Experiments in Fluids 25 (1998) 89—95 ( Springer-Verlag 1998
Application of rainbow schlieren deflectometry for concentration measurements in an axisymmetric helium jet K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, DeVon Griffin
89 Abstract The rainbow schlieren deflectometry technique was used to measure oxygen concentrations in a laminar, isothermal helium jet discharged vertically into ambient air. The concentration distributions were inferred from the color schlieren image by taking into consideration the sampling interval and noise in measurements, especially near the jet center. Excellent quantitative agreement was reached between measurements from schlieren and a continuous sampling probe. This work demonstrates the capability of the schlieren technique for providing accurate, spatially-resolved, nonintrusive, full-field of view measurements of species concentration in an isothermal binary system. Because the basic quantity measured is the refractive index, the present schlieren technique can be extended for quantitative measurements of other scalar flow properties related to the refractive index.
1 Introduction Measurements of concentration in non-reacting and reacting flows have been conducted using several different techniques. The conventional sampling probe (Fristrom, 1995) is one of the simplest and oldest techniques; it has limited spatial resolution and can not be used to measure instantaneous concentrations. The aspirating probe (Brown and Rebello, 1972) is used to measure concentration at high temporal resolution. This probe has a hot-film at the throat of a sonic nozzle; thus the measurements are independent of the velocity. Hot-wire probes developed by Way and Libby (1970) measure concentration and velocity component(s) simultaneously; these probes are not convenient to use because of the elaborate calibration process involved (So et al., 1990). All of the probes
Received: 21 April 1997 /Accepted: 14 November 1997 K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli School of Aerospace and Mechanical Engineering University of Oklahoma, Norman, OK 73019, USA DeVon Griffin NASA Lewis Research Center Cleveland, OH 44135, USA Correspondence to: A. K. Agrawal This work was supported in part by the NASA Microgravity Science and Application Division grant NAG 3-1594. We wish to thank Paul Greenberg at NASA Lewis for comments on this manuscript.
mentioned above are intrusive and as such introduce flow disturbances that are difficult to quantify. Alternatively, optical diagnostic techniques are nonintrusive; they are also capable of measuring data at high spatial and temporal resolutions. Laser light scattering techniques such as Rayleigh and Raman scattering are widely used for local measurements of species concentrations. Several applications of these techniques to measure concentration in non-reacting flows were reviewed by Gouldin et al. (1986). More recently, a detailed study of variable density jets utilizing Rayleigh scattering was reported by Pitts (1991). Long (1993) has reviewed light scattering techniques for combusting flows. Several examples and references applying these techniques for point and planar measurements of species concentration are given by Long (1993). Another class of optical techniques include shadowgraphy, schlieren imaging and interferometry (Goldstein and Kuehn, 1996) that provide a measure of the refractive index of the medium, from which scalar properties of the flow can be determined. The schlieren technique has been used extensively for qualitative flow visualization. Its application to quantitative measurements entails practical difficulties including the inhomogeneous absorption of light by the medium, shadowgraphy, and nonlinearities in the recording medium (Greenberg et al., 1995). These difficulties are overcome by replacing the knife-edge filter in the conventional schlieren apparatus by a continuously graded spectral or rainbow filter (see Fig. 1). Accordingly, the grey scale schlieren image changes to a color image suitable for quantitative measurements. A review of color-coding schlieren techniques for heat and fluid flow applications is given by Settles (1985). The color schlieren technique for quantitative measurements has been limited by the number of colors and discontinuities at the color boundaries on the spectral filter. Fewer colors lead to poor sensitivity of the instrument and color discontinuities degrade the spatial resolution in the schlieren image. A filter with continuous color spectrum was introduced by Howes (1984) who fabricated it by projecting the spectral output of a white arc lamp onto a color slide film and then, by photographically reducing the printed area on the slide to the desired filter size. A clear central region on the filter was sized to match the undeflected rays. Recently, Greenberg et al. (1995) developed quantitative Rainbow Schlieren Deflectometry (RSD) technique utilizing computer-based imaging approach. Following are some of the distinctive features of this approach: (1) the color on the filter is represented by a single parameter, hue, according to the hue-saturation-intensity (HSI) color model. This means that
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a unique hue will be transmitted to the image plane from a given location on the filter plane. Thus, the displacement of a light ray on the filter plane could be related to the hue measured in the image plane. This color representation eliminates problems associated with inhomogeneous absorption of light by the medium and nonlinearities in recording the image, (2) the filter with the desired hue distribution is created in computer software and then printed digitally on a high resolution 35 mm color film recorder, (3) the filter characteristics are evaluated by traversing it at the filter plane in the schlieren apparatus without the medium. Thus, the relationship between the hue transmitted and transverse location on the filter plane (or the filter transmissivity function) is established for a particular setup, (4) an iterative procedure is used to optimize the filter such that the filter transmissivity curve is linear, thereby, providing uniform sensitivity (change in hue for a given ray displacement) in the operating range. Further details of the quantitative RSD are given by Greenberg et al. (1995). In this work, we apply the RSD technique to measure oxygen concentration in an isothermal, axisymmetric laminar jet of helium discharged vertically in air. We demonstrate, by comparing with measurements from a sampling probe, that accurate concentration distributions are obtained by the RSD technique. In the following sections we provide the experimental and analytical details of the procedure, present comparisons between RSD and sampling probe measurements, and discuss factors affecting the accuracy of the schlieren measurements.
2 Experimental A schematic of the schlieren apparatus is shown in Fig. 1. It consists of rail-mounted optical components including a 50 micron wide, 2 mm high laser-machined source aperture, two 63 mm diameter, 490 mm focal length achromatic lenses, a computer generated 35 mm wide slide with color gradations in a 1.3 mm wide strip, and a 3-chip CCD video camera with 50 mm focal length lens. A continuous, 150 W halogen light source connected to a 200 micron diameter fiber optic cable provides the light input at the source aperture. The camera output in the RGB format is digitized by a personal computer with 24 bit color frame grabber. A light ray from the collimating lens is deflected while passing through the test
medium with refractive index gradients. The second lens decollimates the deflected ray to form a displaced image of the source at the filter plane. The camera lens is then used to image the test section onto the CCD array. The rainbow filter shown in Fig. 2a transmits a color uniquely represented by hue, according to the hue-saturation-intensity (HSI) model, from a given position on the filter plane. Thus, a color image of the test section revealing the ray deflections caused by the refractive index gradients in the medium is formed on the CCD array. The schlieren image is digitized to obtain the hue distributions which, in turn, provide the desired measurements as discussed in the next section. The resolution and sensitivity of the rainbow schlieren apparatus depend primarily on the size of the source aperture, the ray displacement at the filter plane, and resolution of color gradients on the filter. Small source size, large ray displacements and finer color gradations on the filter are desired for superior performance. However, the light throughput decreases with decreasing slit size, thereby, placing a lower limit on the slit size. The ray displacement at the filter plane is determined by the path-integrated value of the refractive index gradient in the test media. The rainbow filter in Fig. 2a, sensitive only to the transverse refractive index gradients, was printed on a 35 mm slide film recorder with a high resolution of 115 pixels/mm. Thus, about 150 independent hues were used in making the 1.3 mm wide active region of the filter, and the source image size of 50 microns nominally covered 5.5 hues (or pixels) on the filter plane. The hue transmitted at a filter location is integral of the filter transmissivity over the width of the source image. The filter transmissivity function was evaluated by traversing the filter in steps of 0.01 mm at the filter plane of the schlieren apparatus without the test media. At each step, the hue transmitted by the filter was computed from the background image taken by the camera. Figure 2b shows the mean value and standard deviation of the transmitted hue as a function of the filter position. The filter in Fig. 2a was optimized using the iterative procedure described by Greenberg et al. (1995), and therefore, its transmissivity function in Fig. 2b showed the hue varying linearly in most of the filter width. The standard deviation of the hue varied from 0.5 to 2.0% of the full range of 360 degrees. The maximum value of the standard deviation directly affecting the measurement accuracy is an order of magnitude higher than the corresponding value of 0.2% reported by Greenberg et al. The
Fig. 1. Schematic of the schlieren apparatus
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Fig. 2. a Rainbow schlieren filter; b filter calibration curve
lower resolution of the present system is attributed, in part, to the chromatic errors caused by the refractive optics (as opposed to the reflective optics) and to a smaller filter width providing only 150 independent hues compared to about 400 hues utilized by Greenberg et al. (1995). The flow system consisted of a pressurized helium cylinder, high pressure connection hoses, an MKS Model 258C mass flow meter, a needle valve and a 195 mm long vertically oriented jet tube with 7.2 mm ID and 10.4 mm OD. The experiment was conducted at room conditions for a Reynolds number of 150 based on the tube ID. The jet was laminar and did not display the self-exited behavior observed by Richards et al. (1996). The probe measurement setup consisted of a 3 mm ID 0.5 mm thick quartz tube bent 90 degrees and reduced gradually to 0.76 mm OD and 0.4 mm ID, a 50 W diaphragm vacuum pump, on line dehydrators, a calibrated Gas Tech Model GTI-02-01306 diffusion based oxygen analyzer, and a 3D traversing system with stepper motors. The analyzer had a measurement accuracy of 1% of the reading. A computerized data acquisition and control system with icon-based software was used to digitize the analyzer output and to automate the probe movement. Probe measurements were taken at two axial planes; z\5 mm and 20 mm from the jet exit. The measurements were based on continuous sampling while moving the probe across the jet in increments of 0.5 mm and allowing enough time to reach steady state at each point. The average concentration at each point was computed from 100 samples taken at 10 Hz.
3 Analytical
Fig. 4. Rainbow schlieren image of the helium jet
where d\n[1 is the refractive index difference and n is the refractive index of the test medium normalized by that of the surrounding air. The angular ray deflection is transformed by the decollimating lens to transverse displacement at the filter plane given by
The angular deflection of a light ray by an axisymmetric refractive index field depicted in Fig. 3 is given for small deflections by the following relationship (Rubinstein and Greenberg, 1994):
dr = dd e(y)\2y : dr J(r2[y2) y
Fig. 3. Representation of an axisymmetric refractive index field
d(y)\f tane(y)+f e(y) c c (1)
(2)
where f is the focal length of the decollimating lens. The ray c displacement can be found, as depicted in Fig. 2a, from the hue
H(y) transmitted at the displaced transverse ray location y on the filter plane and the background hue Ho or the hue transmitted at the undisplaced ray location. H(y) and Ho are obtained from the digitized color schlieren image. Once the angular deflections are calculated from Eq. (2), the refractive index field is found by inverting Eq. (1) using the Abel transformation (Rubinstein and Greenberg, 1994):
dy 1= d(r)\[ : e(y) n J(y2[r2) r 92
(3)
Following Vasil’ev (1971), the integral in Eq. (3) is split into a sum of integrals, factoring out the deflection angle. Thus,
rj`1 dy 1 N d(r )\[ + [e ]e ] : j j`1 i 2n j/i rj J(y2[r2i )
(4)
where r \iDr is the radial distance from the centerline, Dr is i the sampling interval, and N is the total number of intervals in the test media. Note that e(y) was approximated by linear interpolation although other schemes such as three-point Abel inversion (Dasch, 1993) have been used in the literature. The integral in Eq. (4) can be performed analytically. After some algebra the result can be expressed as follows:
N d(r )\ + D e i ij j j/i
(5)
where
D \J ij ij
if j\i (6)
\J ]J ij i,j`1
j[i
with
C
j]1][( j]1)2[i2]1/2 1 J \[ ln ij j]( j2[i2)1/2 2n
D
(7)
by volume, were the only gases in the test medium. Equation (9) was used to create a table between the refractive index difference and oxygen mole fraction varying from 0.0 to 0.21. Dale-Gladstone constants were taken from Yates (1993). This table was used to compute the oxygen mole fraction from the refractive index difference found from Equation (5).
4 Results and discussion Figure 4 shows a color schlieren image of the helium jet. Several features including the upstream diffusion and radial spreading of the jet are readily observed. A uniform color or background hue is observed radially far away from the jet centerline. This same hue is also seen at the jet centerline where the transverse component of the density gradient and hence, the transverse light ray displacement is zero. The field of view in Fig. 4 is 63 mm providing a spatial (or pixel) resolution of 94 lm for the image size of 640]480 pixels. The schlieren images were used to obtain hues at an axial location of 5 mm from the jet exit. Figure 5 shows the hue distributions obtained from 3 different schlieren images taken 1 min apart. These distributions are shown for radial locations of up to 15 mm which is beyond the schlieren boundary. Figure 5 shows excellent repeatability and only minor hue variations attributed to the flow fluctuations and/or system response characteristics. With this result one could equally proceed with the analysis using any one of the images. The hue in Fig. 5 varied from 150 to 290 degrees representing a range of 140 degrees or about 40% of the full scale. Note that an optical system utilizing the full scale of the rainbow filter may not be constructed without the prior knowledge of the maximum ray deflection in the test medium. The hue distribution was used to compute angular deflections from the filter calibration curve in Fig. 2b and Eq. (2). The estimated standard deviation of the angular deflection varied from 1 to 3% of the local value. The background hue (H ) taken from Fig. 5 was 214 degrees. The resulting angular o deflection data were Fourier transformed to examine the
Equation (5) has the form similar to that used by Dasch (1993) to invert path-integrated data from interferometric or absorption measurements. In this form the coefficients D are ij independent of the sampling interval. For a mixture of gases the refractive index is given by (Yates, 1993)
n\1]d\1]+ i o i i i
(8)
where the i summation is over all species, i is the Dalei Gladstone constant of a species, and o is the mass density of i the species. Equation (8) can be rearranged to the following form for an ideal gas at constant temperature
P n\1] + i X M i i i RM T i
(9)
Here P is the pressure, RM is the universal gas constant, T is the temperature, M is the molecular weight of the species, and i X is the species mole fraction. In the present work helium and i air, assumed to be a mixture of 21% oxygen and 79% nitrogen
Fig. 5. Hue distributions on the axial plane z\5 mm
frequency spectrum as suggested by Hughey and Santavicca (1982). Figure 6 shows the variation of the power spectral density with frequency computed using the fast Fourier transform algorithm given by Press et al. (1993). Figure 6 indicates that most of the spectral power existed in frequencies up to 0.2 mm~1 (here the frequency has units of mm~1 because the original distribution is in mm) and virtually no signal was present at frequencies higher than 0.3 mm~1. Figure 6 demonstrates that the angular deflection data were bandlimited at a frequency much lower than the Nyquist frequency of 5.3 mm~1 corresponding to the sampling interval of 94 lm in the experiment. Frequencies higher than 0.25 mm~1 were considered noise and were set to zero. An inverse Fourier transform was then taken to yield the filtered data. The unfiltered and filtered angular deflections plotted in Fig. 7 show that filtering removed the high frequency noise although the signal and noise can not be separated completely. According to the sampling theorem the entire information content of a signal can be recorded by sampling it at a rate equal to the inverse of twice the maximum frequency (Press et al. 1992). Thus, the truncated maximum frequency of 0.25 mm~1 required a sampling interval of at least 2 mm or 21 pixels. Clearly, the spatial resolution in the experiment was much higher than that required for adequate sampling. The deflection data thus obtained allow computations of the refractive index difference from the inversion formula given by Eq. (5). Vasil’ev (1971) pointed out that the accuracy of inversion improves with an increase in the number of sampled data if the measurements are noiseless. This is easily explained because the assumption to separate angular deflections from the integral in Eq. (3) is more accurate for smaller sampling intervals. However, the noise in the measurements has an opposing effect as observed by Hughey and Santavicca (1982) and discussed by Dasch (1993). The inversion error because of the measurement noise can be estimated by examining the coefficients D in Eq. (5) which transform the angular deflecij tions to the refractive index. As discussed by Dasch (1993), one
Fig. 7. Distributions of angular deflection on the axial plane z\5 mm
Fig. 6. Variation of the power spectral density with frequency
Fig. 8. Distributions of the refractive index difference on the axial plane z\5 mm
can determine from Eqs. (6) and (7) that D ’s are highest near ij j\i and that their magnitudes increase towards the center i.e., as i]1. This implies that the inversion error at a radial location is caused mainly by the measurement noise near that location. Furthermore, the inversion error due to the measurement noise is highest at the center. The latter conclusion is reflected in Fig. 8 which shows the distribution of the refractive index difference (labeled as Filtered/1) obtained using the filtered deflection data in Fig. 7. The inversion errors have caused a physically unrealistic increase in the refractive index in the middle ^2 mm region; the helium jet is expected to have the minimum refractive index in the center. The signal-to-noise ratio is poor in this region and filtering the deflection data did not completely eliminate the noise. Note that the measured quantity in the schlieren apparatus is the ray displacement at the filter plane which depends on the density gradient in the medium. Small gradients in the central region of the jet cause
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Fig. 9. Distributions of the oxygen mole fraction on the axial plane z\5 mm
Fig. 10. Distributions of the oxygen mole fraction on the axial plane z\20 mm
small ray displacements resulting in poor signal dominated by system noise. As mentioned previously, the inversion error can be reduced by increasing the sampling (or integration) interval. The refractive index difference distribution labeled Filtered/20 in Fig. 8 was obtained when the integration in Eq. (5) was performed using the filtered deflection data at every 20th pixel location by skipping the intermediate measurements. The integration interval of 20 pixels was within the limit of 21 pixels imposed by the sampling theorem. Figure 8 shows that a higher integration interval eliminated inversion errors in the center region. The distribution of the refractive index difference computed using the unfiltered deflection data at every 20th pixel location (labeled Unfiltered/20) is also shown in Fig. 8. This plot suggests that filtering removed some of the noise from the center region. Away from the center the filtered and unfiltered refractive index curves were coincident. Next, we compare in Fig. 9 the oxygen mole fraction profiles resulting from the refractive index distribution and sampling probe measurements. The schlieren reconstruction with 20 pixel integration interval (labeled Schlieren/20) agrees well with the probe measurements. As discussed before, the measurement noise has caused the high resolution schlieren reconstructions (labeled Schlieren/1) to deviate from the probe measurements in the middle ^2 mm region. Regardless, the accuracy of the present schlieren setup is substantial considering the excellent agreement reached between the schlieren and probe measurements. Finally, we compare in Fig. 10 the schlieren and probe measurements of oxygen mole fraction at an axial location of 20 mm from the jet exit. Again, a good agreement is reached between the two measurement techniques.
the sampling interval and noise in the data. Schlieren results were affected most by the measurement noise in the center region where low density gradients causing small angular deflections produced poor signal to noise ratios. Filtering by Fourier transform was beneficial but not completely effective in removing the measurement noise. Accurate schlieren reconstructions were obtained when the sampling interval was increased without introducing undersampling or signal aliasing. An excellent agreement between the schlieren and probe measurements was achieved even though only 40% of the full range of the filter was used in the experiment. Further improvement in the measurement accuracy and sensitivity (change in hue for a given ray displacement) could be achieved by properly matching the optical components including the light source, the source aperture, the collimating, decollimating and camera lenses, and the filter width. This work demonstrates that the rainbow schlieren deflectometry technique is capable of providing accurate nonintrusive measurements in the full field-of-view. Although the present application was to measure species concentration in an isothermal gas jet, the technique can be used to measure other scalar flow properties related to the refractive index. This technique can also be extended for measurements in time dependent flows. In axisymmetric and two-dimensional flows, the temporal resolution is determined by the acquisition rate of the imaging system. Commonly available imaging systems can provide full field data at 60 Hz. However, the temporal resolution in asymmetric, time dependent flows is constrained by the time required to acquire schlieren images at multiple viewing angles.
5 Concluding remarks The rainbow schlieren deflectometry technique was used to measure oxygen concentrations in an axisymmetric helium jet. Reconstructing the concentration field from the path-integrated angular deflection data required proper considerations of
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