The work is aimed at solving the scientific and practical problem of non-disturbing diagnostics of the phase and temperature fields of reacting jets and flames. On the example of an axisymmetric hydrogen-diffusion flame and a hot air flow from a candle flame, a method was developed that is adequate to the problem being solved, based on Hilbert polychromatic visualization of phase optical density fields, measuring the temperature profile in selected areas of the medium under study, pixel-by-pixel processing of RAW images recorded by a photographic matrix in RGB channels. The visualized Hilbert structures carry information about the phase optical density perturbations induced by the temperature field. The phase structure of the probing light field in the axial symmetry approximation of the flame under investigation is analyzed using the Abel transform. Iterative selection of radial temperature profiles, adapted Bezier curves, is performed with the subsequent calculation of the spatial structure of the refractive index and phase function. The reconstruction of the temperature field by the example of the study of a hydrogen-air flame is carried out taking into account the diversity of the partial optical properties of the gas mixture in a model consistent with the Gladstone-Dale dispersion formula. The influence of disturbances in the air surrounding the flame on its axial symmetry is discussed. The criterion for the reliability of the research results is a comparison of the hilbertograms obtained in the experiment and the hilbertograms reconstructed from phase structures induced by temperature fields. Optical diagnostics of a flame, hydrogen-air diffusion flame, Hilbert optics, hilbertogram GraphiCon 2021: 31sh International Conference on Computer Graphics and Vision, September 27-30, 2021, Nizhny Novgorod, Russia ORCID: 0000-0003-2404-526X (V.A. Arbuzov); 0000-0001-9488-8650 (E.V. Arbuzov); 0000-0001-7874-039X (Yu.N. Dubnishchev); 00000003-3486-4459 (O.S. Zolotukhina); 0000-0001-8178-7607 (V.V. Lukashov)
To date, the focus of scientific research is the problems associated with the stabilization of
combustion, the features of heat release in the reacting stream and the formation of harmful emissions
in combustion products [
2021 Copyright for this paper by its authors.
The article [
The aim of this work is to develop and adapt methods of polychromatic Hilbert optics for diagnosing the optical density fields of phases and flame temperature with the reconstruction of the spatial phase and temperature structure.
The diagnostic complex is based on the IAB-451 shadow device [
The lighting module consists of a light source – an RGB LED with operating wavelengths 1 = 636 nm, 2 = 537 nm and 3 = 466 nm, a collimator lens and a slit diaphragm located in the front Fourier plane of the lens, which forms a luminous flux. The Fourier spectrum of phase perturbations induced in the probing light field by a flame is localized in the frequency plane of the Fourier lens, where the Hilbert quadrant filter is located. Depending on the spectral characteristics of the light source, Hilbert filtering converts the light field into an optical analytical signal [11] recorded by a digital video camera connected to a computer [12].
In figure 1 shows the Hilbert images of a hydrogen-air flame obtained by simultaneous shooting at three different wavelengths of the radiation source – 1 = 636 nm, 2 = 537 nm and 3 = 466 nm. The probe radiation was detected by the matrix of digital camera “Canon EOS 650D” with dimensions of 22,3 × 14,9 mm and a number of active pixels are 18 Mpx. The frames were taken at a shutter speed of 1/60 s and with a resolution of 5184 × 3456 px. The fuel mixture 2/ 2 was fed vertically into stationary air through a pipe with an inner diameter of = 5 mm and a length of 500 mm. The tube material is stainless steel. The experiments were carried out in the range of Reynolds numbers = 1000 ... 2000.
At these parameters, a laminar Poiseuille flow was established in the tube. The composition and consumption of the fuel mixture of hydrogen with nitrogen was set using a software generator of calibration gas mixtures UFPGS-2. at three different wavelengths of the radiation source – 1 = 636 nm, 2 = 537 nm and 3 = 466 nm
The phase structure of the probing light field in the study of an axisymmetric flame is determined through the Abel integral as where 2 = 2 + 2;
= 2 / is the wavenumber; is the radius of the burner section; ( , ) is the refractive index as a function of the distance r from the center of the section selected in the = const plane; 0 is the refractive index of the medium undisturbed by the flame (figure 2). The -axis is set by ∆ψ( , ) = 2 ∫ [ ( , ) − 0] √ 2− 2 the direction of the probe light beam. Coordinates 1, 2 limit the size of the flame section in the direction of the probe beam, and indicates the position of the section along the flame axis.
Light beam
When studying the flow induced by the flame of a candle, the axisymmetric distributions of the refractive index ( , ) obtained from the Abel equations determine the temperature fields ( , ) in the selected flame section using the Gladstone-Dale equation [13]:
( , ) = [ ( 0, −)1−1] 0, (2) where 0 is the temperature and 0 is the refractive index of the undisturbed medium surrounding the flame.
In figure 3 shows RGB Hilbert images of heated airflow from a candle flame, chromatically selected on RGB channels.
a b c d Figure 3: (a) – experimental RGB-hilbertograms of the candle flame; (b) – R-channel, 1 = 636 nm; (c) – G-channel, 2 = 537 nm; (d) – B-channel, 3= 466 nm
Reconstruction of the radial distribution of the air flow temperature in the section at a distance of = 26 mm from the end of the candle wick for the R-channel is shown in figure 4. The red line represents the phase function ∆ψ( , ) obtained from the Abel equation (1), the green and black lines represent the interferogram and the hilbertogram, respectively, reconstructed from the phase function ∆ψ( , ). The blue line is the experimental hilbertogram.
The method used consists in the iterative selection of the radial temperature profile represented by the Bezier curve (a special case of B-splines), followed by the calculation of the refractive index ( , ) and the phase function ∆ψ( , ). Further, the hilbertogram (interferogram) is calculated from the phase function. The iterative algorithm is repeated until the points of the local minima of the Hilbert bands on the experimental and reconstructed hilbertograms coincide. The adequacy of the phase functions obtained from the solution of the Abel equation and the real one is confirmed by the coincidence of the reconstructed temperature fields in the selected areas.
In figure 5.a shows the reconstructed radial temperature distributions of the air flow from the candle flame in the section = 26 mm for three RGB channels. There are good matches. The distributions of the refractive index ( , ) in the section = 26 mm for each of the three RGB channels are shown in figure. 5.b. In figure 5.c shows the corresponding distributions of the phase function ∆ψ( , ) in the section = 26 mm, obtained from the solution of the Abel equation (1).
Temperature 5 10
15 Phase function 20 25
X[mm] 40 6 0 0 30
X[mm]
Verification of the results obtained is confirmed by solving the direct problem: from the reconstructed temperature field (figure 6), the Hilbert images are reconstructed. They are compared with the images obtained in the experiment (figure 7). The experimentally obtained and reconstructed structures are of a similar nature. This confirms the reliability of the results. Some discrepancy is associated with the influence of dynamic perturbations of the air surrounding the flame on its axial symmetry in a real experiment.
According to the dispersion formula of Gladstone-Dale [14, 15]
( ) = (1 + 2 + 4 ), (3) the refractive index of the -th component of the burning mixture depends on the wavelength of the radiation source and parameters , , and , which are determined empirically by measuring ( ) for three different values . When calculating by formula (3), we can restrict ourselves to a term with a quadratic dependence on the wavelength.
The refractive index of the entire mixture (hydrogen-air flame) is defined as
− 1 = . . . . ∑ (1 + 2) ∙ , (4) where is the pressure; . . – atmospheric pressure under normal conditions (101 325 Pa); – temperature; . . – temperature under normal conditions (0°C ~ 273 K).
The method for reconstructing the temperature distribution from the measured values of the refractive index is based on equation (4), which relates n to the temperature and composition of the gas mixture at a point. When hydrocarbons are burned in air, the refractive indices of the reacting components differ slightly from each other. A hydrogen flame, in contrast to a hydrocarbon flame, is characterized by a wide variety of partial optical properties of the gas mixture. In this case, an assessment of the content of the main components 2, 2, 2 and 2 is required. The sufficiency of such a partial structure for determining the refractive index of a mixture was verified using experimental data [16]. It should be noted that when working with a hydrogen-oxygen flame, taking into account ionization can affect the accuracy of determining the temperature [17].
An image of the structure of a hydrogen-air flame visualized by Hilbert, obtained synchronously at three different wavelengths ( 1 = 636 nm, 2 = 537 nm, and 3= 466 nm), is shown in figure 8.a, the division of the frame into RGB channels is shown in figure 8.b-8.d.
a b c d Figure 8: (a) – RGB-hilbertogram of a hydrogen-air flame ( 2/ 2, hydrogen volume fraction 25%); (b) – R-channel, 1 = 636 nm; (c) – G-channel, 2 = 537 nm; (d) – B-channel, 3 = 466 nm; (white line – 6 mm cross-section from the end of the burner tube)
For solve the inverse problem – to restore the flame temperature and molar fractions of fuel combustion products, it is necessary to restore the value of the phase function ∆ψ from the experimental hilbertogram and, having solved the Abel equation, determine the refractive index of the medium. Thus, by carrying out measurements at different wavelengths and using formulas (3)-(4), it is possible to restore the values of molar concentrations of fuel combustion products and, accordingly, the temperature distribution in the flame.
The area of combustion of a hydrogen-air flame can be conditionally divided into two zones, consisting of 3 main chemical elements of combustion products. Inner zone – 2, 2 and 2; outer zone 2, 2 and 2. The zone boundary is the area with the maximum temperature (flame front), since in its vicinity both the fuel and the oxidizer tend to zero. To determine and in each zone is sufficient to make measurements at three different wavelengths of the probing field.
In figure 9 is a graph illustrating the initial recovery of the phase function ∆ψ( , ) at 6 mm from the end of the burner tube for the R-channel. The red curve represents the phase function ∆ψ( , ), the blue curve is the section of the experimental hilbertogram, the black curve is the hilbertogram reconstructed from the phase function ∆ψ( , ). 0 mm is the center of the burner tube.
The method for determining the phase function consisted in iterative sequential selection of the shape and height of the curve represented by the Bezier polynomial (a special case of B-splines), and the subsequent calculation of the refractive index ( , ). Further, according to the distribution ( , ) for the selected area, the hilbertogram is reconstructed, which is compared with the experimental hilbertogram. The iterative algorithm is repeated until the coordinates of the local extrema of the experimental and reconstructed hilbertograms coincide. The coincidence of the coordinates of the extrema of the experimental and reconstructed hilbertograms means checking the results obtained. When restoring the phase function ∆ψ( , ), the values and were taken from the reference data.
Phase function 80 60 40 20
00 400 300 200 100 X[mm] b
X[mm] Figure 12: Reconstructed radial temperature field of an axisymmetric section of a hydrogen-air flame in a 6 mm section: solid line – reconstructed temperature ; black dots are the temperature measured by the thermocouple
The maximum temperature in the flame front is close to the adiabatic combustion temperature, which is 1490 K for an 2/ 2 mixture (25%) in air. The discrepancies in the experimental and reconstructed data are due to the violation of the axial symmetry of the flame in a real experiment due to the influence of dynamic disturbances of the air surrounding the flame. The difference between the reconstructed temperature values and the thermocouple measurements can also be introduced by the influence of the thermocouple on the flame or possible variability of the flame.
The method of the multiwave RGB-optical Hilbert diagnostics of phase and temperature fields in the approximation of axial symmetry of the object under study (in an air heat flow from a candle flame, hydrogen-air diffusion flame) has been theoretically and experimentally substantiated. The verification of the results is provided by comparing the hilbertograms obtained in the experiment and the hilbertograms reconstructed from phase structures using the Abel transform. One of the important advantages of the proposed method for flame diagnostics is the potential for measurements with a spatial resolution of up to several microns, which makes it possible to study the entire range of scales of the reacting flow up to Kolmogorov. The issue of visualization of asymmetric temperature fields is considered in perspective. The tomographic attachment will be added to the diagnostic complex in this case. This will allow visualization of the phase function in four different directions and estimate the flame temperature.
The work was carried out within the framework of the state assignment of IT SB RAS No. 121031800217-8. 7. References [10] Yu. N. Dubnishchev, V. A. Arbuzov, E. V. Arbuzov, O. S. Zolotukhina, V. V. Lukashov, Optical diagnostics of temperature field of an axisymmetric flame, Scientific Visualization 11, No 4 (2019) 130–139. doi: 10.26583/sv.11.4.11 [11] Yu. N. Dubnishchev, Theory and conversion of signals in optical systems, SPb. Lan Publishing
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