Numerical Study on Premixed MethaneâAir Flame Propagation in a

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Numerical study on premixed methane-air flame propagation in a confined vessel at low initial temperature Gan Cui, Zili Li, Hongbo Li, Zhenxiao Bi, and Shun Wang Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b03433 • Publication Date (Web): 28 Dec 2017 Downloaded from http://pubs.acs.org on December 31, 2017

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Numerical study on premixed methane-air flame propagation in a confined vessel at low initial temperature Gan Cui*, Zili Li*, Hongbo Li, Zhenxiao Bi, Shun Wang College of Pipeline and Civil Engineering in China University of Petroleum (East China), Qingdao 266580, China Shandong Provincial Key Laboratory of Oil & Gas Storage and Transportation Safety, Qingdao 266580, China Qingdao Key Laboratory of Circle Sea Oil & Gas Storage and Transportation Technology, Qingdao 266580, China

ABSTRACT: In this paper, premixed methane-air flame propagation in a confined vessel at low initial temperature was simulated using a multi-step chemical reaction mechanism. The confined vessel was a cylinder with aspect ratio of 3 with asymmetrical position of the ignition source near the side cover. The equivalence ratio and the initial temperature of the premixed unburned combustible gas were 1.0 and 150 K, respectively. The overall evolution of the flame and the flame dynamics were obtained respectively. Through the entire flow field variation, vortex movement and pressure wave propagation characteristics during the whole process of combustion, the flame propagation mechanism of methane combustion at low initial temperature was established finally. Results indicate that five stages are divided during the methane combustion in a confined vessel: spherical flame propagation, “fingertip” shaped flame propagation, flame “skirt edge” contacts the side wall, “crescent” flame propagation and typical “tulip” flame propagation. In the process of flame propagation, the reverse of the flame front and formation of the “tulip” flame can be immediately contributed to the interaction of the flame front, flame induced reverse flow and vortex motion. However, the pressure wave propagation back and forth along the flame propagation direction has no obvious effect on the formation of “tulip” flame. When the distorted “tulip” flame is formed, vortex motion is not observed. The formation of the distorted “tulip” flame is caused by the superposition of the secondary pressure wave formed by the contact of the flame with side wall. However, because of the low intensity of pressure wave, RT instability is weak, and the distortion of flame front is not obvious. Flame propagation velocity and pressure wave are interacted with each other. In the process of combustion, the variation of flame propagation velocity and pressure rise rate show almost the same phase. The increase in flame propagation velocity directly leads to the increase in pressure rise rate, whereas the pressure wave propagation back and forth in the confined vessel leads to the oscillation of propagation velocity. Keywords: Methane; Premixed flame propagation; Flame dynamics; Low temperature 1. Introduction With the increasing emphasis on environmental protection and demand for energy, and the increasing shortage of traditional energy such as oil, the development and utilization of LNG and oxygen-bearing coal-bed methane show significant

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economic and social benefits 1-9. However, in the LNG storage and oxygen-bearing coal-bed methane liquefied production, the methane concentration in the gas phase may be within the flammability limits, causing a potential explosion hazard and resulting in significant economic losses and casualties 10-12. Regardless of the combustion caused by LNG leakage or low temperature liquefaction of oxygen-bearing coal-bed methane, there is a common ground: the initial temperature of the combustion is cryogen (typically lower than 150 K). Under low temperature conditions, the combustion characteristics and mechanism are bound to be very different from those at room temperature, and there is a lack of research on gas combustion at low initial temperature. Therefore, the study of methane combustion at low initial temperature is of great safety significance for the storage of LNG and the liquefied production of oxygen-bearing coal-bed methane. The flame propagation characteristics of gas combustion in confined space are a very important area in combustion science and technology, which have been studied extensively by experiments, theories and numerical simulations 13-23. However, most of the researches are only about the dynamic instability of the flame and the interaction between the flame and the acoustic wave. The study of the flame acceleration mechanism (the interaction effect of the flame velocity and the pressure wave) and the shape variation of the flame during the propagation process are relatively limited. Clanet and Searby et al. 14 studied the combustion process in a gas pipeline, and observed the "tulip" flame in the flame propagation process. Then, they divided the flame propagation process into four stages, and gave the corresponding empirical formulas based on the characteristic time of different stages. Based on the study of Clanet and Searby et al. 14, by analyzing the flow field and flame dynamics, Bychkov et al. 17 obtained the theoretical model of flame formation and the time prediction model in the "tulip" flame formation. This theory provided the theoretical calculation formulas for the characteristic time corresponding to the four stages divided in the flame propagation process, the propagation distance of the flame front, the variation of acceleration in flame propagation, and the variation of the flame front area, respectively. Xiao et al. 24 studied the flame propagation characteristics of hydrogen-air mixture in a horizontal closed tube using experimental tests and numerical simulations (using single-step chemical reaction models). The results showed that the flame propagation was divided into four stages: spherical flame, "finger-tip" flame, flame "skirt" touching side wall and "tulip" flame. Then, Xiao et al. 25 studied the propagation characteristics of hydrogen-air premixed combustion flames in a sealed combustor using experimental measurements and numerical simulations. In the numerical simulation, a detailed 19 steps chemical reaction mechanism and flame thickening model were used. The results showed that "tulip" flames and distorted "tulip" flames appeared during flame propagation, and Taylor's instability was the cause of the formation of distorted "tulip" flames. Furthermore, Xiao et al. 26 studied the flame propagation characteristics of a stoichiometric hydrogen-air mixture in a closed channel using a fully compressible Navier-Stokes equation, a high-order numerical simulation method and a single-step Arrhenius reaction model. The main contents of the study included: (1) Formation, propagation


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and effects of pressure waves on the flame propagation; (2) Formation and disappearance of the "tulip" flame, and the effect of the disappearance of "tulip" flame tip on the flame propagation; (3) Distorted "tulip" flame formation and propagation mechanism. As described above, Xiao et al. 24-26 studied the variation of flame shape using experimental and numerical simulation methods, and explained the formation mechanism of each propagation stage. However, for methane and hydrogen, the detailed chemical reaction mechanism, the thermophysical parameters and the chemical reactivity are quite different. Thus, the flame propagation process of methane is bound to be very different from that of hydrogen. About the "tulip" flame and distorted "tulip" flame formation mechanism, Xiao et al. gave their own explanations. However, different researchers have different explanations on this mechanism. For instance, viscous and quenched effects 27,28, interaction of flame front and pressure waves 29, DL/hydrodynamic instability 30-32, circulating flow of burned gases 33-35 and Taylor instability 14. To date, which mechanism can be an accurate explanation is not conclusive. In addition, the different initial temperature results in the difference of flame propagation dynamics. Therefore, the flame propagation characteristics of methane combustion at low initial temperature are worthy studying further. The purpose of this paper is to study premixed methane-air flame propagation in a confined vessel at low initial temperature using simulation method. The confined vessel is a cylinder with aspect ratio of 3 with asymmetrical position of the ignition source near the side cover. The equivalence ratio and the initial temperature of the premixed unburned combustible gas are 1.0 and 150 K, respectively. The overall evolution of the flame and the flame dynamics (such as combustion pressure, flame propagation velocity and flame front position) are obtained respectively. Through the entire flow field variation, vortex movement and pressure wave propagation characteristics during the whole process of combustion, the flame propagation mechanism of methane combustion at low initial temperature is discussed finally. 2. Mathematical model 2.1 Governing equations Many chemical elementary reactions, physical flow and mass transfer process occur during combustion. The combustion process is described by the following equations 36, 37.

ρRT M

(1)

∂ρ ∂ ( ρu ) ∂ ( ρv ) + + = 0 ∂t ∂x ∂y

(2)

p =

∂( ρu ) ∂ ( ρu 2 ) ∂ ( ρuv) ∂τ xx ∂τ yx ∂p + + = + − ∂t ∂x ∂y ∂x ∂y ∂x


(3)

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∂ ( ρv) ∂ ( ρv 2 ) ∂( ρuv) ∂τ yy ∂τ xy ∂p + + = + − ∂t ∂y ∂x ∂y ∂x ∂y

(4)

∂ ( ρE ) ∂[u ( ρE + p )] ∂[v( ρE + p )] ∂ (uτ xx + vτ yy − q x ) ∂ (vτ yy + uτ yx − q y ) + + = + + qw& (5) ∂t ∂x ∂y ∂x ∂y

∂ ( ρY ) ∂ ( ρuY ) ∂ ( ρvY ) ∂ ∂ Y ∂ ∂ Y + + = ( ρD )+ ( ρD ) - w& ∂t ∂x ∂y ∂x ∂x ∂y ∂y

(6)

τ xx =

4 ∂u 2 ∂υ µ − µ 3 ∂x 3 ∂y

(7)

τ yy =

4 ∂v 2 ∂u µ − µ 3 ∂y 3 ∂x

where,

τ xy = τ yx = µ

qx = −K

E=

∂u ∂v +µ ∂x ∂y

∂T ∂T , q y = −K ∂x ∂y

p 1 + (u 2 + v 2 ) (γ − 1) ρ 2

k = K /(qc p )

(8)

(9)

(10)

(11) (12)

c p = γR /(γ − 1)

(13)

µ = µ 0T n , D = D0T n , k = k 0T n

(14)

w& = dY / dt = AρY exp(

- Ea ) RT

(15)

2.2 Physical model In this paper, the calculation model is a cylindrical confined vessel, which can be simplified as a two-dimensional model to deal with, as shown in Fig.1 (a). The two-dimensional model has been proved to be effective to model the flame propagation 28, 38-40. The length X is 0.3 m and the width Y is 0.1 m. The positive direction of the X axis represents the direction of the flame propagation. In the simulation, the size of the grid directly affects the accuracy of the calculation results and the calculation time. If the grid is too rough, the calculation time is greatly shortened, while the variation of the flame shapes and pressure waves cannot be well captured. If the grid is too dense, and also because of the detailed chemical reactions


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considered in the calculation, the computational time is exponentially increased. Therefore, before the calculation, the effect of grid size on the calculated results should be studied first. A uniform grid is used and four grid sizes are set: 0.5 mm, 1 mm, 2 mm and 3 mm, respectively. The meshing of 1 mm size is shown in Fig.1 (b). Fig.2 shows the variation of the combustion pressure with different grid size. It can be seen from Fig.2 that the different grid size leads to a large different variation of the combustion pressure with time. However, when the grid size is reduced from 1 mm to 0.5 mm, the evolution of combustion pressure varies little, but the calculation time increases a lot. Therefore, in order to ensure the accuracy of computation and decrease the calculation time, the final selection of the grid size is 1 mm.

(a) (b) Figure 1. (a) Computational model; (b) Mesh generation

Figure 2. Variation of the combustion pressure with different grid size

2.3 Numerical method A 20-step detailed chemical reaction mechanism 41 is employed in the numerical simulation. For the combustion problem in a small confined vessel, we believe that the combustion is completed in a very short time. Therefore, the four walls of the computational model shown in Fig.1 (a) are set to adiabatic boundaries. At the initial moment, the pressure is 0.1 MPa and the temperature is 150 K. In the case of the equivalence ratio of 1, the mass fraction of methane is 0.053, the oxygen mass fraction is 0.21, and the nitrogen mass fraction is automatically calculated in FLUENT. The mass fractions of the remaining components are zero. The initial velocity in the X


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and Y directions is also zero. The ignition source can be set as a high temperature air mass. A circle with a radius of 0.005 m is created, and the gas in the circular area is composed of the reactants CH4, O2 and N2. In the initialization step, the circular area is patched to a high temperature (such as 1400 K) to achieve the ignition source setting, as shown in Fig.1 (a). As the combustion is completed in a very short time, the buoyancy effect is neglected which is proved by Liu et al. 42 and Xiao et al. 24. Another aspect, at the initial stage of combustion the Froude number Fr=u2/(gH)≈2, and Fr≈28 at the ignition time of 20 ms. The Froude number Fr is larger than 1 which indicates in flame propagation, inertial force is greater than gravity. Therefore, under the research condition of this paper, it is reasonable to neglect the gravity effect, which is also proved by Xiao et al. 38, 43. The relevant governing equations are discretized by the finite volume method. The turbulence equation is based on the standard k-ε model, and the non-equilibrium wall function method and the volume reaction are adopted. In this paper, the combustion process of methane is accompanied by the motion of vortices, and the combustion chemical reaction mechanism of methane is coupled in the calculation of hydrodynamics. Therefore, the eddy dissipation concept (EDC) model is chosen to calculate. The grid size is set as 0.001 m and the time step is 1e-05 s. The SIMPLEC method is used for coupling the pressure and velocity fields. Second-order upwind scheme is used for the convective terms, and the time is advanced by the fully implicit Euler method. The time stepping method is successive over-relaxation. The relaxation factor of pressure, density, momentum and energy is set to 0.3, and the relaxation factor of each component is set to 1. 3. Results 3.1 Variations in shape during flame propagation In general, due to hydrodynamic instability, the shape of the flame throughout the combustion process will continue to change. Fig.3 shows variation of the flame shape of methane combustion at low initial temperature, where the flame corresponds to the temperature field. At the beginning of the ignition, corresponding to 0 ms and 2 ms in Fig.3, the flame core is small which is not affected by the upper and lower ends of the wall. The flame is free to spread around in spherical shape. As the flame area increases, the flame propagation begins to accelerate. As the flame front gradually approaches the upper and lower ends of the wall, the propagation of the flame in the Y direction is limited, while in the X positive direction, the flame is still free to spread. At this moment, the flame is gradually elongated in the X direction and becomes "fingertip" shape. The above stages correspond to the 5 ms, 10 ms and 17 ms in Fig.3. Subsequently, the "skirt" of the elongated flame contacts the side wall, and the flame side is quenched on the wall. Correspondingly, the flame surface area is gradually decreased. The reduction of the flame surface area directly leads to a decrease in the propagation velocity of the flame center. As the flame continues to propagate, the radius of curvature of the flame front increases gradually until the flame front


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becomes a plane, corresponding to 21 ms, 25 ms, 30 ms, and 35 ms. After the flame front becomes flat, the center of the flame front begins to sink, and the flame front becomes "crescent" and propagates forward, corresponding to 38 ms, 45 ms, 50 ms and 60 ms. As the flame continues to spread, the tip of the flame front center continues to extend to the burned area, while the tip of the two symmetrical "flames" near the wall continues to extend to the unburned zone. Thus, a typical “tulip” flame is formed. Thereafter, the flame propagates in a "tulip" shape until the end of the combustion, corresponding to 70 ms, 80 ms and 90 ms. In summary, according to the flame shape variation, the flame propagation of methane combustion at low initial temperature is divided into five stages: spherical flame propagation, “fingertip” shaped flame propagation, flame “skirt edge” contacts the side wall, “crescent” flame propagation and typical “tulip” flame propagation.


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Figure 3. Variation of the flame shape of methane combustion under low temperature

Fig.4 shows an enlarged view of the flame front at different combustion times. The red area is the burned area, the blue area is the unburned area, and the boundary between the red area and the blue area is the flame front. At t=60 ms, the flame front is a smooth surface; t=65 ms, a small wrinkle appears at the tip of the flame front. However, the wrinkle is not obvious. As the flame propagates, the wrinkle of the flame front tip becomes noticeable. At t=70 ms, 75 ms, 80 ms and 85 ms, there is a significant wrinkle at the tip of the flame front, as shown in the ellipse in Fig.4. As the flame continues to propagate, the wrinkle begins to disappear at t=87 ms and disappears completely at t=88 ms. Then, the flame front becomes smooth again. In summary, in the process of flame propagation, after the formation of “tulip” flame, wrinkle will appear in the smooth flame front, which is called the distorted “tulip” flame. The formed mechanism of the “tulip” flame and the distorted “tulip” flame will be discussed in the subsequent chapter.


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Figure 4. Evolution of the distorted “tulip” flame

3.2 Flame propagation dynamics 3.2.1 Evolution of the combustion pressure Fig.5 shows the variation of combustion pressure with time at different monitoring points. The monitoring points are selected as the central position on the left side wall (X=0 m), the central position of the confined vessel (X=0.15 m) and the central position on the right side wall (X=0.3 m). In Fig.5, the pressure continuously increases with ignition time. At the beginning of the ignition, the pressure rise is slow; when t=60 ms, the combustion pressure increases rapidly with time; when t=110 ms, the combustion pressure reaches the maximum. Since the model is set to adiabatic boundaries, the combustion pressure will remain unchanged after combustion is completed. It also can be seen that the pressure rise curves at different monitoring points are completely coincident with each other, indicating that the combustion pressure in the whole confined vessel reaches uniform in a very short period of time. Thus, in subsequent studies, only the pressure at the central position on the right side wall is selected.


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Figure 5. Variation of the combustion pressure at different monitoring points

Fig.6 shows the variation of the combustion pressure rise rate with ignition time. In the whole combustion process, there appear two peaks in the pressure rise rate curve. After ignition, the flame propagates in a sphere and a "fingertip" shape, and the flame is continuously stretched. The flame area increases, and the combustion rate increases accordingly, resulting in a gradual increase in the pressure rise rate. Subsequently, due to the elongated flame "skirt" contacts the side wall and quenches, the flame area decreases and the combustion rate slows down. Correspondingly, the pressure rise rate gradually decreases. When the flame front becomes a plane, the flame front begins to reverse and the flame surface area gradually increases again. The combustion rate starts to increase, resulting in an increase in the pressure rise rate. The pressure rise rate reaches the maximum at t=96 ms. Thereafter, due to the flame front reaches the right side of the confined vessel and quenches, the flame area is greatly reduced and the pressure rise rate drops to zero in a relatively short period of time. There are two special areas in the pressure rise rate curve: the oscillation of the curve in the range of 0 to 40 ms and the large oscillation in the red box A in Fig.6 (a), i.e. in Fig.6 (b). Before the flame front reverses, the pressure rise rate is slow and the flame front is far away from the right side wall of the confined vessel. Thus, the pressure wave travels to the right side wall for a longer period of time. The pressure wave is reflected after reaching the wall, which results in the oscillation of the pressure rise rate at the start of the ignition. After the flame front reverses, the pressure rise rate increases, and the distance between the flame front and the right side of the wall decreases. The time when the pressure wave propagates to the right side of the wall and reflects is greatly reduced, which is much less than the monitoring time interval 0.5 ms. As a result, the pressure rise rate curve recorded after the reversal of the flame front does not oscillate significantly. More interestingly, the pressure rise rate curve has a large oscillation, as shown in Fig.6 (b). This oscillation occurs only in a short time interval (1 ms) and has a large amplitude, which is due to the sudden disappearance of the distorted “tulip” flame. This will be discussed in detail in later chapter.


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(a) (b) Figure 6. (a) Variation of the combustion pressure rise rate; (b) Variation of the combustion pressure rise rate at the box A in (a)

3.2.2 "Tulip" Flame Front Propagation Distance Fig.7 shows evolution of the flame tip and “tulip” flame cusp positions and the distance between them after the reversal of flame front. We define the flame tip as the two tips of "tulip" flame near the side wall, while the “tulip” flame cusp as the center of the flame front. As can be seen from Fig.7, the flame front becomes a flat at t=40 ms after ignition, and the distance between the flame tip and “tulip” flame cusp is zero. As the combustion continues, the flame tip and “tulip” flame cusp position gradually increase, and the distance between them also gradually increases. When t=80 ms, the distance between the flame tip and “tulip” flame cusp reaches the maximum, which is about 0.072 m. At this moment, "tulip" flame completely forms. As the flame continues to propagate, the flame tip and “tulip” flame cusp position still increase. However, the distance between them gradually decreases, indicating that the “tulip” flame cusp propagation speed is larger than that of the flame tip. As the time continues to increase, the distance between the flame tip and the “tulip” flame cusp continues to decrease until the end of the combustion.

Figure 7. Evolution of the flame tip and “tulip” flame cusp positions and the distance between them


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3.2.3 Relationship between flame propagation velocity and flame front position Fig.8 shows the relationship between the flame front center position and the propagation velocity. In general, the propagation velocity of the flame front center has two peaks in the whole combustion process, and the peak value is basically the same, which indicates that the maximum flame area in the “fingertip” propagation process is basically equal to that in the “tulip” flame propagation process. There is an inflection point in the variation curve of the flame center position with time. The appearance time of the inflection point is 30 ms and the corresponding position is 0.12 m. On the left side of the inflection point, the increase rate of the flame center position is significantly greater than that on the right side of the inflection point. At the initial moment of ignition, the flame propagation speed is 1 m/s when the initial temperature is 150 K. After ignition, the flame propagation velocity increases rapidly with time, and because of the oscillation of pressure wave, the flame propagation velocity also appears obvious oscillation. The flame center velocity reaches the first peak at t=20 ms at a speed of 3.7 m/s. At this moment, the flame center position remains on the left side of the inflection point. The reason for the acceleration of the flame propagation at this stage is: flame propagates in a spherical and a "fingertip" shape, which is stretched in the X direction. The flame area increases, resulting in acceleration of the combustion rate, and correspondingly, the flame propagation velocity increases. As the combustion continues, the flame propagation velocity decreases rapidly with time and reaches zero at t=35 ms. The decrease in flame propagation velocity is due to the contact of the elongated flame "skirt" with the side wall. The combustion rate decreases, which eventually results in a decrease in the flame propagation velocity 14, 17. There is an intersection between the flame propagation velocity curve and the flame center position curve, and the intersection is exactly the inflection point of the flame center position curve. This indicates that when the flame propagation velocity begins to decrease, the flame center position increase rate does not decrease, and the flame front does not reverse. After the flame center velocity reaches to zero, the flame front begins to reverse, and the "tulip" flame begins to form. Thereafter, the flame front area continues to increase, leading to further increase of the flame propagation speed. At t=93 ms, the flame propagation velocity curve reaches the second peak, where the "tulip" flames are completely formed. Subsequently, due to the disappearance of the "tulip" flame, the flame center propagation velocity decreases slowly.


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Figure 8. Relationship between the location and velocity in the middle of the flame front

Fig.9 shows the relationship between the location and velocity of the flame tip after the flame front is reversed. In this study, the flame tip is defined as the tip of the "tulip" flame near the side wall. Overall, the flame tip propagation velocity decreases with time. The position of the flame tip increases continuously with time. However, due to the gradual decrease in propagation velocity, the rise rate of the flame tip location is gradually decreasing. As also can be seen from Fig.9, in the flame propagation process, the flame tip velocity shows severe oscillation. In order to explore the reason of this large oscillation, variations of the pressure field with time when the flame surface contacts the side wall is studied. Fig.10 shows the evolution of pressure fields with time after flame contacts the side walls. In the figure, the black curve represents the flame front. At t=60 ms, the flame front tip contacts the wall and a hemispherical pressure wave centered on the tip is formed. The magnitude of this hemispherical pressure wave is the minimum in the entire pressure field. As the combustion continues, t=60.05 ms, 60.10 ms to 60.15 ms, the hemispherical pressure wave expands outward and the pressure wave intensity gradually increases. At t=60.20 ms, the pressure wave at the tip position reaches the maximum in the entire pressure field. Thereafter, as the flame tip continues to propagate (t=60.25 ms), a tip-centered hemispherical pressure wave is formed again. The value of the pressure wave is the lowest in the entire pressure field. Based on the above analysis, it can be seen that during the propagation of the flame tip, a pressure wave is generated at the tip position, and the pressure wave is oscillating in the whole pressure field. The oscillation of the pressure wave has an effect on the propagation velocity of the flame tip. Therefore, the propagation velocity of the flame tip also oscillates significantly.


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Figure 9. Relationship between the location and velocity of the flame tip

(a)

(b)

(c)

(d)


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(e) (f) Figure 10. Evolution of pressure fields when flame contacts the side walls (a) t=60 ms; (b) t=60.05 ms; (c) t=60.10 ms; (d) t=60.15 ms; (e) t=60.20 ms; (f) t=60.25 ms

4. Discussion 4.1 Formation of the “tulip” flame Fig.11 shows the relationship between the velocity vector field and the flame shape of methane combustion at low initial temperature in confined vessel. In the figure, the red area represents the burned area, and the boundary of the red area is the flame front. The white arrow represents the velocity vector. The direction of the arrow indicates the direction of the velocity, and the length of the arrow represents the magnitude of the velocity. In this study, if the direction of the flow is opposite to the direction of the flame propagation, it is defined as the reverse flow. Studies have shown that the presence of baroclinic effect makes the fluid flow in the burned zone spin 22, 33. At this point, the Navier-Stokes equation can be expressed by the vortex equation shown in equation (16). r r r r r ∇p dΩ − (Ω ⋅ ∇v ) + Ω(∇ ⋅ v ) = −∇ × ( ) (16) dt ρ

r r Where, Ω = ∇ × v is the vortex. The source term in equation (16) can be expressed by equation (17). −∇×(

∇p

ρ

)=

∇p × ∇ρ

ρ2

(17)

It can be seen that the right side of the equation (17) is the product of the pressure gradient and the density gradient vector. Thus, if the pressure gradient and the density gradient are at an angle, there will be a vortex generated in the burned area near the flame front. That is, if the flame surface is wrinkled or tilted due to disturbance, a small vortex will occur near the flame surface. Many small vortices are superimposed on each other to form a large scale vortex, and the movement of the large scale vortex causes the shape of the flame front to change constantly in the process of propagation 34. It can be seen from Fig.11 that the velocity vector direction in the flow field


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coincides with the flame propagation direction before the flame "skirt" contacts the side wall, corresponding to t=10 ms and 20 ms. In the burned area and the right side of the confined vessel, the velocity vector is very small, indicating that the fluid has almost no movement. In the flame front, the velocity vector is large and reaches the largest in the flame front center, indicating that the flame is accelerating in propagation. Due to restrictions by the upper and lower ends of the wall, the flame is slowly stretched. After the flame "skirt" contacts the wall, corresponding to t=30 ms in the figure, the velocity vector direction is perpendicular to the wall surface near the point where the flame "skirt" contacts the wall, and a weak reverse flow is formed in the burned zone. At the same time, a small vortex is formed at the burned zone far away from the flame front (X=0.03 m). At this moment, the velocity vector at the center of the flame surface is still consistent with the flame propagation direction, while the velocity is small. However, the flame propagates faster near the point where the flame "skirt" contacts the wall. From t=30 ms to 36 ms, the small vortices formed in the burned zone gradually increases and propagates forward rapidly. Later, two large vortexes behind the flame front are gradually developed and the radius of curvature of the flame front also gradually increases. At the same time, the reverse flow velocity of the fluid in the burned zone is increased. At t=36 ms, the velocity of the flame front is almost zero, and the reverse flow velocity behind the flame is large. As the flame continues to propagate, t=45 ms, the direction of the velocity field in the flame front center is opposite to the flame propagation direction, and the velocity is fast. At the same time, the velocity vector direction of the flow field near the wall is the same as the flame propagation direction, and the propagation velocity is also fast. Therefore, the propagation velocity of the flame front center is much smaller than the velocity of the flame near the wall, which directly leads to the formation of the "tulip" flame. In the period after the formation of the "tulip" flame, corresponding from t=36 ms to 45 ms, the vortex continues to move toward the flame, and the burned zone is gradually occupied by the reverse flow. As the flame continues to propagate, corresponding to the process from t=45 ms to 90 ms in the figure, the "tulip" flame shape becomes more and more obvious and the burned area is completely reverse flow with large velocity vector. At the same time, the vortex is no longer moving to the direction of flame propagation, while moves backward. When the "tulip" flame is close to the right end of the confined vessel, the vortex moves to a position close to the left side of the confined vessel and gradually disintegrates. At this point, the flow field in the entire confined vessel is reverse. In summary, in the flame propagation process of methane combustion at low initial temperature, the reverse flow of the fluid and the vortex movement are induced in the burned area. The kinematic properties of the fluid have a significant effect on the flame dynamics. The numerical results show that in the flame propagation process, the reverse of the flame front and formation of the “tulip” flame can be immediately contributed to the interaction of the flame front, flame induced reverse flow and vortex motion.


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Figure 11. Relationship between velocity vector fields and flame shapes

4.2 Formation of the distorted “tulip” flame Fig.12 shows the variation of the streamlines in the process of formation, propagation and disappearance of the distorted “tulip” flame, where the white lines are the streamlines. At t=60 ms, the streamlines of the flame front at the tip of the "tulip" flame is directed to the unburned zone, and the reversal of the streamline occurs in the burned zone. At t=65 ms, the reversal of the streamline moves toward the flame front, and the flame tip front begins to reverse. At t=70 ms, it is clear that a streamline directs from the unburned zone to the burned zone at the flame tip front. The flame front where the streamline passes through will be reversed, resulting in a depression. With the propagation of the flame, at t=75 ms, 80 ms and 85 ms, the reversal of the streamline gradually coincides with the flame front. The streamline at the depression of flame front still points to the burned zone and the streamline at the flame tip points to the unburned zone, which results in the elongation of the flame tip and increases the degree of distortion in the flame. As the flame continues to propagate, the boundary of the reverse line exceeds the flame front, and the streamlines passing through the flame front point to the burned areas, causing the disappearance of the distortion of flame front. Since the direction of the streamline is consistent with that of the velocity vector, the reverse flow induced at the flame tip front is the direct cause of the formation of distorted "tulip" flame. However, during


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the formation of the distorted “tulip” flame, vortex is not observed to form behind the flame tip. Therefore, the formation mechanism of the distorted “tulip” flame is different from that of the typical “tulip” flame. It has been mentioned that during the flame propagation process, a pressure wave is generated when the flame front tip contacts the side wall. The interaction between the pressure wave, the flame front and the propagation velocity vector is the root cause of the formation of distorted "tulip" flame. In Fig.6, the time of the large oscillation in the pressure rise rate curve is between 87 and 89 ms, and in Fig.12, the disappearance of the flame front distortion occurs at about 88 ms. In the literature 26, it is pointed out that when the boundary is a thermal wall, the disappearance of the distorted "tulip" flame tip corresponds to the contact of the flame with the side wall. Furthermore, in Fig.10, secondary pressure wave will be generated when the flame contacts the side wall, and it is oscillating in the flame propagation process. Thus, the large oscillation of the pressure rise rate in Fig. 6 (b) is due to the disappearance of the distorted "tulip" flame.

Figure 12. Variation of the streamlines during the evolution of distorted “tulip” flame


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4.3 Interaction analysis of flame propagation velocity and pressure rise rate Fig.13 shows the relationship between the velocity of the flame front and pressure rise rate. It can be seen that the flame propagation velocity and the pressure rise rate curve show a good consistency: there are two peaks in the whole combustion process. The difference is that the time when the flame propagation velocity reaches the peak is shorter than that when the pressure rise rate reaches the peak. After ignition, the increase in the flame area leads to an increase in the combustion rate, and the increase in the combustion rate leads to an increase in the combustion pressure rise rate. Due to the limitation of the wall, the pressure wave propagates back and forth in the confined vessel, resulting in the oscillation of the flame propagation velocity. As the flame "skirt" contacts the side wall, the flame area decreases, resulting in a decrease in flame propagation velocity. At this time, due to the secondary pressure wave generated by the contact of the flame with the side wall, the pressure rise rate will not immediately decrease, but continues to increase. As the flame continues to contact the wall, the flame surface area is further decreased. Then, the combustion rate is greatly reduced, eventually leading to a decrease in the pressure rise rate. At the initial stage of the second increase in flame propagation velocity, the flame propagation velocity is almost zero. At this time, the secondary pressure wave generation rate is low, resulting in further decrease in the pressure rise rate. As the combustion continues the flame propagation velocity increases gradually, resulting in an increase in the generation rate of secondary pressure waves, which eventually leads to the second increase in the pressure rise rate. Near the end of combustion, as the flame area is greatly reduced, the flame propagation velocity and the pressure rise rate almost simultaneously decrease. In summary, in the process of combustion, the variation of flame propagation velocity and pressure rise rate show almost the same phase. The increase in flame propagation velocity directly leads to the increase in pressure rise rate, whereas the pressure wave propagation back and forth in the confined vessel leads to the oscillation of propagation velocity. Therefore, the flame propagation velocity and the pressure wave are interacted.

Figure 13. Relationship between the velocity of the flame front and pressure rise rate


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4.4 The mechanism of pressure wave Throughout the combustion process, the pressure wave can be generated in the following ways: ignition, flame contacts side wall and distorted "tulip" flame disappears. These pressure waves are presented in the form of expansion wave and reflected by the wall and the flame front, resulting in the back and forth propagation in the confined vessel. In general, the interaction between the pressure wave and the flame causes flame instability (including variations in the flame shape and propagation velocity) 44-48. Here, we analyze the mechanism of the pressure wave to gain a deeper understanding of the interaction between the pressure wave, the flow field and the flame front. Fig.14 (a) shows the vertical pressure profiles at X=0.225 m and different times. In order to show the pressure fluctuation more clearly, the pressure is normalized, that is, the pressures in the Y direction are divided by the pressure P0 which is the pressure at the lower surface of the side wall. The selected time is the moment when the flame front tip reaches X=0.225 m. As has been shown in Fig.10, it is known that a pressure wave is generated when the flame contacts the wall. Corresponding to Fig.14 (a), at t=61.05 ms, the flame tip begins to contact the wall and two large pressure peaks are generated on the wall, which is equivalent to occurrence of a slight explosion. The pressure wave is initially propagated in a spherical shape, and the two pressure peaks move together toward the center. At t=61.15 ms, the two pressure peaks coincide at the center position. As the flame propagates forward, the amplitude of the pressure peak decreases gradually, and at t=61.25 ms, the pressure in the vertical direction tends to be consistent. Throughout the pressure wave propagation process, the largest intensity of the pressure wave is 0.45 kPa. It has been found by Teerling et al. 45 and Shalaby et al. 46 that when the pressure wave intensity is 1 kPa or larger, through the RT instability (Rayleigh-Taylor instability), the pressure wave can lead to large wrinkles on the flame surface and intermittent changes in the flame shape. In the low temperature conditions of this study, the pressure wave intensity is less than 1 kPa, and the flame surface does not appear large wrinkles (obvious distorted "tulip" flame). Although the intensity of the pressure wave decreases with time, the combination effect of many pressure waves has a relatively large effect on the flame surface, leading to the subsequent formation of the distorted "tulip" flame, as shown in Fig.4. Fig.14 (b) shows the horizontal pressure profiles at Y=0.05 m and different times. In order to show the pressure fluctuation more clearly, the pressure is normalized, that is, the pressures in the X direction are divided by the pressure P0’ which is the pressure at the center of the right side of the wall. The selected time is the moment when the flame front begins to reverse. At t=37.5 ms, the pressure wave front is located in the center of the vessel, that is, X=0.15 m. As time passes, the pressure wave front moves to the left (X negative direction) and reaches the left wall at t=37.6 ms. As a result of the reflection of the wall, the pressure wave front begins to move to the right (X positive direction) and reaches the right side of the wall at t=38.2 ms. Thus, the pressure wave is oscillating in the direction of the flame propagation. When the pressure wave front passes through the flame front, the propagation velocity of the flame front is oscillated. Since the pressure wave intensity is 0.24 kPa which is less


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than 1 kPa, the RT instability is weak, and the bending of the flame surface caused by the pressure wave is not obvious. Therefore, at low temperature, the pressure wave is not the main factor leading to the formation of the typical "tulip" flame.

(a) (b) Figure 14. (a) Vertical pressure profiles at X=0.225 m and different times, P0 is the pressure at the lower surface of the side wall; (b) Horizontal pressure profiles at Y=0.05 m and different times, P0’ is the pressure at the center of the right side of the wall

4.5 Comparison with existing results from a normal temperature combustion Through the above research, the overall evolution of the flame, the flame dynamics and the flame propagation mechanism of methane combustion at low initial temperature are obtained, respectively. Then, it is important to compare the results with existing results from a normal temperature combustion. Here, we will summarize the difference as well as the causes of the difference from a low temperature combustion process. In our studies, at low temperature, five stages are divided during the flame propagation in a confined vessel: spherical flame propagation, “fingertip” shaped flame propagation, flame “skirt edge” contacts the side wall, “crescent” flame propagation and typical “tulip” flame propagation. From the existing results obtained by Clanet et al.14 and Xiao et al.24, 43, at normal temperature combustion, four stages are divided during the flame propagation in a confined vessel: hemispherical shape flame propagation, “fingertip” shaped flame propagation, flame “skirt edge” contacts the side wall, and typical “tulip” flame propagation. As can be seen, “crescent” flame propagation is not involved at normal temperature combustion. This is because at low temperature, after the flame front reverses, the “tulip” flame tip extends to the burned zone a short distance, and the flame presents a “crescent”-shaped. However, at normal temperature, after the flame front reverses, the “tulip” flame tip extends to the burned zone a long distance, and the flame presents a typical “tulip”-shaped. The main reason for the above phenomenon is the different formation mechanisms of “tulip” flame at low temperature and at normal temperature. At normal temperature, Clanet et al.14 think that the formation of the “tulip” flame is because of the Taylor instability. Bychkov et al. 17 points out that Darrieus–Landau (DL) instability is the main cause of the ‘tulip” flame formation. Ponizy et al. 49 indicates that the purely hydrodynamic induces the formation of the “tulip” flame, and the intrinsic instabilities such as


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Rayleigh–Taylor, Richtmyer–Meshkov or Darrieus–Landau are not the main factor. However, at low temperature, the formation of the “tulip” flame can be immediately contributed to the interaction of the flame front, flame induced reverse flow and vortex motion. This explanation mechanism is very similar to that raised by Xiao et al. 26 . At low temperature, the pressure wave propagation back and forth along the flame propagation direction has no obvious effect on the formation of “tulip” flame. However, at normal temperature, according to Xiao et al.'s 24, 26, 43 research, also proved by Shen et al. 50, pressure waves play an important role in “tulip” flame formation. Quantitative terms, in Fig.14 (b), due to the low combustion rate caused by low temperature, the pressure wave intensity is 0.24 kPa which is less than 1 kPa, and the RT instability is weak. However, in Xiao et al.'s 26 research, the intensity of the pressure wave along the flame propagation direction is as high as 20 kPa. Therefore, the interaction effect of the pressure wave, reverse flow and vortex movement leads to a more obvious phenomenon of “tulip” flame. On the other hand, according to the studies by Xiao et al 26, five stages are divided during the flame propagation in a confined vessel: hemispherical shape flame propagation, “fingertip” shaped flame propagation, flame “skirt edge” contacts the side wall, typical “tulip” flame propagation, and obvious distorted “tulip” flame propagation. Obviously, the obvious distorted “tulip” flame propagation is not involved in our study. As also can be seen in Fig.4, at low temperature, the distortion of flame front is not obvious. However, in Xiao et al.'s 26, 38, 51 research, the distorted “tulip” flame is strong and constantly in the process of formation and disappearance. The main reason for the above phenomenon is the different formation mechanisms of distorted “tulip” flame at low temperature and at normal temperature. Firstly, Xiao et al. 26, 38, 51, 52 indicate that behind the distorted “tulip” flame front, there is a vortex formation which leads to the occurrence of the secondary flame distortion. However, from Fig.12 in this paper, when the distorted “tulip” flame is formed, vortex motion is not observed. Secondly, Xiao et al. 25, 26, 38, 51, 52 and Shen et al. 53 study that the pressure wave is the incitation factor of the distorted ‘tulip” formation. However, in this paper, the pressure wave is not the main factor. Quantitative terms, as shown in Fig.14 (a), due to the low combustion rate caused by low temperature, the largest intensity of the pressure wave is 0.45 kPa which is less than 1 kPa, and the RT instability is weak. However, in Xiao et al.'s 26 research, the pressure wave intensity is larger than 3 kPa and RT instability is strong, resulting in constant disintegration and generation of the distorted "tulip" flame. In all, at low temperature, the formation of the distorted “tulip” flame is caused by the superposition of the secondary pressure wave formed by the contact of the flame with the side wall.

5. Conclusions In this paper, the flame propagation characteristics of methane combustion at low initial temperature are studied using numerical method. Some main conclusions are drawn below. (1) At low temperature, five stages are divided during the flame propagation in a confined vessel: spherical flame propagation, “fingertip” shaped flame propagation,


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flame “skirt edge” contacts the side wall, “crescent” flame propagation and typical “tulip” flame propagation. In the process of flame propagation, the reverse of the flame front and formation of the “tulip” flame can be immediately contributed to the interaction of the flame front, flame induced reverse flow and vortex motion. Due to the low intensity of pressure wave, the pressure wave propagation back and forth along the flame propagation direction has no obvious effect on the formation of “tulip” flame. However, at normal temperature, the effect of the pressure wave leads to a more obvious phenomenon of “tulip” flame. (2) In the process of flame propagation, after the formation of “tulip” flame, wrinkle will appear in the smooth flame front, which is called the distorted “tulip” flame. The formation mechanism of the distorted “tulip” flame is different from that of the typical “tulip” flame. When the distorted “tulip” flame is formed, vortex motion is not observed. The formation of the distorted “tulip” flame is caused by the superposition of the secondary pressure wave formed by the contact of the flame with the side wall. However, because of the low intensity of pressure wave, RT instability is weak, and the distortion of flame front is not obvious. However, at normal temperature, the vortex and the pressure wave are the incitation factors of the obvious distorted ‘tulip” formation. (3) In the whole combustion process, there appear two peaks in the pressure rise rate curve. At the initial stage of combustion, an obvious oscillation occurs in the pressure rise rate curve. From t=87 to 89 ms, there is a large oscillation of the pressure rise rate due to the disappearance of the distorted “tulip” flame. (4) In general, the propagation velocity of the flame front center has two peaks in the whole combustion process, and the peak value is basically the same, which indicates that the maximum flame area in the “fingertip” propagation process is basically equal to that in the “tulip” flame propagation process. At the initial stage of combustion, the propagation velocity oscillates due to the oscillation of the pressure. After the flame front reverses, a secondary pressure wave is generated, resulting in a large oscillation of the flame tip velocity. However, the overall trend of propagation velocity is gradually reduced. (5) Flame propagation velocity and pressure wave are interacted with each other. In the process of combustion, the variation of flame propagation velocity and pressure rise rate show almost the same phase. The increase in flame propagation velocity directly leads to the increase in pressure rise rate, whereas the pressure wave propagation back and forth in the confined vessel leads to the oscillation of propagation velocity.

Nomenclature A pre-exponential factor [mol·cm-3·s-1] cp constant-pressure specific heat [J·Kg-1·K-1] D mass diffusivity [m2·s-1] E total energy [J·Kg-1] Ea activation energy [kcal·mol-1] k thermal diffusivity [m2·s-1] -1 M molar mass [kg·mol ] n constant, 0.7 p pressure [Pa] q heat release [J·m-3·s-1] qx, qy

heat flux [w·m-2]

R

gas constant [J·mol-1·K-1]


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t u x

time [s] horizontal velocity [m·s-1] abscissa

Y

mass fraction [-]

T v y

temperature [K] vertical velocity [m·s-1] ordinate

Fr

Froude number

2

g gravity acceleration, 9.8 m/s H half of the height of the confined vessel, 0.05 m Greek Symbols τxx, τyy, τxy stress tensor [Pa] µ viscosity [Pa·s] ρ density [kg·m-3] ω reaction rate [kg·m-3·s-1] γ Specific heat capacities [J·Kg-1·K-1]

AUTHOR INFORMATION Corresponding Author *Tel: +86 18954831115. Email:[email protected] *Tel: +86 15053293355. Email:[email protected]

Notes The authors declare no competing financial interest.

ACKNOELEDGEMENTS This investigation has been supported by the Fundamental Research Funds for the Central Universities (No. 18CX02005A).

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