometer method at 145.4 dB ......................81 C4 Error bar in the phase using the deflectometer at 145.4 dB ....................................82 C6 Error bar in the amplitude using the deflectometer at 145.4 dB .............................83 C7 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 flowvisualization 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. Twopoint crossspectral 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 datareduction method. In addition, the twomicrophone 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 modernday 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 nonoptical 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 refractiveindex variation in fluids. Common methods [3] include shadowgraph, schlieren and interferometric techniques. Since these techniques are nonintrusive, 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 highdensity 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 flowvisualization 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 refractiveindex 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 setup consists of optics to produce a pointlight source, two lenses, a knifeedge 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 KNIFEEDGE POINT LIGHT SOURCE TEST AREA LENS 1 LENS 2 SCREEN Figure 11: Simple schlieren setup. At this position, a knifeedge (oriented perpendicular to the desired density gradient component) cuts off a certain portion of the lightsource 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 photosensor 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 reinvented 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 crosscorrelating the signal from each beam, information on the behavior of scatterer numberdensity 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 crosscorrelation 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 crosscorrelation of the two signals and the local meansquare density fluctuation in the beam intersection point are determined. Davis [8] used a singlebeam 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 fiberoptic sensor was embedded in a schlieren image to determine the convective velocities of largescale 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 highReynolds 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 largefield focusing schlieren system. The system was used to examine complex two and threedimensional flows. Diffuse screen holography was used for threedimensional 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 threedimensional 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 lowspeed 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 twodimensional, 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 lightintensity fluctuations in a realtime schlieren image to obtain the quantitative flowfield 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 hotwire 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 higherorder 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 selfsustaining 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 flowinduced cavity oscillations, are the results of modeswitching 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 cuton 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 21, 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 21: 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 (21) where p, and p, are the incident and reflected pressure respectively. Here p, = Ae ek and p = B ekl are phasors. The timeharmonic dependence, e'", is implicit in p The complex reflection coefficient R is defined as R=p. (22) P, Substituting Equation 22 in Equation 21, the expression for p(d) becomes p(d)= p,(ekd + Re kd). (23) 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). (24) 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) (25) 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 (27) 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 (28) 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. (28) The reflection factor R can also be split into real and imaginary components as R = Rea + jRmg (29) where RmgR = R sin and Re, = R cos , The reflection coefficient R can be measured [19] [20] using the twomicrophone method. H eks R = R e = e2k(l+s) (210) e jks H where s is the centertocenter 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, (211) 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 29 in Equation 27 as 1R 2 2 Zreal real ,mg2 (212) (1 reail )2 + Rmg and 2R Z'g =( R 2 +mg (213) (1Rel) + 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 setup 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 (213) The pressure gradient in the x direction is =c2 (214) ax ax From the above equation, it can be seen how the density varies in the x direction in an impedance tube. Substituting Equation 23 in Equation 214 and taking the derivative w.r.t d, we obtain the density gradient as ap pjk(ekd Rekd) O2(215) 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 GladstoneDale equation [3] (Equation 216). n1= kp. (216) where k = 2.259 x 104 m3/kg is the GladstoneDale 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 nonvacuum media, with the velocity given by c = c n, (217) 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 refractiveindex 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 (220) x n ox e' = z (221) 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, (222) 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, (223) Hno x 1 an = n dJz. (224) no y Since the refractive index of air is approximately equal to 1, the expressions can be written as =1az. (225) On EY = dz. (226) In the case of twodimensional flow in a tunnel of width W, the expressions become = Wn (227) ax' S=Wn. (228) 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 knifeedge. As shown in Figure 22 an image of the slit source produced at the knifeedge, of which only a part of height a is allowed to pass over the knifeedge. 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,, (229) 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 knifeedge and the screen IKEAKE I ... sAscreen (230) or Ien= IKEAKE (231) screen where 'KE is the intensity of light passing the knifeedge. In the case of half knifeedge a cutoff, the image on the rear side of the knifeedge has a height b and breadth and 2 hence the area a AKE = b. (232) 2 Equation 231 can be expressed as IKab Ie IKEab (233) S 2ACee 2A When the light source image shifts by an amount Aa IKE + Aa b screen + AI = (234) screen And the intensity change on the screen can be expressed as A I Aab A screen (235) KNIFEEDGE IMAGE WITH OPTICAL DISTURBANCE Ab b UNDISTURBED IMAGE OF THE SOURCE Aa Figure 22: Light source in the plane of the knifeedge 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 233 and Equation 235 as Al C _ I screen 2Aa 2f2 a x. a a (236) r The contrast sensitivity is given by S dC 2 (237) 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 photosensor can detect absolute light intensity change. Hence the quantitative sensitivity is defined as d(A) 2f screen (238) dE a Substituting the expression for Iscren we obtain d(AI) f2IKEb S d (239) dE A Using the value of angular deflection from Equation 227 we obtain / 2 f2W cn f n (240) I a ~c screen a 9 Using the GladstoneDale Equation 216 we get AI 2kf2W Op 2k p (241) Screen a (2 Alternatively, the relative intensity change in they direction can be obtained by turning the knifeedge by 90, and the relative intensity change is Al 2kf2W (p (242) 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 knifeedge. 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 241. 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 setup will be discussed in detail in Chapter 3. Figure 23 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 nolonger 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 .. .... ....KNIFEEDGE RECTANGULAR : (1 APERTURE f' f1 f2 f, 1 1 2 2 " Figure 23: 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 knifeedge is placed. The image is further magnified and viewed on the screen placed at a distance x from the knifeedge. 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). (243) It is assumed that the power is conserved through out the optical setup for a case without the knifeedge, therefore P = IA is constant. IsiAslit = Iscreen woAscreen Iso ,dh or Isceen wo (244) screen where Iscreen wo is the intensity of the image on the screen with out knifeedge. 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 23, 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, (245) 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 = (246) 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 (247) 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 (248) f22h where x is the distance between the knifeedge and the traverse. Hence the total transverse magnification on the screen is .screen ]x+1} (249) d2 2f 'sce eni w [f i (249) Substitituting Equation 249 into Equation 245 the area of the screen can be expressed as =. [ d2 fh +l (2 2] 2 f, IKeAn dh. (250) The resulting intensity on the image plane is kn e.... wo I'ht (251) 1 screen/wo (251) ^A2 A2 2 / f i fi In practice however, a knifeedge is placed at the focal point of Lens 2. There is no power loss prior to the knifeedge location. Hence IKEAKE = source .... Source (252) where IKE is the light intensity, and AKE is the area illuminated on the front side of the knifeedge. AKE = (mKE2 dh. (253) Also, the magnification at the knifeedge can be calculated as m, = m1 m2 f, Substituting in Equation 252, the light intensity at the knifeedge can be expressed as IKE ='s.. (254) The knifeedge blocks a portion of the light as shown in Figure 22. In the case of halfcutoff, the area of the image on the rear side of the knifeedge AKE' (where the prime superscript denotes a position after the knifeedge) has already been derived in Equation 232. As in section 2.2.2, the power can be assumed to be conserved from the knifeedge to the screen and Equation 230 can be applied. Also, the constants As,,,. and IK can be obtained from Equation 250 and Equation 254 respectively. The expression for the intensity of the image on the screen (Equation 233) can be modified as Screen Isourceab (255) Sd2 f J 222 f2 X +I dh f  and the quantitative sensitivity (Equation 239) of the instrument in the x axis can be expressed as S, Iource f2 2 (256) f dr2 f2h f Since f J\ LENS 2 TEST AREA ID S x2 2 Figure 24: Schlieren head with the conjugate plane. Equation 256 can be written as Sf surce2_f2 2 ! f / (257) 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 (258) s Hence if the screen is placed at the conjugate plane the sensitivity can be expressed as S, Is..u f2 (259) ]3L2 2hJ Shs The equation can be nondimentionalized as Sq_ (260) 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 31. 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 8inch 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. LIGHT SOURCE E LENS 1 LENS 2 REEN REFERENCE PHOTODIODE PHOTODIODE TRANSIMPEDANCE AMPLIFIER HIGH PASS FILTER MULTIMETER FILTER 16BIT A/D Figure 31: Deflectometer setup. The testsection 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 flowfield. The knifeedge 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 32 is straight and is of a constant square crosssection of width 25.4 mm rigid, smooth, nonporous 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 32: Normal impedance tube the test section. The metal used for the construction of the planewave tube is aluminum It has a length of 0.724 m with a cutoff 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 testsection 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 soundinsulating 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 2827002) 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 2D TRAVERSE KNIFEEDGE LIGHT SOURC TESTSECTION PLANEWAVE TUBE Figure 33: 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 outputoffset voltage was 8.0 mV and the control voltage of the photodetector 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 twodimensional traverse (Velmex Model MB4012P40JS4). 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 knifeedge was placed on a onedimensional traverse (Newport model ESP100), which has a resolution of 1 umn. 3.3.3 SignalProcessing Equipment The output signal of the photodiode and the microphone are filtered using a computer optimized filter (Kemo Model VBF35 MultiChannel Filter/Amplifier System) with a flat pass band and linear passband phase, that operates as a high pass filter with a cutin frequency of 850 Hz. The filtered signal passes to a 16bit AD 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 knifeedge deflection. Subsequently, the crossspectral 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 227 and Equation 228. Using the GladstoneDale relationship in Equation 216, assuming twodimensional flow field, and no & 1 for the surrounding air gives E = kW (41) 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 lightintensity fluctuations. It has already been derived (Equation 230) 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 22, and the parameters are described in Chapter 2. For a knifeedge at half cutoff, DEFLECTED I IMAGE ak A A  ^ b UNDISTURBED IMAGE KNIFEEDGE Figure 41: Light source in the plane of the knifeedge. a k = (42) 2 The maximum deflection in the light ray, which can be detected by the system, occurs at a Aa = (43) This case has been shown in Figure 41. Substituting Equation 43 in Equation 229, Aa = f2, gives a E (44) 2f2, To determine the maximum density gradient that can be detected by the system, combining Equation 44 and Equation 41 gives P a (45) max 2f2kW Also, the magnification of the source at the knifeedge is mKEf2 (46) and a = mKEd, (47) where d is the width of the rectangular aperture. Using Equation 47, the density gradient can be rewritten as Op) d (48) max 2fkW Substituting the value of the focal length f = 100 mm, the width of the testsection 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 215. 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) (49) 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 knifeedge, the light intensity at the image plane is represented as Imax and, when fully blocked by the knifeedge, the intensity is 0, assuming that diffraction effects are negligible. Thus the light intensity varies from 0 Ima When the knifeedge 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 (410) a When light is refracted due to the density gradient in the test section the expression for intensity is modified as k + Aa Imax (411) a where Aa is the knifeedge deflection as described in Chapter 2 and is given by Equation 229. From the above expression, the static sensitivity with respect to the deflection of the image at the knifeedge is given by K = 0 max (412) 8Aa a Since the light intensity is linearly related to the knifeedge deflection via Equation 411, a direct calibration can be done in an undisturbed (noflow) case, by recording the lightintensity for several knifeedge positions ranging from no cutoff to full cutoff. The calibration curve which gives the voltage variation (directly proportional to the light intensity) vs knifeedge position can be used to determine the angular deflection of light rays that pass through a flow with a density gradient for a fixed knifeedge. In order to ensure that all the measurements are taken in the linear range, the schlieren system was operated at half cutoff, in which case the knifeedge blocks half the image of the source.  4 12.5  1.5T       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 Knifeedge position (mm) Figure 42: Knifeedge calibration of photodiode sensor. A typical knifeedge calibration at the center of the image plane is shown in Figure 42. It shows the output voltage of the photodetector as the knifeedge 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 photodetector that measures the source intensity directly. Figure 43 shows the calibration curve after the photodetector signal has been normalized using the reference detector signal. It can be seen that the temporal nonuniformity of the light source shown in Figure 42 has been corrected by the reference photo detector in Figure 43. 0.2 i9  I * A* 01. >   ^  ~ ^ '& 4     1.25 0.75 0.25 0.25 0.75 1.25 Knifeedge position (mm) Figure 43: Photodiode knifeedge calibration. The experiments also account for any zero offsets in the photodetectors. A dimensionless parameter D is defined which takes into consideration the two corrections mentioned above. S= d pd dark (413) Vref pd ref pd dark where Vpd is the DC voltage of the photodetector, Vef pd is the voltage of the reference photodetector and Vd _dark and Vref pd dark are the "nolight" voltage offset of the photodetector and reference photodetector, 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 44. The plots show the effect of nonuniform cutoff in the image plane. This leads to nonuniform 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 44. Nonuniform 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. d6.0c 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 Knifeedge position (mm) Figure 44: Photodiode knifeedge calibrations at three locations. to another as can be seen in Figure 44. 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 (414) 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 45. 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 Knifeedge position (mm) Figure 45: Linear region of the calibration curves at three locations after regression analysis. It was determined that the nonuniform cutoff is mainly due to the finite filament size of the tungstenhalogen lamp. The condenser lensslit combination could not produce uniform illumination over the entire area slit. This effect was mitigated by placing ground glass behind the rectangular slit. Figure 46 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 41. 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 Knifeedge position (mm) Figure 46: Calibration curves after the ground glass was inserted. Table 41: 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 47: 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 47 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 noflow case is defined in Equation 413. For a case where there is light intensity fluctuation, this parameter is modified as SVpd + V'pd dark (415) Vref pd ref _pd dark where V' is the fluctuating term in the photodetector 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 415 as V SAO=OO= (416) 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 photodetector 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 knifeedge. This optical signal is converted to an electrical signal using a photodetector. The signal is then filtered and amplified using a filter/amplifier and digitized at high speed by a 16bit AD 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 crossspectral analysis, the coherent power in the photodetector signal and the relative phase difference between the photodetector signal and the microphone signal are determined. The density fluctuation [12] can be represented as px (x,t)= Re (x)ect (417) The spatially dependent term in the above equation is a complex quantity, which can be expressed as S(x) = (x) e). (418) 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, (419) 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 photodetector at any location can be represented as V '(x, co) = Re [V '(x) el(x)e"'t ], (420) and the reference microphone signal at a distance d, in the planewave tube is given by P(d,co) = Re IP(d) e'"(le't ]. (421) In Equation 419, 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 twomicrophone 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 416 to obtain the corresponding values of AO The slope of the calibration curve can be written as AO slope = (422) 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 knifeedge, 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 422. Subsequently, the angular deflection E" corresponding to the magnitude of light intensity fluctuation can be obtained using Equation 229. Finally, the density gradient is obtained by substituting E, in Equation 41. 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 48. detector Ox kW Figure 48: 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' (423) 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,. (424) Gyy,coh Gyy. (424) It is a direct measure of the powerspectral 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 photodetector at the operational gain of the system. The pulse input duration was found to be much shorter (D 20 nsec) than the photodetector system time constant. Hence, the input was treated as an impulse. Figure 49 shows the measured impulse response of the photodetector. 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 49: Impulse response of the experimental photodetector. 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 410: Frequency response of the experimental photodetector 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 (425) 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 overdamped secondorder system, it can be concluded that there is a time delay and an amplitude attenuation. The frequencyresponse characteristics obtained from the damping ratio and the undamped natural frequency are shown in Figure 410. 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 twomicrophone method [25]. The entire experimental procedure was automated using a LabVIEW program. The program simultaneously acquired data from two photodetectors and two microphones, controlled a 2D 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 photodetector 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 photodetector signals is used to correct the temporal nonuniformity in light source as discussed in section 4.1. For the twomicrophone 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 210 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 23 pressure distribution for R = 1 is p(d) = p, (ekd e kd). (51) From Equation 213, the density distribution can be obtained as p(d) = (ekd +ekd). (52) And using Equation 215, the density gradient for R = 1 is derived as Op p jk(eJkd e kd) Od c2 Figure 51 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 51: 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 52. 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 51: 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 52. 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~~~~~ ~ ~ rr 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  :   :  . .      100Ir  TTT 0 1 2 3 4 7 8(54) 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 21), using 54, 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 215 as kd d2. The density gradient amplitude is derived from Equation 215 as 8 0 .. . . . ... .  i r 0 S_0 1 2 3 4 5 6 7 8~~~L~~~~~I~~~ I I I I kd Figr 52 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 /(oskl) r cos(r kd kl) +(inkrl))+ r sin( rkdkl)2 I I I (54) 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. (55) 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 215 S= tan I (cos(k(d )) R cos( kd kl)) (56) (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], (57) 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. Crossspectral 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 jantinod ) 9(0 2700 0  4 I I  ~ ~~ ~ ~         2  Prt 3    )800 0 2 4 6 8 10 kd Figure 53: 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 54: Pressure waves at various phases for R = 0. 54 4 4 Iiinideint w I incident wave 3     : o0 900 1800 2700 S.       12 10 8 6 4 2 0 kd Figure 55: 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 53 shows sample snapshots of the movie for R =1 case. This represents a rigid termination or soundhard 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 26 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 54. 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 onadbudr odto.Pesr Simulations were also obtained for a general resistive termination (see Figure 55). 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 planewave tube: the optical deflectometer and the twomicrophone 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 setup 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 planewave tube. The noise floor of the detector for a nolight case was determined and expressed as power spectral density in Figure 56. The power spectra of the detector for lighton cases, with various values of the gain are plotted in Figure 57. 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 photodetector. Hence the operating amplifier gain was set at 0.35. Figure 58 and Figure 59 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 56: Noise floor of the experimental photodetector at an operational gain of 0.35. 1010 0 0.5 1 1.5 2 2.5 3 3.5 4 Frequency (Hz) x 104 Figure 57: Noise floor of the experimental photodetector (lighton) for various gains. Figure 510 shows the coherence between the photodetector and the reference microphone signal. Although the coherence value is low (0.19) at the operating frequency, the coherent power of the photodetector at 5 kHz is approximately three ordersofmagnitude above the noise floor (Figure 511). 107 108 m 10'9 101 o "_ 10 1 10 10 1014 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure 58: Example of photodetector 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 59: Example of reference microphone power spectrum at 145.4dB. 0 50 100 150 200 250 300 350 400 450 Frequency (Hz) Figure 510: Example of coherent spectrum at 145.4dB. 0 1000 2000 3000 4000 5000 6000 7000 Frequency (Hz) 8000 9000 10000 Figure 511: Example of photodetector coherence power at 145.4dB. x 106 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 512: 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 513: Density gradient amplitude along the length of the tube at 145.4dB.    I  I   I            I  * * I I   Il C ,    Light intensity fluctuations were measured at 20 equally spaced locations. The frequency response function gain has been plotted in Figure 512. 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 513. 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 514, for example, the power of the photodetector 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 516). The density gradient distribution at 126.38 dB is shown in Figure 518. 10 10 10 a 109 10 10 1011 10'12 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure 514: Example of photodetector power spectrum at 126.4dB. 1000 2000 3000 4000 5000 6000 7000 Frequency(Hz) 8000 9000 10000 Figure 515: Example of reference microphone power spectrum at 126.4dB. 10s6 10 107 10l 1011 10 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure 516: Coherent power of the photodetector at 126.4dB. 10o E !5 S105 10 Co 1010 101 L 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure 517: 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 518: 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 photodetector signal and the microphone signal is obtained from the frequency response function (Figure 519). The phase is found to remain constant with a 1800 shift at the densitygradient 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 twomicrophone 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 519: Phase difference between the photodetector 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 520: 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 521: Density gradient phase along the length of the tube using the microphone method at 145.4dB. The reflection coefficient is obtained from the twomicrophone 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 55 and Equation 56 and are plotted in Figure 520 and Figure 521, respectively. From Figure 521, 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 AppendixC), in Figure 522 and Figure 523 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 522: 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 523: 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 524: Phase of the density gradient using the two methods at 145.4dB . 50 0o 150 L 2 8 10 12 Figure 525: 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 526: 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 524. 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 525 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 onedimensional 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 123157 dB. The amplitude and phase distributions of the density gradient were measured using the deflectometer. A second method based on the twomicrophone 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 lensbased system. Hence, a mirrorbased 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 nonparaxial 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 photodetectors were characterized as over damped second order system after the completion of the experiments. A more sensitive photodetector with lower noise and increased bandwidth, such as a cooled photomultiplier 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 KNIFEEDGE GEOMETRY The magnification between the focus of Lens 2 and the screen can be derived as follows. d 2 k 2 A4 Figure Ai: 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 Ai that ZAOB = ZCOD. Hence y= x, Magnification y+k m3k 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 Bl: 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 twomicrophone 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 416) and the slope of the calibration curve (Equation 417) were derived in section (4.1.1). Substituting, Equation 416 in Equation 417 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 229, 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 41 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 C1: 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 ly2 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 Cl for a SPL of 145.4 dB. The uncertainties in various parameters have been tabulated in C1. Table Cl: 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 twomicrophone method. The expression for the absolute pressure in the normal acoustic impedance tube is derived in Equation 54. 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 215 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' ikezki (ezkd Re ekd) A = (C16) 8A c2 ap Aike1k 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, kdlkl))2 (C21) A0 = 0; cA _A3 R sin(, 2kdl) Odl 2G,, Table C2: 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 C2: Error bar in the amplitude using the microphone method 145.4 dB . U.Jb II 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 031method: 0.25  02 < 0.15  c3 2 0.05 0.05 0 2 4 6 8 10 12 kd Figure C3: 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 C2. In Figure C3, the error bars obtained from the microphone method and the deflectometer method are compared. The uncertainties in various parameters are shown in Table C2. 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 C4: 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 twomicrophone method. 0 2 4 8 10 12 Figure C6: Error bar in the amplitude using the deflectometer at 145.4 dB. 100 150 2     :r           r         I _ L : r __ _ ,rr .  J   ,  100  50  100  150 L 2 2 4 8 10 12 Figure C7: 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 C6 and the comparison in Figure C7. 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 D2 can be expressed as p(d) = Aeki(ekd + R e e"kd). (D2) Using Equation 28, 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 215 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 D5 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 3540661557 SpringerVerlag 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. 631656. [7] Wilson LN; Damkevala RJ (1969) Statistical Properties of Turbulent Density Fluctuations. J.Fluid Mech,vol. 43, part 2, pp. 291303. [8] Davis MR (1972) Quantitative schlieren measurement in a supersonic turbulent jet. J.Fluid Mech. (1972), vol. 51, part 3, pp. 435447. [9] Davis MR (1974) Intensity, Scale and Convection of Turbulent Density Fluctuations. J.Fluid Mech (1975), vol. 70, part 3, pp. 463479. [10] Weinstein LM (1991) An improved largefield focusing schlieren system. AIAA Paper 910567. [11] Alvi FS; Settles GS (1993) A SharpFocusing Schlieren Optical Deflectometer. AIAA Paper 930629. [12] Weinstein LM (1993) Schlieren system and method for moving objects. NASA CASE NO. LAR 150531. [13] Garg S; Settles GS (1998) Measurements of a supersonic turbulent boundary layer by Focusing schlieren deflectometry. Experiments in Fluids 25 254264. [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. 123124. [16] Michael A. Kegerise; Eric F. Spina; Louis N. Cattafesta 111 (1999) An Experimental Investigation of Flowinduced Cavity Oscillation. AIAA993705. [17] Cattafesta LN; Kegerise ma; Jones GS (1998) Experiments on Compressible FlowInduced Cavity Oscillations. AIAA982912. [18] Blackstock DT (2000) Fundamentals of Physical Acoustics. New York: John Wiley & Sons, Inc. [19] ASTME105090, "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 twomicrophone randomexcitation 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 472475 [23] Settles GS (2001) Schlieren and Shadowgraph Techniques. Visualizing Phenomena in Transparent Media. [24] Leonard M. Weinstein (1991) LargeField HighBrightness Focusing Schlieren System.AIAA Paper 910567 [25] Schultz T; Louis N. Cattafesta 111; Nishida T; Sheplak M (1998) Uncertainty Analysis of the TwoMicrophone 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. 