Buoyant Unstable Behavior of Initially Spherical Lean Hydrogen-Air Premixed Flames

Buoyant unstable behavior in initially spherical lean hydrogen-air premixed flames within a center-ignited combustion vessel have been studied experimentally under a wide range of pressures (including reduced, normal, and elevated pressures). The experimental observations show that the flame front of lean hydrogen-air premixed flames will not give rise to the phenomenon of cellular instability when the equivalence ratio has been reduced to a certain value, which is totally different from the traditional understanding of the instability characteristics of lean hydrogen premixed flames. Accompanied by the smoothened flame front, the propagation mode of lean hydrogen premixed flames transitions from initially spherical outwardly towards upwardly when the flames expand to certain sizes. To quantitatively investigate such buoyant instability behaviors, two parameters, ―float rate (ψ)‖ and ―critical flame radius (Rcr)‖, have been proposed in the present article. The quantitative results demonstrate that the influences of initial pressure (Pint) on buoyant unstable behaviors are different. Based on the effects of variation of density difference and stretch rate on the flame front, the mechanism of such buoyant unstable behaviors has been explained by the competition between the stretch force and the results of gravity and buoyancy, and lean hydrogen premixed flames will display buoyant unstable behavior when the stretch effects on the flame front are weaker than the effects of gravity and buoyancy. OPEN ACCESS Energies 2014, 7 4939


Introduction
With the concerns upon the depletion of fossil fuels and the deterioration of the global environment, investigations on alternative fuels have become a hot topic in the energy science field.Due to its preeminent advantages in burning velocity and emitted products, hydrogen gas has been considered as a promising alternative fuel for the future [1,2] and has been employed in spark-ignition (SI) internal combustion engines throughout the world [3][4][5].For a SI internal combustion engine, the combustion performance is strongly affected by the propagation process, and hence an in-depth understanding of the characteristics of propagating hydrogen-air premixed flames is essential for designing and optimizing hydrogen fuelled SI engines [6,7], and it is widely believed that the intrinsic instabilities are of the utmost importance for premixed flames [8,9].
Over the past decades, the knowledge about the intrinsic instabilities of hydrogen premixed flames has been remarkably enriched by previous scholars' efforts.According to the previously reported results [10][11][12][13], the flame front of premixed hydrogen-air flames will be continuously wrinkled towards cellular instability during the propagation.There exist three intrinsic factors to induce an unstable premixed flame-hydrodynamic effects, thermal-diffusive effects, and buoyant effects [14].The hydrodynamic instability is induced by the thermal expansion across the flame front [15], which has been proved to be present for all premixed flames and whose intensity is always indicated by the density ratio between the two sides of the flame front [8][9][10][11][12][13][14]; the thermal-diffusive instability is induced by the competing effects between heat conduction from the flame and the reactant diffusion towards the flame, and the intensity is always indicated by the Lewis number (which is defined as the ratio of the thermal diffusivity to the mass diffusivity) [16]; and the buoyant instability is commonly regarded as be relevant to body-force.
According to the results obtained by the previous scholars, for lean hydrogen-air premixed flames, with the decline of fuel concentration (usually indicated by the equivalence ratio φ, defined as the ratio of actual fuel-air ratio to stoichiometric fuel-air ratio), the density ratio across the flame front is reduced but the Lewis number is increased [8], which means the hydrodynamic instability will be weakened and the thermal-diffusive instability will be enhanced.As can be seen in Figure 1 (flame images during the propagation obtained in the present investigation), for the lean hydrogen premixed flames whose φ falls into the range of 0.410 to 0.997, with the decrease of φ, the flame front with a similar size is more wrinkled and the cellular behavior is more obvious.The enhancement of unstable behavior is the result of the enhanced thermal-diffusive instability rather than a declined hydrodynamic instability, and such a conclusion has been widely recognized by others [17][18][19][20][21][22][23][24].However, from the comparison between the lean hydrogen premixed flames with φ = 0.265 and φ = 0.410, a distinct phenomenon can be observed whereby the flame front with a leaner fuel concentration is relatively smoother, which means the further enhanced thermal-diffusive instability has not made the leaner hydrogen premixed flame more cellular unstable.Furthermore, for the lean hydrogen premixed flame with φ = 0.265, the initial spherical outward propagation will transit towards an upward displacement where the geometrical center of the flame is lifting up when it has propagated to a certain size, namely, the hydrogen premixed flame has become buoyant unstable.Although some scholars have investigated buoyant unstable flames by numerical methods [25][26][27][28], few of their conclusions have been verified by experimental results.Therefore, making experimental observation and quantitative analysis on such buoyant instability can provide more information to help understand the properties of lean hydrogen premixed flames.In the present article, the buoyant unstable behavior has been observed experimental in lean premixed hydrogen-air flames within a range of φ values from 0.193 to 0.304, initial pressures (P int ) from 0.060 MPa to 0.500 MPa, and room temperature (300 ± 3 K).Two parameters: -float rate‖ and -critical flame radius‖ have been proposed to quantitatively investigate the buoyant unstable behavior.Finally, the mechanism of such buoyant unstable behavior has been analyzed and discussed in terms of stretch force and the combined effect of gravity and buoyancy.

Experimental Setup and Procedure
The experimental rig employed is mainly composed of six parts: a closed combustion vessel, discharge system, ignition system, optical access system, high-speed camera, and data acquisition-control system, as shown in Figure 2. The closed combustion vessel is made of stainless steel and has an inner volume of 2.744 liters.Two quartz windows with an effective diameter of 100 mm are oppositely mounted in the vessel.The discharge system is designed as a fork type, whose handle is connected to the vessel and four points are respectively connected to the hydrogen bottle, air bottle, vacuum pump, and the environment.Two electrical pressure transmitters are employed for monitoring the partial pressure during the discharge process, one is a DPA01M-P with a band from −100.0 kPa to 100.0 kPa, and the other is a DPA10M-P with a band from −0.001 MPa to +0.999 MPa (both from Taida, Taiwan, China).The ignition system includes a 12 V transistorized automotive ignition coil, and one pair of tungsten-copper electrodes which are oppositely mounted on the other two sides of the vessel beside the windows.The ignition energy is set as 37 mJ.The optical access system is a Z-shape arranged Schlieren system, which includes a xenon lamp, two flat reflectors, two slits, and two concave mirrors with a diameter of 200 mm and a focal length of 2000 mm.The high-speed camera is an IDT Motion Pro-Y4 (IDT, Newark, NJ, USA), operating at 10,000 frames per second with a resolution of 256 × 256 pixels.The data acquisition-control system is composed of a pressure transducer (6052C, Kistler Instrument AG, Winterhur, Switzerland), a charge amplifier (Kistler 5018A), a data acquisition module (NI PCI-6123, National Instruments, Austin, TX, USA), and a control platform compiled in NI LabVIEW language.
Procedurally, the vessel is evacuated, and then the mixture is prepared according to Dalton's law of additive pressure.The discharge progress can be divided into three stages: (i) filling hydrogen into the vessel until the data shown on the panel of DPA01M-P reaches the desired value, and then evacuating the fork-type pipe after closing Valve A; (ii) releasing air to the pipe after closing Valve B, and then filling air into the vessel after opening Valve A; and (iii) waiting 15 min to ensuring the homogeneity and quiescence of the mixture.Finally, a control signal is sent to synchronously activate both the ignition system and high-speed camera, and the time delay is set as 4 ms.Prior to refilling for the next set of experiments, the vessel is flushed twice with air to remove any residual products and then it is pumped to vacuum again.It should be noted that absolute vacuum cannot be realized within the vessel, and -99.7 kPa (shown on the panel of the DPA01M-P) is regarded as a unified signal for the vacuum condition and the residual gas will be accounted for in the volume fraction of air.The hydrogen employed is pure (99.995%) hydrogen, and the air employed is compressed air.

Analysis of Uncertainties
Within the present investigation, the uncertainties mainly originate from three issues: (a) the accuracy of equivalence ratio; (b) the clearance effects of the fork-shaped pipe; and (c) the accuracy of the measured flame radius.
The equivalence ratio is controlled by the partial pressure of hydrogen gas and initial pressure, and thus both the accuracies of φ and P int are affected by the truncation errors due to the minimal scale of the pressure transmitters' reading.Within the present discharge system, the uncertainties involves three aspects: (a) the true value of the -prescribed vacuum pressure‖ (−99.7 kPa); (b) the true value of the amount of filled hydrogen gas; and (c) the true value of the -initial pressure‖.Taking the case of φ = 0.265 and P int = 0.06 MPa for an example, the first error source is the truncation error on the reading of -prescribed vacuum, the true value of the set -−99.7 kPa‖ falls the range of −99.74 kPa to −99.66 kPa; the second error source is the truncation error on the reading of pressure transmitter after filling the hydrogen gas, the expectant reading should be -−93.7 kPa‖, but the true value falls the range of −93.74 kPa to −93.66 kPa, namely, the actual partial pressure of the discharged hydrogen gas falls into the range of 5.92 kPa to 6.08 kPa; and the third error source is the truncation error on the reading of the -initial pressure‖, the true value of the expectant reading -−40.0kPa‖ falls into the range of −40.04 kPa to −39.96 kPa, namely, the actual partial pressure of the discharge air falls into the range of 53.62 kPa to 53.78 kPa; and hence, the actual value of φ falls into the range of 0.26209 to 0.269977, namely, the maximal uncertainties on such case is 1.88%.Figure 3 shows the maximal uncertainties of all the investigated cases within the present article, as can be seen, the maximal uncertainties for the present article is less than 2.00%.The clearance effects of the fork-shaped pipe can be neglected in the present investigation due to the reasonable discharge approach.In the flame radius measurements, at least five measurements have been performed for each set of conditions to obtain the average values, the repeatability is generally satisfactory, and the uncertainties decrease as the flame radius increases.

Observation of the Morphology of Buoyant Unstable Behavior
Figure 4 shows the Schlieren images of hydrogen-air premixed flames with different φ (from 0.193 to 0.304) under atmospheric pressure and room temperature, in which, R sch is the flame radius of the Schlieren image of the flame front, and it is the calculation result from the flame front's projection area (A) given by: As can be seen, during the initial period of propagation (within which R sch ≤ 10 mm), nearly all the investigated lean hydrogen premixed flames give a similar propagation behavior in that the flame front propagates outwardly from the ignition position in a spherical manner associated with a smooth surface.For the lean hydrogen premixed flames whose φ falls into the range of 0.265 to 0.304, as flame further propagates, some obvious cracks occur and develop on the flame front (as can be observed on those -relatively richer‖ lean hydrogen premixed flames shown in Figure 1); but for the lean hydrogen premixed flames with φ = 0.193 and φ = 0.235, the flame front is always smooth during the subsequent propagation.It can be summarized that, for the lean hydrogen premixed flames within the investigated range of fuel concentrations, the -cellularization‖ of the flamefront has been suppressed with the decrease of fuel concentration, which is distinct from the traditional understanding of the trends of the variation of the cellularization of lean hydrogen premixed flames which is dominated by thermal-diffusive instability.
Accompanying the suppression of cellularization on the flame front, the manner of flame propagation, flame shape, cells' development and cells' distribution are noticeably changed from what is observed in Figure 4. Firstly, regarding the aspect of propagation's manner, the flame has begun to show a tendency of upward propagation when it has propagated to a certain size, and the leaner the flame is, the more obvious the upward tendency becomes, and even lean hydrogen premixed flames with φ = 0.193 and φ = 0.235 have been totally lifted up during the propagation when R sch ≤ 20 mm.Secondly, on the aspect of flame shape, due to the remarkable upward propagation, the upper part of the flame front has been elongated, and the curvature of the upper part becomes more arc-like while that of the lower partial becomes more flat.Thirdly, on the aspect of cells' development, for those lean hydrogen premixed flames for which cells can also be observed on the flame front, the cells on the lower part still uniformly split into new cells as the flame further propagates, but the cells on the upper part just expand rather than continuously splitting when the flame front has begun to propagate upwards.Fourthly, on the aspect of cells' distribution, for those cellular hydrogen premixed flames, due to the different manner of cell development, the cells' distribution in the upper part is distinct from the uniform distribution on the lower part, and the cells on the upper part are larger in size and less in the number.Since the mentioned four behaviors observed from Figure 4 are distinct from what has always been reported for lean hydrogen premixed flames dominated by thermal-diffusive and hydrodynamic instability, we summarize them as distinct behaviors of buoyant instability, and it also can be concluded that such buoyant instability behaviors are enhanced with the decrease of fuel concentration for lean hydrogen premixed flames under normal pressure and room temperature.
Figure 5 shows the Schlieren images of hydrogen-air premixed flames with φ = 0.193 at room temperature under different P int conditions.From the comparisons shown in the figure, three phenomena can be easily observed.Firstly, the hydrogen premixed flame with φ = 0.193 will be totally lifted up during the propagation under all the investigated P int conditions, but the flame still propagates outwardly with a spherical shape within the initial period of propagation under reduced, normal, and certain elevated pressures (such as 0.150 MPa).Secondly, for a similar flame size, the lift height under reduced and normal pressure are lower than those of flames under elevated pressures; under elevated pressures, the higher the P int is, the higher the lift height becomes.Thirdly, the flame front is very smooth under reduced and normal pressures, but there exists some obvious -signets‖ on the flame front under elevated pressures, and the higher the P int is, the -deeper‖ the signet can be observed.Based upon the behaviors observed in Figure 5, it seems that the buoyant instability behavior can be enhanced by increasing P int .Figure 6 shows the Schlieren images of hydrogen-air premixed flames with φ = 0.304 at room temperature under different P int conditions.As can be seen, the investigated lean hydrogen premixed flames can maintain the initial spherical outward propagation under elevated pressures, and the upward propagation tendency can just be observed for relatively larger flame sizes under reduced and normal pressure, and the tendency is indeed unremarkable.Under the investigated elevated pressures, the cells formed on the flame front are distributed and develop uniformly, but under reduced and normal pressure, the gradient distribution in size of those cells on the upper part of flame front can still be observed.The higher the P int is, the more cellular the flame front with a similar flame size becomes, and the less remarkable the tendency of upward propagation becomes, that is, the weaker the buoyant instability behaviors become.Although the observed phenomena are distinct from what is observed in Figure 5, but it is normal considering the traditional understanding of the behavior of lean hydrogen premixed flames.With the absence of buoyant instability for hydrogen premixed flames, increasing P int will enhance the intensity of the hydrodynamic instability while the thermal-diffusive instability remains unchanged, and the enhanced hydrodynamic instability could enhance the cellularization of flame front; therefore, it seems such phenomena can be easily explained.However, the hydrogen premixed flames with φ = 0.193 under a different P int also suffer the similar variation trends of hydrodynamic and thermal-diffusive instability, but why do they show distinct phenomena (as shown in Figure 5)?Through the comparisons between the observations on the flame images, it can be found that the two different variation trends of unstable behaviors are accompanied by the distinct phenomenon that the hydrogen premixed flames with φ = 0.193 had been lifted up prior to the formation of cells while a -floating‖ of the hydrogen premixed flames with φ = 0.304 (if it happened during the propagation) occurs after the formation of cells.Therefore, when the lean hydrogen premixed flames begin to float the trend of such floating is essential to the variation regulations of the unstable behavior of lean hydrogen premixed flames.Therefore, one should make quantitative studies on such buoyant unstable behaviors by examining the -floating‖ trends of the investigated flames.

Studies on the Buoyant Instability Trends
Based upon the abovementioned and discussed phenomena, the extent of upward propagation is the most remarkable behavior of the buoyant instability and is also an essential aspect to determine how the unstable buoyant behaviors happen; therefore, it can be taken as the indicator for quantitatively describing the -intensity‖ of the buoyant instability.Within the present investigation, the extent of upward propagation is indicated by a parameter -Float Rate, Ψ‖ which is defined as: where D top is the distance from the ignition position to the flame's top; D bottom is the distance from the ignition to the flame's bottom; and ΔL is the difference between D top and D bottom .It can be clearly understood that, for a specific R sch , larger the ΔL is, the more obvious the buoyant instability behavior becomes; while, for a specific ΔL, the smaller the R sch is, the more obvious the buoyant instabilty; therefore, the parameter -float rate‖ provides a good comprehensive characterization of the buoyant instability.
Figure 7 shows the variation of Ψ with R sch under different conditions.As can be seen, under all the investigated conditions, the value of Ψ increases as R sch increases, namely, such buoyant instability will be continuously enhanced as flame expands once the buoyant instability has ocurred, which means the buoyant instability behaviors have a self-enhancing nature.From Figure 7a, it can be found that, during the period of constant pressure, Ψ increases as the flame expands with an accelerating growth rate, and the rate of acceleration at a specific R sch is increased with the reduction of φ, which indicates that the leaner the hydrogen premixed flame is, the stronger the buoyant instability becomes.When the flame front enters a zone with an increasing pressure, Ψ still increases as the flame expands but the growth rate of Ψ the decreasing noticeably.Compared to the influences of φ on the buoyant instability, the influences of P int on the variation trends of Ψ are relative complex.As can be seen from Figure 7b, for the lean hydrogen premixed flame with φ = 0.193, as P int increases, Ψ at a similar R sch firstly decreases and then increases, and the minimal value is attained at atmospheric pressure; which indicates the variation of P int (relative to atmospheric pressure) could enhance the intensity of the buoyant instability.For the lean hydrogen premixed flames with φ = 0.265 and φ = 0.304, the values of Ψ at a specific R sch monotonously increases as P int increases.Furthermore, no matter whether promoting or restraining, the effects of P int on the variation of Ψ become weaker as φ increases.
As mentioned above and discussed in the previous subsections, when the lean hydrogen premixed flames have begun to be lifted up is essential to the variation trends of the buoyant instability behavior; therefore, we propose the term -critical flame radius, R cr ‖ (which is defined as the flame radius at which Ψ has increased up to 10%) as the indicator to the onset of the buoyant instability.As shown in Figure 8, for the lean hydrogen premixed flames with φ ≤ 0.235, R cr generally decreases as P int increases, and the extent of decrease is enlarged as P int increases for a specific φ, and the variation extent decreases as φ increases.For the flames with φ ≥ 0.265, R cr increases as P int increases, the extent of R cr varies noticeably with the increase of P int for a specific φ and/or the increase of φ for a specific P int .Furthermore, combined with the observed flame images, for the lean hydrogen premixed flames with φ = 0.193 and φ = 0.235, R cr is less than the flame radius at which cells can be observed on the flame front (actually, for most investigated cases in the present article, no cells have been formed on those flames while the flame was still propagating within the field of view); for the lean hydrogen premixed flames with φ = 0.265, φ = 0.279 and φ = 0.304, R cr is larger than the flame radius at which cells have been formed.Based upon the observation and trends mentioned above, it can be supposed that decreasing φ could promote the buoyant instability under a specific P int , but the effects of P int on the buoyant instability are different for different φ values.

Discussion on the Dominating Role of the Buoyant Instability
An outwardly propagating flame mainly suffers three forces, besides its own thermal expansion, which are respectively the stretch force, buoyancy, and gravity, as shown in Figure 9.For a spherical outwardly propagating flame, in the absence of the influence of electrodes, the stretch force upon the right semi-part should be equal to that upon the left semi-part due to the symmetrical structure, and thus the flame movement in the vertical direction is dominated by the combined effect of buoyancy and gravity.It should be noted that the differences in geometric parameters (including the length, radius, roughness, etc.) and physical characteristics (such as materials, carbon laydown on the surface of electrodes) of electrodes can induce differences in the stretch forces on the right and left semi-parts, and these unbalanced stretch forces on both semi-parts can be observed from the flame shape if such differences are sufficiently large.Due to the good symmetrical characteristics observed in the flame images, such differences were neglected in the present article.

Gravity field
According to the Archimedes' principle, the buoyancy of a flame is the product of gravity acceleration, the flame's volume, and the unburned reactants' density; therefore, the combined effect of buoyancy and gravity can be indicated by the flame's volume and the density difference across the flame front (Δρ, the difference on the density between the unburned and the burned fuel).If we consider combustion occurs at an adiabatic temperature and composition at constant pressure, the value of Δρ is a constant value during the flame's propagation, and hence the global result of the competition between buoyancy and gravity is being increased during the propagation due to the flame's increasing volume, which explains well why Ψ increases as R sch increases under any conditions.Figure 10 shows the variation trend of Δρ with φ under different P int , where both the densities of unburned and burned mixtures are calculated using the chemical equilibrium program GASEQ [29].As can be seen, for lean hydrogen premixed flames, with the increase of φ, the value of Δρ firstly increases and then decreases, and the maximal value is attained near φ = 0.4, but the difference in Δρ From Figure 11b, it can be found that, for the hydrogen-air premixed flames with φ = 0.193, the values of α at a same R sch are distinct under different P int during the initial period of propagation, but such differences decline noticeably as flame propagates further, and the difference can even be neglected when R sch is more than 15 mm.For the initial period of propagation, as P int increases within the investigated range, α at a same R sch firstly increases and then decreases, and the maximal value is attained at atmospheric pressure.The results indicate that for such lean hydrogen premixed flames under normal and elevated pressures, the higher the P int is, the weaker the intensity of the global stretch force becomes; furthermore, the competition between buoyancy and gravity is enhanced (as reported in Figure 10).Therefore, the intensity of the competition between buoyancy and gravity may exceed the intensity of the global stretch force, and such a difference will be enhanced with the increase of P int .Under reduced pressure, both the intensities of the global stretch force and the competition between buoyancy and gravity are weakened compared to the normal pressure, but the difference between the two aspects may also weaken, which means the composite effects of such a difference may be stronger than those under normal pressures; hence, the buoyant instability behavior would be enhanced compared to that seen under normal pressure.Such conclusions can better explain the buoyant instability behaviors observed in the present article for the lean hydrogen premixed flames with φ = 0.193 and φ = 0.235 (which have similar variation trends for both Δρ and α).
From Figure 11c, it can be found that for the premixed hydrogen-air flames with φ = 0.304, the value of α at a same R sch increases monotonously as P int increases within the investigated range, the difference on α is relatively smaller during the initial propagation period, but such a difference increases remarkably as the flame propagates further.For the initial period of propagation, the difference of α is not obvious under different P int values, which means verifying the initial pressure won't induce any noticeable variation of the global stretch force acting on the flame front; however, the competition between buoyancy and gravity will be enhanced by increasing P int , namely, the buoyant instability potential has been enhanced.However, from the flame images shown in Figure 6, few related phenomena can be observed.Actually, by making a comparison between the cases of φ = 0.193 and φ = 0.304, it can be found that the influence of P int on the value of Δρ is highly similar (shown in Figure 10), namely, the competition between buoyancy and gravity under a specific P int is mostly kept at a similar level; however, the values of α at a same R sch are distinct, the minimal value of α for the case of φ = 0.304 is even higher than the maximal value of the case of φ = 0.193 for a specific P int , that is, the intensity of the global stretch force on the flame front has been significantly enhanced.Therefore, it can be said that for lean hydrogen premixed flames with φ = 0.304, increasing P int induces an obvious enhancement of the global stretch force rather than the competition between buoyancy and gravity, and the role of global stretch force on maintaining the flame spherically outwardly dominates the effect of the competition between buoyancy and gravity driving the flame upwards, and hence the flame will not display any obvious buoyant instability behaviors.However, when the flame has propagated to a certain size, the increased competition between buoyancy and gravity (induced by the flame's increasing volume) will exceed the declining global stretch force, and then the tendency of upward propagation will appear again (as shown in Figure 6).Figure 12 shows the variation trend of flow rate with global stretch rate under different conditions.As can be seen, the flow rate will not be observed when the stretch rate is still sufficiently large, and the value will be remarkably increased when the stretch rate further decreases from certain smaller values.Comparing the cases of lean hydrogen premixed flames with φ = 0.193 and with φ = 0.304, it can be seen that under a specific P int , the value of Ψ for the leaner flame is lower than that for the richer, which is similar to the correlation of Δρ.The results again indicate that the buoyant instability is related to Δρ (the competition between buoyancy and gravity) and α (the global stretch force).
Based upon the above discussions, it can be concluded that whether the buoyant instability can be seen or not is determined by the result of the competition between the stretch force acting on the flame and the combined resultant of buoyancy and gravity, and the flame won't show an obvious buoyant instability if the stretch force plays a dominant role in the propagation; otherwise, the flame will propagate upwards.

Figure 2 .
Figure 2. Schematic diagram of the experimental apparatus.

Figure 3 .
Figure 3. Maximal uncertainties of the cases reported within the present article.

Figure 4 .
Figure 4. Schlieren images of hydrogen-air premixed flames with different equivalent ratios at atmospheric pressure and room temperature.

Figure 7 .
Figure 7. Variation of Float Rate with flame radius: (a) with different equivalence ratios under atmospheric pressure; (b) with φ = 0.193 under different initial pressures; (c) with φ = 0.265 under different initial pressures; and (d) with φ = 0.304 under different initial pressures.

Figure 8 .
Figure 8. Critical flame radius effect on the buoyant instability for hydrogen-air premixed flames with initial pressure for different equivalence ratios.

Figure 9 .
Figure 9. Schematic diagram of the forces acting upon a spherical propagating flame.

Figure 10 .
Figure 10.Density difference of hydrogen-air premixed flames with equivalence ratio under different initial pressures.

Figure 11 .
Figure 11.Global stretch rate of hydrogen-air premixed flames with flame radius: (a) with different equivalence ratios under atmospheric pressure; (b) with φ = 0.193 under different initial pressures; and (c) with φ = 0.304 under different initial pressures.

Figure 12 .
Figure 12.Flow rate of hydrogen-air premixed flames with global stretch rate: (a) with different equivalence ratios under atmospheric pressure; (b) with φ = 0.193 and φ = 0.304 under different initial pressures.