The possibility of using the Gauss-Newton optimization algorithm in Hilbert diagnostics of reacting media (flames) is considered in the paper using the example of data on the temperature and main compounds of a laminar methane-air Bunsen flame. The method of reconstructing phase optical density fields is used to solve the inverse problem of Hilbert optics (it was previously proposed and tested on numerical models): the hilbertogram is calculated in the considered flame axisymmetric section, from which the initial values of the phase function and the medium refractive index are reconstructed. The applications scope expansion of the obtained results in the experimental data processing will be the work development.

The possibility of using the Gauss-Newton optimization algorithm in the Hilbert diagnostics of reacting media (flames) is considered in the work using the example of temperature data and the main connections of a laminar methane-air Bunsen flam. The previously proposed and tested on numerical models technique for reconstructing phase optical density fields was used to solve the inverse problem of Hilbert optics. The hilbertogram distribution was calculated in the considered axisymmetric flame section, from which the initial value of the phase function and the refractive index of the medium was then restored. The work development will be the scope expansion of obtained results applications to the experimental data processing.

В данной работе представлены результаты изучения влияния разбавления метана водородом на физико-химические и электрические характеристики пламени диффузионной струи СН4/Н2. Было обнаружено, что для смесей с молярной долей метана выше 40% величина протекающего тока линейно зависит от состава. Разбавление водородом при молярной доле метана в смеси менее 40% приводит к тому, что зависимость тока от состава становится нелинейной. В этом случае граница перехода от линейной зависимости к нелинейной не зависит от скорости потока и формы электрода. Результаты измерений хемолюминисценции радикала СН* выявили совершенно аналогичную зависимость свечения от объёмной доли водорода. Полученные результаты указывают на существование двух режимов, в которых возможно существенное различие в кинетических механизмах диффузионного горения смеси СН4/Н2.

The possibility of processing small-view hilbertograms by the Gerchberg–Papoulis method to restore the spatial distribution of the refractive index was studied. The method consists in iterative transitions from the function estimates in the frequency and coordinate spaces with correction based on a priori information. Numerical modeling of the refractive index reconstruction for various test functions by the Gerchberg–Papoulis method using the Radon data, known for four angles, has been carried out. Using a four-angle tomographic complex implemented on the basis of a modified IAB-463M shadow device, experimental studies were carried out by the example of optical Hilbert diagnostics of reacting media in a high-speed shooting mode (up to 2000 frames per second).

The problem of reconstructing the phase function in the Hilbert diagnostics of gaseous, condensed and reacting media is discussed in the paper. The method for restoring the phase disturbances structure of a probing light field based on the iterative Gauss-Newton algorithm is proposed. It does not require the definition of second derivatives and significantly reduces the number of computes. The method consists in successive selection of the phase profile given by the Bezier curve and calculation of the hilbertogram. The reliability of the results is the coincidence of the reference and reconstructed hilbertograms local extrema. The Jacobian matrix of the nonlinear integral operator of the Hilbert visualization is calculated. The proposed algorithm has been certified on test functions. The development of the method and its applications is associated with the processing of experimental results, including complex structures, where the phase function is represented by several Bezier polynomials.

Hilbert-optics and signal conversion methods form the basis for modern information and measurement technologies. The Hilbert transform in the frequency space is physically reduced to a certain phase transformation of the Fourier spectrum of the signal in a wide frequency band. The article discusses the possibility of Hilbert tomography using the example of studying the isothermal flow of a freon-22 jet. An optical system based on the IAB-463M modernized wide-aperture shadow device has been developed. This system makes it possible to perform four-angle diagnostics of the studied phase object using probing light beams. The beams orientation is defined at angles of 0, π/2, π/4 and 3π/4 to the optical axis of the shadow device. This makes it possible to simultaneously register all four tomographic projections in the form of Hilbert images in separate sections of the CCD-matrix of the high-speed camera. The use of the iterative Gershberg-Papulis scheme for finding the parameters of the medium under study based on Hilbert tomography data is further proposed. This algorithm for reconstructing a function from its Radon data obtained by a limited number of projections consists in iterative transitions from estimating the function in the Fourier plane to its estimate in the spatial domain, corrected based on a priori information. The spectrum of the restored function is determined on the entire frequency plane as a result of an iterative process, which makes it possible to determine the values of the required function.

The problem of phase function reconstructing in Hilbert diagnostics of gaseous, condensed and reacting media is discussed in the work. The method for reconstructing the phase disturbances structure of a probing light field, based on the iterative Gauss-Newton al- gorithm, is proposed.This method does not require the second derivatives determination and greatly reduces the number of calculations.It consists in the sequential selection of a complex phase profile, which is specified by the sum of third degree Bezier curves, and the hilbertogram calculation in order to minimize the root-mean-square error between the experimental and reconstructed hilbertograms. The Jacobi matrix for the nonlinear integral operator of Hilbert visualization is calculated. The proposed algorithm was tested on test functions.The devel- opment of the method and its applications is associated with the application of the algorithm to the processing of experimental results.

A method for reconstructing phase disturbances of a probing light field using the iterative Gauss-Newton algorithm is discussed as part of the Hilbert diagnostics development of gaseous, condensed and reacting media. In this case, the need to determine second derivatives is eliminated, which simplifies the calculations. The method consists of selecting a phase profile, which is specified by a Bezier curve, and hilbertogram calculating. The coincidence of the reference and reconstructed hilbertograms serves as a criterion for the results reliability. The Jacobian matrix for the nonlinear integral operator of Hilbert visualization is obtained. The algorithm is analyzed using a test function. The method development is associated with the algorithm application to the processing of experimental results, including the reconstruction of complex structures in which the phase function is described by several Bezier polynomials.

The method of multiwave RGB-optical Hilbert diagnostics of the phase and temperature spatial structure of the flame (in an air flow heated by a candle flame) at a fixed moment of time is discussed. The object of study satisfies the model of axial symmetry of the torch associated with the vertical orientation of the candle. The reliability of the received results is confirmed by comparing the reconstructed radial temperature by experimental hilbertogram with data measured by a thermocouple. The flickering effect of the flame was not taken into account in the work, this is the subject of further research.

The method and possibility of implementation of isotropic Bessel–Hilbert diagnostics of phase optical density fields of gaseous, condensed, and reactive media are discussed. The probing field in such systems is formed as a mono- or polychromatic configuration of Bessel beams the spatial-frequency Fourier spectrum of which is optically conjugate with the concentric phase structure of an isotropic Hilbert filter, and the color structure corresponds to spectral sensitivity of the photomatrix of the video camera connected to a computer which analyzes the frame-by-frame sequence of the obtained images. The structure of such a probing field is resistant to diffraction perturbations induced by the medium under study and their influence on isotropy of the Hilbert visualization of the phase optical density. The areas of possible applications are experimental fluid and gas dynamics, thermophysics, oceanology, and the corresponding industrial technologies.

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.The method based on polychromatic Hilbert visualization of the phase optical density fields induced by the temperature field in the studied medium, pixel-by-pixel processing of RAW images recorded by the photomatrix in RGB channels, and Hilbert verification of the results is presented with an example of studying an axisymmetric hydrogen-air diffusion flame.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.

The work is motivated by the scientific and practical significance of the problem of non-disturbing diagnostics of phase and temperature fields induced in a gas medium by a flame of a torch (candle). The spatial conditions in which the fields are studied satisfy the model of axial symmetry of the torch associated with the vertical orientation of the candle. A method adequate to the problem to be solved has been developed, based on polychromatic Hilbert visualization of phase optical density fields, measurement of the temperature profile in selected sections of the medium under study, registration and selection of RAW images recorded by the photomatrix in RGB channels. The visualized Hilbert structures contain information on the phase optical density perturbations induced by the temperature field. The reliability of the results is confirmed by comparing the experimentally obtained hilbertograms and those reconstructed from phase structures using the Abel transform.

Work motivation – adaptation of optical Hilbert diagnostic methods for visualization and study of optical density and phase temperature fields in the structure of an axisymmetric diffusion hydrogen-air flame. The diagnostic complex is implemented on the basis of the IAB-451 device with modified blocks of optical filtering, information source and processing. A laminar jet flame H2/N2 in still air is considered. The investigated torch is oriented vertically. Visualization of phase disturbances induced by the medium under study in a multi-wavelength probing (λ1 = 636 nm, λ2 = 537 nm and λ3 = 466 nm) light field is performed using polychromatic Hilbert and Foucault-Hilbert transformations in combination with registration and pixel-by-pixel processing of the dynamic RGB image structure. The dynamic phase structure of the diffusion flame is visualized. The initial temperature approximation, based on the assumption of an air mixture, is corrected so that the calculated hilbertogram matches the measured one as closely as possible. The data obtained are in good agreement with the results of thermocouple measurements. The temperature was recorded by thermocouples at reference points. The phase function is reconstructed in axisymmetric sections from RGB-hilbertograms. The reliability of the results is confirmed by comparing the experimentally obtained hilbertograms and hilbertograms reconstructed from phase structures using the Abel transform.

The problem of reconstructing interference and Hilbert structures from a numerical model of the evolution of the thermal field of convective flows in a water layer bounded by flat heat-exchanging surfaces under unsteady boundary conditions in the monotonic cooling mode and taking into account density inversion at a temperature of +4 °C was solved. The simulation of the thermal fields of convective flows in the form of the dynamic structure of isotherms was carried out taking into account the nonlinear dependence of thermal conductivity and water density on temperature. The field of the phase function, its Hilbert image and the interference field, which are compared with the results of the interference and Hilbert visualization of the fields of phase optical density obtained in the experiment, were reconstructed from the isotherms field supplemented by calculating of the velocity field and of the temperature gradients field. The presented films illustrate the qualitative adequacy of the coevolution of numerical models and real processes.

The structure of the temperature field of an axisymmetric flame using an example of a candle, spiritlamp, and propane-air torch was studied.The optical diagnostics adapted to the study of combustion problems is based on the visualization by methods of Hilbert-optics of phase disturbances induced by the medium under study in a probe light field.The diagnostic complex is implemented on the basis of the IAB-463M device with modified blocks of optical filtration, light source, registration and information processing.The dynamic phase structure of the candle flame and spiritlamp was visualized as a classic object for approbation the diagnostic method.The dynamic phase structure of the propane-air torch was investigated.The temperature was measured using thermocouples at the reference points.The phase function was restored on axisymmetric sections from the obtained hilbertograms, and the temperature field of flame was reconstructed using the inverse Abel transform.

The problem of reconstructing interference and Hilbert structures from a numerical model of the evolution of the thermal field of convective flows in a vertical water layer bounded by flat heat-exchanging surfaces under unsteady boundary conditions in the monotonic cooling mode and taking into account density inversion at a temperature of +4 degrees Celsius was solved.The simulation of the thermal fields of convective flows in the form of the dynamic structure of isotherms was carried out taking into account the nonlinear dependence of thermal conductivity and water density on temperature.The field of the phase function, its Hilbert image and the interference field, which are compared with the results of the interference and Hilbert visualization of the fields of phase optical density obtained in the experiment, were reconstructed from the isotherms field supplemented by calculating of the velocity field and of the temperature gradients field.The presented films illustrate the qualitative adequacy of the coevolution of numerical models and real processes.

The evolution of the crystallization wave front and convective structures in a horizontal layer of supercooled water bounded by tempera- ture-controlled flat surfaces is visualized using methods of Hilbert optics. The phase transition is manifested by the occurrence of a crys- tallization wave and is accompanied by a positive energy release, which, in turn, affects the dynamic distribution of the optical phase density gradient in supercooled water and induces phase perturbations in the probing light field. The results of measurements of the phase velocity and the shape of the crystallization front approximated by Bezier curves are presented. The wave front velocity is obtained using modified time-of-fight method. The phase velocity field is found to exhibit spatio-temporal quasi-periodicity that can be related to the existence of oscillatory phenomena in the crystallization process.

The dynamics of the crystallization front induced by the temperature gradient at the upper boundary of a horizontal layer of water bounded by flat thermostatted surfaces is studied. The formation and evolution of convective structures are visualized by methods of the Hilbert optics and digital video recording. The difference in the temperatures of the upper (T1) and lower (T2) thermostats satisfies the condition T1 < T2. In this system, the temperature of the cooled upper surface is an order parameter. Reaching a critical value of this parameter leads to a bifurcation phase transition from the liquid state (supercooled water) to the solid state (ice). The velocity and shape of the crystallization wave front are determined. The spatial-temporal state of the crystallization wave is found, in which the wave front shape is a line of equal velocities. The dynamic profile of the isothermal surface bounding the spatial shape of the solid phase is visualized.