Methods
of Hilbert-optics and interferometry constitute one of the directions of unperturbing
diagnostics of reacting jets and flames [1, 2]. They are based on visualization
and analysis of phase disturbances induced by the studied medium in the probing
light field. In [3, 4] studied diffuse combustion jet of hydrogen, by methods
the Hilbert-optics. An optical diagnostic complex based on the IAB–463M
instrument [5] equipped with modified Hilbert filtering, interference, light source,
Hilbert image registration and information processing modules was used. In [6]
a method for estimating the temperature distribution in an asymmetric flame
using high-contrast stereoscopic photography is described. Spectral
reconstruction of temperature fields using pyrometry of color ratios and
interferometric tomography is reported [7]. A jet flame of premixed propane-air
mixtures has a wide range of applications in scientific research and practical
applications, the successful development of which requires the development of
optical diagnostic methods with the possibility of reconstruction of the spatial
phase and temperature structure of the burning torch.

The
adaptation of the methods of Hilbert-optics to the solution of diagnostic
problems with the reconstruction of the spatial phase and temperature structure
of the flame is the goal of this work. The studies motivated by the scientific
and practical significance of the problem, which consists in the search for
control methods of the structural and thermodynamic parameters of the torch
[8].

The optical diagnostics
complex was created on basis of the IAB–463M [5] device with modified for the
experiments modules of the probing field source, of Hilbert filtering,
registration and processing of the optical signal.

Fig.1. Optical diagnostics complex.

The
optical flame diagnostics complex sketch is shown in Fig. 1. It contains a
lighting module consisting of a light source 1, a lens 2, and a slit diafragm 3
located in the front Fourier plane of the lens 4, which forms a probe field.
The Fourier spectrum of the phase disturbances induced by the medium under
study 5 is localized in the frequency plane of the lens 6, where the quadranted
Hilbert-filter 7 is located, the orientation of which corresponds to the slit
diafragm 3. The lens 8 performs the inverse Fourier transform of the filtered
field, forming, depending on the spectral characteristics of the light source,
analytical or Hilbert-conjugate optical signal. These signals are recorded by a
digital camcorder 9 connected to the computer 10.

In
the frequency plane for the
Fourier spectrum of the phase optical density of the light field perturbed by
the medium (flame) under study, we have immediately after filter 7:

(1)

where is the
transfer-function of the filter, is the Fourier
spectrum of the Hilbert-conjugate optical signal:

Phase shift is a function
of wavelength of the probe
light field, . The lens 8
performs the inverse Fourier transform of the filtered signal (1):

(2)

The recorded by the photo matrix of the
camcorder 9 signal intensity (2):

(3)

where is a
coefficient taking into account the sensitivity of the photo matrix. The
Fourier filter performs a one-dimensional Hilbert transform at a wavelength of satisfying
condition .

The Hilbert transform has the properties of redistributing energy
from the region of low spatial frequencies to the high-frequency region. The
extrema and gradients of the phase optical density of the medium under study transformed
into visualized structures of the Hilbert bands. The spatial distribution of the
Hilbert bands carries information on the perturbations of the phase optical
density induced by the temperature field.

The flame of a candle and spirit lamp was chosen as a classic
object for approbation the diagnostic method. Measurements of the temperature
profile in flame sections at various distances from the end of the wick have been
fulfilled using an Chromel–alumel thermocouple (Fig.
2). The Hilbert images of flames are shown in Fig. 3a and 3b.

a) b)

Fig. 2.
Temperature profiles of the flame: (a) - candles in cross sections at distances
of mm from the
wick; (b) - spirit lamps at distances of mm from the
wick.

The
phase structure of the probe light field disturbed by the medium under study is:

(4)

where is the wave
number of the probe field; - refractive
index of the medium in the spatial structure of the flame; - refractive
index of an unperturbed medium. The axis is
determined by the direction of the probe light beam, the flame torch cross
section is described in coordinates. The
choice of the section position is determined
by the coordinate. The
coordinates , specify the
size of the flame section in the direction of the probe beam.

Eq.
(4) is turned into the Abel equation in the case of axial flame symmetry:

(5)

where – the radius of the cross section of the
considered zone, *– *refractive
index at a distance of from the axis
of the torch. The Abel equation is solved by an approximate method based on
approximation of the experimental temperature data by Bezier curves (a special
case of B-splines).

a) b) c)

Fig.
3. Experimental hilbertograms: (a) – the candle flame, (b) – the spirit lamp
flame; (c) – section diagram of the flame under study in the plane.

The total phase shift for
the light beam in section depends
on the refractive index on
the segment (, *)*, (Fig.3c).

The
axisymmetric distribution of the refractive index found from the
Abel equation allows one to determine the radial fields of temperature in the cross
sections . To do this,
can use the Gladstone – Dale equation:

(6)

Graphs
illustrating the restoration of the radial distribution of the temperature of
the flame of a candle and an spirit lamp in sections mm and mm from the
end of the wick using equations (5) and (6) are presented in Fig. 4. The red
line (Fig. 4 (2)) shows the phase function obtained from
the Abel equation in the selected sections, the green and black lines represent
the interferogram and the hilbertogram recoverded from the phase function . The blue line
is the experimental hilbertogram.

A
comparison of the experimental and reconstructed hilbertograms in the selected
sections shows that the points of local minima of these functions coincide.This
means that the phase function obtained from the solution of the Abel equation
coincides with the real phase function. Therefore, the real and the
reconstructed temperature fields in the selected flame sections coincide.The
criterion for the correctness of the results obtained is the phase identity of
the reconstructed and experimental hilbertograms, which is achieved by
repeating the described procedure.

a)

b)

Fig. 4. Reconstruction
of the phase structure and temperature of the candle flame (a) and the spirit lamp
(b) in the sections mm and mm: (1)

the red line - restored
temperature (small circles - thermocouple data); (2) the red line - phase
function, the green line -interferogram reconstructed from the phase function,
the black line - reconstructed hilbertogram, and the blue line - experimental
hilbertogram.

The
temperature fields reconstructed from the results of measurements using
thermocouples are shown in Fig. 5a and 6a. The temperature fields reconstructed
from the Hilbert images are shown in Fig. 5b and 6b. The similarity of
reconstructed and original temperature fields is quite satisfactory.

As
a verification of the obtained results, the inverse problem has been solved:
from the reconstructed temperature fields (Fig. 5b and 6b) have been restored
Hilbert images (Fig. 7b and 7d), which were compared to the images obtained in
the experiment (Fig. 7a and 7c).

From Fig. 7 it can be seen
that the structures obtained in the experiment and reconstructed have a similar
character. By this confirms the reliability of the results. Some discrepancy is
due to the distortion of the axial symmetry of the flame in a real experiment
due to the influence of dynamic disturbances of the air environment that
surrounding the flame.

a) b)

Fig. 5. The
temperature field of a candle flame: (a) – the reconstructed according to
thermocouple data, (b) – the reconstructed from a Hilbert image.

a) b)

Fig. 6. The
temperature field of the spirit lamp flame: (à) – the reconstructed according to
thermocouple data, (b) – the reconstructed from a Hilbert image.

a) b)
c)
d)

Fig. 7. Flame
of a candle and spirit lamp: (a) and (c) - the hilbertograms of the flame of a
candle and spirit lamp obtained in the experiment; (b) and (d) - the
hilbertograms of the flame of a candle and spirit lamp numerically modeled from
the reconstructed temperature fields.

The
developed method of optical diagnostics was successfully applied in studies on
the jet combustion of premixed propane-air flame.

The
Hilbert image of the jet combustion of a premixed propane-air mixture (propane
concentration 25%) in a still atmosphere (air) is shown in Fig. 8a. The
gradient phase structure of the probe light field induced by the temperature
field in the torch was visualized. The results of measuring the temperature
profile of the flame in cross sections at distances of 0 ÷ 20 mm from
the end of the tube are shown in Fig. 8b. The measurements were carried out
using a TPR thermocouple. A chart illustrating the restoration of the radial
temperature distribution in the flame cross section mm from the
end of the tube is shown in Fig. 8c.

Fig.
9a and 9b– the temperature fields obtained from the data thermocouple
measurements and reconstructed from the hilbertograms, respectively.

Fig. 10a and 10b
illustrate the comparison of Hilbert images of the reconstructed and
experimentally obtained temperature fields.

a) b)

c)

Fig. 8. Propane-air
flame: (a) –Hilbertogram of jet combustion of a premixed propane-air mixture
of 25% in air; (b)– temperature profiles of jet combustion of a propane-air
mixture of 25% at distances of 0÷20 mm; (c) –reconstruction of the phase
structure and torch temperature in the cross section mm.

a) b)

Fig.9. Temperature
field of the propane-air flame: (a) – the temperature field reconstructed from the
thermocouple measurements, (b) – the temperature field reconstructed from the Hilbert
image.

a) b)

Fig. 10. Hilbertograms
of the propan-air torch: (a) – the hilbertogram obtained in the experiment;
(b) – the hilbertogram numerically modeled from the reconstructed temperature
field.

The
study of the flame of a candle, spirit lamp and premixed propane-air torch was
performed in the presented work. For this, the methods of Hilbert optics in the
approximation of axial symmetry using the Abel transform were used. The
reliability of the results is confirmed by comparing the hilbertograms obtained
in the experiment and reconstructed from the phase structure according to Abel.
The comparison results are used as a quality criterion in modeling of the phase
structure and temperature field in the study of the combustion process.

The
authors thank N. S. Bufetov for help in the work.

This work was supported by
the Ministry of Education and Science of the Russian Federation (AAAA-A17-117030310010-9)
and by Siberian
Branch of the Russian Academy of Sciences (complex
program of fundamental scientific research SB RAS II.1 project 0314-2018-0010).

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