Numerical Study of Flame Propagation Morphology for Deflagration in the Pipeline Using Proper Orthogonal Decomposition

A multilevel independent spatial modal analysis of flame propagation characteristics of a deflagration in a specific pipeline was performed using the proper orthogonal decomposition (POD)method, in order to research the evolution process of the explosion which is closely related to flame propagation speed and front rupture pressure. ,e CFD results indicated that the full-order calculation results well agreed with the normal combustion propagation characteristics of premixed methane-air for the flame propagation with the unbroken thin layer. ,e POD analysis results showed that the static temperature gradient of the 1st order mode of initial and subsequent stages both exhibited a range of continuity change from left to right, and the frontal curvature of the cooling area decreased as the flame propagated in all stages. ,e number of the low-temperature interval regions displayed an expanding form of a staircase with the increase of the mode order, especially for subsequent flame in which the interval areas becamemore andmore slender.Moreover, the level of information content in themultilevel modal space wasmostly concentrated in the first 3 modes, especially in the 1st order mode, and the flame propagation pattern at the initial stage was more complicated than the subsequent based on the relational information content features.


Introduction
e underground mining is the mostly major style of China, in which the wicked environments and geological conditions with complex and varying situation often induce such grave disastrous accidents.e deflagration heads the list of those events, resulting in the destruction of the coal mine ventilation system and the spread of toxic and harmful gases, even inducing that multiple subsurface explosions [1].e personal security of the staff related and later relief efforts are both dangerously threatened in the complicated circumstances.erefore, in order to effectively prevent and control these accidents of deflagration, the domestic and overseas scholars have taken many studies about the involved explosion mechanism [2].
Xu et al. [3] investigated the propagation property of deflagration in the roadway with a square cross section of 7.2 m 2 via an experimental test.e results showed that the length of the flame zone was longer than the area of gas accumulation in the explosive evolution and such ratio of differences was up to three to six times.ere was a delay interval before the pressure value of monitoring points increased to a maximum value along the propagation path of deflagration.Cheng and Lin [4] studied the influence upon the flame transmission regular pattern in deflagration for tube furcation and pointed out that the additional turbulent flow was produced under the condition of the branch tube, leading to improve the flame transmission speed in the explosion.Lin et al. [5] carried out the experimental research of the influence of barriers for flame transmission and explosion wave in deflagration.e results indicated that the barriers had a significant impact to the flame transmission and explosion wave, in which the velocity of flame transmission took a sharp increase in the situation with barriers, reaching its maximum value of 20 times L/D ratio.e reason is that the transmission of flame and the turbulent extent of explosive flow can mutually promote each other in the existence of barriers, even resulting in shock wave of explosion to enhance the destructive power in the coal mine tunnels.Li and Gui [6] discussed the flame transmission of deflagration and its acceleration mechanism by experimental and numerical analysis.e results showed that the temperature gradients of front of the flame changed quickly but smoothly in back regions and also manifested that there was a positive feedback for turbulent combustion caused by barriers.Liang and Zeng [7] built a computational model of deflagration to a constant volume bomb through modifying the SENKIN code of CHEMKIN III chemical kinetics package.Using such model, they studied the mole fraction profiles of reactants and the sensitivity of the reaction mechanism of deflagration.e analysis results illustrated that the induced explosion time was prolonged, in which the mole fractions of reactant species such as CH(4) and O(2) and catastrophic gases such as CO, CO(2), and NO were all decreased when water had been added to the mixed gas.Furthermore, the sensitivities of reactions to species of CH(4) and CO(2) were also depressed with the water fraction increasing.Lukashov [8] proposed the sequential use of two mathematical models to compute the deflagration of a coal mine.One of the models was applied to obtain the propagation of air shock waves and another for the computation of the air distribution in the mine situation.e given approach can more precisely estimate the situation of explosion in the mine and identify the regions affected by shock waves and the distributed explosion products.Nie et al. [9] conducted the experimental study of a rectangular explosion test pipe to substantially suppress the shock waves of deflagration for coal mines.e results demonstrated that the foam ceramics can buffer the explosion overpressure by up to fifty percent and the interconnected micronetwork structure of the foam ceramics can help us to quench the flame of explosion.Yu et al. [10] experimentally investigated the effects of the hollow-square obstacle in deflagration by constructing a semiconfined transparent chamber.e research showed that the triangular hollow-square obstacle caused the highest turbulent intensity, flame propagating velocity, and overpressure, whereas the lowest values were for the circular hollow square.Pei et al. [11] discussed the synergistic inhibition effect of nitrogen and ultrafine water mist for deflagration in a ventilated pipe.e experimental data displayed that propagation speed and overpressure of the peak flame declined significantly with the increase of fraction of nitrogen and spraying time of water.However, the explosion inhibition efficiency was gradually decreased as the continuous increase for water mist and nitrogen.Liu and Jia [12] established the overpressure prediction model for deflagration shock wave using the theory of explosion similarity, and the model was well verified by means of relevantly experimental data.Other domestic and international research progress can be found in the literatures [13][14][15][16][17].
Although scholars have carried out a variety of studies for deflagration, the traditionally experimental and numerical analysis are still limited for the further investigation of the mechanism of deflagration.In the research of the deflagration mechanism, the flame propagation pattern is a very important feature, which can greatly affect the speed of flame propagation and the membrane rupture pressure to flame front.erefore, it is necessary to employ more advanced method to study such morphological changes of explosion flame.In this paper, the proper orthogonal decomposition (POD) method is used to research the evolution process of combustion explosion propagation in a pipeline.
e paper has five more sections.e basic theories of deflagration and POD method are introduced in Sections 2 and 3.And the CFD model and the data interface program are developed in Section 4. e numerical results are discussed in Section 5, and some conclusions are drawn in the last section.

Physical and Mathematical Models for Deflagration in a Pipeline
It is an essential part to the analysis of deflagration in the underground passage of a mine, which includes the physical model and the description of mathematics, respectively.

Physical Model.
In the light of such deflagration theory [18], there are two courses of the burning explosion in a pipeline, including the initial ignition stage of hot products and subsequent propagation stage for premixed reactants for coal mine gas in the underground operation channel.e essential process of an entire explosion can be understood as the propagation of combustion and shock waves from ignition and burning new premixed gas areas to the regions of unburned-premixed areas.e physical model of deflagration in the pipeline is shown simplistically in Figure 1.
It is worth noting that the deflagration process is in an ideal condition without obstacles in the coal mine passage.For more complicated situations, one can refer to the literatures [19,20].

Mathematical Model.
e deflagration inside the pipeline is a rapid reaction process of combustion, in which the related equations of control should be satisfied, including the mass conservation equation, momentum conservation equation, energy conservation equation, and the chemical composition equilibrium equation.e description of turbulence follows the standard k − ε model, which is shown in the following equations: 2 Advances in Civil Engineering and ( According to the above transport equations, the turbulence kinetic energy, k, and its rate of dissipation, ε, can be obtained, respectively.G k represents the generation of turbulence kinetic energy due to the mean velocity gradients, and G b is the generation of turbulence kinetic energy due to buoyancy.Y M represents the contribution of the uctuating dilatation in compressible turbulence to the overall dissipation rate.ose C 1ε , C 2ε , and C 3ε are constants.σ k and σ ε are the turbulent Prandtl numbers for k and ε.
e turbulent viscosity, μ t , is computed by combining the turbulence kinetic energy and dissipation rate [21], in which the values of the model constants have been determined from experiments for fundamental turbulent ows including frequently encountered shear ows like boundary layers, mixing layers, and jets as well as for decaying isotropic grid turbulence.
e conservation equations for chemical species takes the following general form, which predicts the local mass fraction of each species, Y i , through the solution of a convection-di usion equation for the species: where R i is the net rate of production of species i by chemical reaction and S i is the rate of creation by addition from the dispersed phase plus any de ned sources.J ⇀ i is the di usion ux of species i, which arises due to gradients of concentration and temperature [22].

Proper Orthogonal Decomposition for Deflagration
e POD method can quantitatively analyze the essential information of the ow at multiple levels so that the main features of the ow can be more accurately captured for further analysis.e method is generally divided into continuous type and discrete type.However, the discrete POD method is more commonly used because of the di culty in obtaining the analytical basis function in practical research.erefore, only the discrete POD method is introduced here, combining de agration parameters in a pipeline.

Discrete-Type Proper Orthogonal Decomposition.
e de agration parameter values of a series of discrete points, obtained by the full-order numerical method under the same transient moment conditions, are arranged in a speci c order to obtain the required sample vector g(x, t m ).e corresponding form is as follows: where N is the total number of discrete points in such a physical model.e sample matrix G can be formed by arranging the sample vectors in the sequence of time evolution, in which the size of this sample matrix is worthy for further discussion, especially for the time dimension: where M is the total number of sample vectors, also named the total number of snapshots in a large number of related studies.
Under the same number of snapshots, mathematical transformations and decompositions are needed in order to nd the best orthogonal basis functions in discrete form.erefore, the covariance matrix should be calculated rst [23]: It is worth noting that such C is a real symmetric matrix and has nonnegative eigenvalues.erefore, the problem of nding the optimal orthogonal basis function is transformed into the solution of the following eigenvalues: where H [r] is the rth eigenvector for the de agration parameter, and the quantity λ r is the eigenvalue corresponding to the rth-order POD mode of explosion ow parameter: e rth-order POD mode of the explosion ow eld can be de ned by where such eigenvalue corresponding to each eigenvector is sorted in decreasing order for the information content contribution to the explosion ow eld.erefore, the major ow characteristics of de agrations can be extracted using low-order POD ow modes, in order to analyze the most important ow essential information.

De nition of Information Contribution for De agration in a Pipeline.
e quantitative analysis of the multilevel ow features of the POD method is mainly re ected in the feature value level corresponding to the feature projection direction.Advances in Civil Engineering e global average information content to explosion ow may be described as where E is the global average pulsating information content for a burning explosion ow eld, E r is the information content of one speci ed such ow mode, and λ r is the eigenvalue of the corresponding spatial mode.
In the light of Equation (10), there are two quantitative indicators of the information content of the explosion ow mode, de ned as follows: where η r refers to the single-order contribution of information content for the rth order mode and κ r represents the cumulative contribution of information content for the modes up to order r. at quantities η r and κ r are representative of quantitative indicators obtained upon averaging the pulsating energy contribution for the velocity-type parameters in the physical eld.In this investigation, such propagation characteristics of de agration in the designated barrier-free pipeline were studied, making use of the discrete-type POD method based on the results of full-order numerical calculation.erefore, it is necessary to introduce the corresponding CFD model for the explosion ow.

CFD Model for Deflagration in the Pipeline
In order to improve the e ciency of full-order numerical computations, the two-dimensional ow area was adopted, combined with a layout strategy of the isotropic structured grid.e length of the pipe model is 18 m, and the width is 0.2 m. e rectangular channels were 9.5 percent methaneair premixed gas lled, ignoring the pipe thickness.

Grid Layout for the Pipeline.
ere are 30 points in the width direction and 2700 points in the length direction, which is a total number of 81000 with quadrilateral elements in such domain.e CFD model is shown in Figure 2.

Initial and Boundary Conditions.
For the CFD model of uniform premixed de agration, the four boundaries of the numerical calculation area are all set to the adiabatic wall, with room temperature 300 K. e relative roughness of the walls are all set to 0.05 mm and the related roughness height in such rough wall formulation is set to 0.02, taking into account the actual coal mining channel situation.In addition, the heat ux per unit area of the wall is set to 0, and the species boundary condition for CH(4), O(2), CO(2), CO, and H(2)O are designated as zero di usive ux type.e initial gas mixture state included the temperature of 300 K, the pressure of a standard atmosphere, and both velocities speci ed to be zero along X and Y directions.When methane-air premixed gas concentration is 9.5 percent, the mass fraction of each component was 5.3 percent of CH(4), 21 percent of O(2), 73.7 percent of N(2), and both 0 percent of H(2)O and CO(2), respectively.For the stability and reliability of subsequent calculations, the computation of 50 time steps under this condition was performed and checked.
en, the model needs to set ignition initial conditions for de agration.According to the thermal ignition theory, the region of high-temperature gas can be set as the ignition zone during the explosion simulation.Here, the left end of the pipeline was set to the ignition position, and the temperature of the local area and the mass fraction of each component at the beginning of the ignition were the temperature of 1400 K, both 0 percent of CH(4) and O(2), 73.7 percent of N(2), 14.5 percent of CO(2), and 11.8 percent of H(2)O, respectively.
It is worth noting that, in order to be closer to the actual situation, the initial ignition area is set to a smaller semicircular area.Its center coordinates are (0, 0.1) with radius 0.02 m, which corresponds to the center of the left end of the channel.

Interface Program to Realize POD-CFD Analysis.
In order to achieve POD multilevel analysis of ame propagation patterns of de agration in the pipeline, an interface program for the POD-CFD calculation of the speci ed eld variable was developed to enable subsequent analysis using full-order CFD numerical calculation results.
e logical framework of the interface program is shown in Figure 3.
e interface program consists of three parts.(1) It is data structure conversion which is for binary output les generated by high-con dence CFD code.(2) In this step, the secondary conversion of the basic data for POD analysis is realized by the in-house Matlab code.Among them, the homology processing corresponding to the data structure is  Advances in Civil Engineering included.
(3) Tecplot template and related Matlab functions were used to visualize the POD analysis results.Among them, for the built-in related macrofunctions, the corresponding subroutine for loop calling is developed.

Full-Order Computational Results for Pipeline Explosion.
According to the condition settings of de nite solution of de agration, combining such methane combustion reaction equations, the results were manifested at the di erent characteristic moments for the main reaction phase upon the ame propagation in Figure 4.
In the light of Figure 4, the development trend of the initial static temperature eld and the subsequent propagation of the ame are coherently presented based on advancement of de agration time.e average speed of ame propagation for the unbroken thin layer is approximately 0.47 m/s, which meets the normal combustion propagation characteristics of premixed methane-air.In order to more clearly characterize the ame propagation process in the early stage of an explosion, the following analysis is performed in conjunction with Figure 5.
According to the ame propagation pattern of Figure 4, the average ame propagation velocity between the adjacent monitoring moments can be calculated.e corresponding situation is shown in Figure 5. Due to the energy excitation that initiates the ignition, the average ame propagation velocity at the relevant initial stage is faster.In the later stage of stable ame propagation, the corresponding average axial propagation velocity is kept within 0.5 m/s.
In Figure 6, the nonblue area on the left indicates that the gas has started to burn rapidly and the maximum static temperature has reached more than 2400 K. e blue area on the right indicates that the original premixed gas has not yet started to burn and is only a ected by the disturbance of the left-side explosion combustion shock wave, and the temperature is relatively low.
Moreover, after the left-hand premixed gas is ignited, the static temperature rises rapidly and forms a spherical ame.Over time and with the special setting of the above ignition zone, the combustion ame gradually appears hemispherical to the right in the pipeline, which speci cally corresponds to the time points of 1, 2, and 3.At the same time, considering the constraints of the upper and lower walls of the pipe itself and the re ection e ect, the axial velocity of ame propagation is much greater than the radial velocity.Due to the di erence of the axial and radial propagation speeds, the combustion ame gradually stretches and lengthens in the axial direction and turns into an ellipsoidal shape.As the ame propagates, the curvature of the ame front and the highest local static temperature both decrease in the pipeline, obviously corresponding to time points 4 to 7.
For subsequent ame propagation characteristics, shown in Figure 4, the local maximum static temperature regions corresponding to time points 8, 9, and 10 are further reduced during the ame spread.In order to intuitively re ect the changing characteristics of subsequent ame propagation speeds, the time points of equal time intervals were used, from 11 to 25. Due to the impacts of the combustion shock wave and wall surface, it clearly shows that, as the ame continues to propagate backwards in the pipeline, its speed of propagation continues to decrease, and the curvature of the ame front is still decreasing in Figure 7, especially for the time points 23 and 25.In addition, the change regions of the high static temperature gradient are also concentrated near the thin layer of the ame front for the di erent followup moments of combustion.

POD Analysis for Computational Results.
In order to understand deeply the morphological characteristics of the ame propagation for de agration in the pipeline, the POD method is used here, speci cally for the initial stage of explosive combustion and typical subsequent propagation stages.
For the initial stage of POD analysis, the start time of the sample data is the time corresponding to the time point 1 in Figure 4, that is, the time of the explosion combustion at 0.001 s. en, taking 0.001 s as the equal time interval, a total of 600 full-order numerical transient solutions were selected in the order of the evolution of the explosion time to form the sample matrix for the initial phase analysis.
For the analysis of such stably subsequent propagation of ame, the initial time is selected as the 13 time point shown in Figure 4, the time interval is still set to 0.001 s and the same total of 600 full-order transient solutions.e sample matrix is in the same form as the initial stage.Because the follow-up ame propagation patterns of de agration have high similarity with each other, no other subsequent stages are selected for POD correlation analysis here.
e dimensionless ame propagation modes in the pipeline in the initial stage for the rst 9 orders are manifested in Figure 8, and the subsequent rst 9 orders dimensionless modes are shown in Figure 9.

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According to Figure 8, the static temperature gradient of the 1st-order mode of initial stage exhibits a wide range of continuity changes from left to right, and the cooling gradient intensi es as the rightward propagation distance increases.Furthermore, as the ame propagates backward, the frontal curvature of the cooling area decreases.e 2nd and 3rd modes both begin to show a low-temperature interval area in the right direction of the pipeline, and the left parts of such 2nd and 3rd modes exhibit the characteristics of opposite changes with orthogonality.From the 4th mode, the increase in the number of static temperature variation interval area in the pipeline occurs.With the increase of the mode order, the number of the interval regions shows an expanding in the form of a staircase, in which the 4th and 5th modes are 2 low-temperature interval regions, and the 6th and 7th orders are 3 such interval regions, as well as the 8th and 9th orders are 4 low-temperature interval regions.It is obvious that the ranges of the low-temperature interval and According to Figure 9, the subsequent ame propagation frontal morphology can be decomposed into multiple levels of independent spatial existence.e 1st-order mode still shows continuous gradient cooling characteristics; however, because the subsequent ame propagation front has a smaller impact range than the initial stage of combustion, the 1st-order mode static temperature variation range is also smaller than the initial stage of the same order mode.e 2nd-order mode has a low-temperature interval region, and it displays the orthogonality of the region corresponding to the right part of the 1st-order mode.e 3rd-order mode has a high-temperature cooling region after the low-temperature interval area.It is very interesting that   the total number of low-temperature interval and hightemperature cooling regions increases linearly as the mode order heightens from the 3rd to the 9th order.e further observations found that such 2 kinds of interval areas become more and more slender with the increase of the mode order.
To further analyze the quantitative e ect of these multilevel independent spatial modalities on the overall ame propagation morphological characteristics, according to such formula (11), the single-order content information contribution level of initial explosion and subsequent combustion propagation stages were both obtained, shown separately in Figures 10 and 11.
In the light of Figure 10, the in uence of the 1st-order mode information ratio in the initial combustion stage is almost 50 percentage; therefore, it has the greatest impact on the overall ame propagation pattern.e 2nd-order to the 9th-order modes also have an important in uence on the integral ame propagation shape, in which the smallest modal e ect ratio of 9th order has exceeded 1.2 percentage; especially for the 2nd and 3rd modes, they approximately reached 10.4 and 6.69 percentages, respectively.e in uence of such 50th-order is relatively negligible, only about 0.08 percentage.
en, with the analysis of Figure 11, the multilevel spatial modal information in the subsequent propagation stage is generally consistent with the distribution of the in uence characteristics in the initial stage.However, the proportion of the 1st-order modal information exceeds 78.3 percentage, which is obviously higher than the in uence of the same order proportion in the initial stage.Moreover, the levels of the 2ndorder to the 9th-order modal information content all are  Order number for the modes Single-order content information contribution 1st mode: 78.32% 2nd mode: 6.27% 3rd mode: 3.02% 4th mode: 1.89% 5th mode: 1.33% 7th mode: 0.77% 8th mode: 0.61% 9th mode: 0.49% 50th mode: 0.03% 6th mode: 0.99%  Advances in Civil Engineering lower than the same order modes of the initial stage, respectively.For the 50th-order mode, the information content is also signi cantly lower than the initial explosion stage.e modal cumulative content information contribution characteristics of the initial explosion stage and subsequent propagation stage are shown in Figures 12 and 13, respectively.For the rst 9 modes, the latter 93.7 percentage is signi cantly higher than the former 81.89 percentage.To the rst 50 modes, the latter 97.39 percentage is still higher than the former 95.54 percentage.
Based on the above information content analysis, the level of information content in the multilevel modal space of de agration is mostly concentrated in the rst 3 modes, especially for the 1st-order mode.Moreover, the modal information content concentration level of such subsequent propagation stage is higher than the initial explosion stage.erefore, it shows that the ame propagation pattern at the initial explosion stage is more complicated than the subsequent stage, and for more detailed research, the initial stage needs to involve more high-information content modalities, not only the rst 9 modes.

Conclusions
A multilevel independent spatial modal analysis of the ame propagation characteristics of a de agration in a speci c pipeline was performed, using such proper orthogonal decomposition method.In order to implement the POD-CFD de agration analysis, an interface program was developed, which can realize the conversion from the CFD binary

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output files to the POD calculation basic data structure and also includes those visualizations of numerical results.e full-order numerical calculation results manifested that the local maximum static temperature exceeded 2400 K in the initial stage of the explosion combustion, and the combustion flame gradually stretched and lengthened in the axial direction and turns into an ellipsoidal shape because of the difference of the axial and radial propagation speeds in the pipeline.Moreover, the curvature of the flame front and the highest local static temperature both decreased as the flame propagated.For subsequent flame propagation characteristics, the local maximum static temperature regions were further reduced and the corresponding speed of propagation and the curvature of the flame front continued to decrease, especially for the late stage of combustion.In general, the full-order calculation results well agreed with the normal combustion propagation characteristics of premixed methane-air for the flame propagation with the unbroken thin layer.erefore, the above numerical calculation results are reasonable and effective.
e POD analysis results showed that the static temperature gradient of the 1st-order mode of the initial stage exhibited a wide range of continuity changes from left to right and the frontal curvature of the cooling area decreased as the flame propagated backward.
e higher modes all showed the low-temperature interval area in the right direction of the pipeline and some of them with opposite changes in orthogonality.
e number of the lowtemperature interval regions displayed an expanding in the form of a staircase with the increase of the mode order.For the subsequent flame propagation frontal morphology, the 1st-order mode still showed continuous gradient cooling characteristics; however, the 1st-order mode static temperature variation range is smaller than the initial stage of the same order mode.e further observations found that the interval areas became more and more slender with the increase of the mode order.According to the modal information content analysis, the level of information content in the multilevel modal space of deflagration is mostly concentrated in the first 3 modes, especially for the 1st-order mode, and the flame propagation pattern at the initial explosion stage is more complicated than the subsequent stage.
Furthermore, this type of interface program is also suitable for POD analysis of deflagration in pipelines with obstacles, which is a next plan for further analysis of deflagration.Due to the influence of the obstacle on the fine structure of the deflagration flame in the pipeline, it will cause secondary deflagration with the coal dust cloud and strengthen the turbulence of the internal flow field of the flame, thereby accelerating the propagation of the flame in the pipeline.erefore, it is necessary to make a deeper analysis of the relevant situation.In the subsequent analysis, the interface program with practical value of engineering analysis will also be further developed.

Figure 1 :
Figure 1: Physical model of de agration process in the pipeline.

Figure 2 :
Figure 2: Grid layout of de agration for the pipeline.

Figure 4 :Figure 5 :
Figure 4: Flame propagation process of de agration upon static temperature in the pipeline.

Figure 6 :
Figure 6: Flame propagation characteristics of de agration at the initial stage.

Figure 7 :Figure 8 :Figure 9 :
Figure 7: Flame propagation characteristics of de agration in the subsequent stages.

Figure 10 :
Figure 10: Single-order content information contribution for the rst 50 modes in the initial stage of ame propagation.

Figure 11 :
Figure 11: Single-order content information contribution for the rst 50 modes in the subsequent stage of ame propagation.

Figure 12 :Figure 13 :
Figure 12: Cumulative content information contribution for the rst 50 modes in the initial stage of ame propagation.