Influence of hydrodynamic processes on apparent flame velocity in accidental explosions

. The main impact factor of an accidental explosion is explosive pressure, the numerical value of which is determined by the magnitude of the apparent flame velocity during explosive combustion of an explosive mixture. Therefore, the issues related to predicting the apparent flame velocity in deflagration explosions are the main ones when considering the problem of ensuring explosion resistance of buildings and structures and investigating accidents accompanied by explosions. Relationships are presented that confirm the unambiguous dependence of explosive pressure on apparent flame velocity. It is shown experimentally that in a deflagration explosion in free space, laminar combustion of the mixture occurs at first, so the flame front is flat and smooth, and at a certain distance from the ignition source, there is turbulization of the mixture moving in front of the flame front, and the explosive combustion becomes turbulent. The dependence determining the flame acceleration in an accidental explosion is given. It is experimentally shown that there is no rapid cessation of explosive combustion and flame arrest at the central initiation of a spherical explosive cloud of stoichiometric concentration. The methodology described in this article for determining the apparent flame velocity in deflagration explosions allows a fairly accurate prediction of the potential threat posed by explosive facilities and installations in the surrounding area.


Introduction
Before describing the results of experimental and theoretical studies aimed at investigating the issue of the influence of hydrodynamic processes on the apparent flame velocity of a deflagration explosion, it is necessary to note the following features of accidental explosions.First, explosive combustion in most accidents is deflagration rather than detonation.Second, the main impact factor of an accidental explosion is the explosive pressure, the numerical value of which is determined by the magnitude of the apparent flame velocity, which is realized as a result of the deflagration combustion of the explosive mixture.Therefore, the issues related to predicting the apparent flame velocity in deflagration explosions are the main ones when considering the problem of ensuring explosion resistance of buildings and structures and investigating accidents accompanied by explosions.

Materials and methods
We consider a technique for calculating the dynamic parameters of compression waves generated by deflagration explosions in the spectral region to substantiate the dependence of explosive pressure on the apparent flame velocity of a deflagration explosion.The possibility of using the Fourier method in the analysis is based on the fact that most deflagration explosions are characterized by relatively small flame propagation velocities (relative to the speed of sound), which allows to use linear equations of fluid motion to describe the wave flows arising from the explosion.The linearized equations of fluid motion admit the superposition principle, allowing us to use Fourier analysis in their study.This is the so-called acoustic (linear) approximation.When using the acoustic approximation to determine the dynamic characteristics of compression waves arising from deflagration explosions, the known solution for a zero-order acoustic radiator (monopole) in a boundless space is used [1][2][3][4][5].
A monopole in acoustics is a sphere with radius a, making pulsating oscillations with frequency ω symmetrically about the center.
The relations given in [1][2][3][4][5] confirm the unambiguous dependence of the explosive pressure on the apparent flame velocity.Therefore, reliable prediction of the flame front kinematics in a deflagration explosion guarantees the correct determination of the explosive pressure dynamics and, consequently, the real destructive capacity of the deflagration explosion under consideration.

Results
We have conducted experimental studies of deflagration explosions in the atmosphere to confirm the relationship between apparent flame velocity and explosive pressure.The experiments were carried out on a unit with the following general principle of operation.A thin film created an explosive mixture in a cubic volume.The mixture was ignited in the center of the explosive cloud.If necessary, turbulizers were installed inside the cube to accelerate the explosive combustion of the mixture.Figure 1 shows several instantaneous photographs of the deflagration explosion that were obtained by high-speed motion picture photography.The pictures shown correspond to the following time instants: 20 ms, 30 ms, and 60 ms.The shooting was done at a speed of 600 frames per second.
The results of the velocity filming allowed a reasonably accurate determination of the kinematics of the apparent flame velocity.An interpolation curve was constructed from the experimental points describing the time dependence of the flame front position.The interpolation was carried out by a 12th-degree polynomial to reveal local variations of the apparent flame velocity in the experimental dependencies.The derivative of the resulting interpolation relationship, shown in Figure 2, represents the apparent flame speed.In addition, in Figure 2, the dots indicate the numerical values of the flame velocity obtained by formal calculation of the time derivative obtained from the experimental values of the flame front position recorded during photography.On the time axis of the upper graph of Figure 2, the time instants to which the explosion photos in Figure 1 correspond are marked.Figure 3 shows the experimental oscillogram of the explosive pressure and the calculated time dependence of the explosive pressure obtained from the experimental kinematic parameters of the flame front.On the time axes of the experimental and computational plots of Figure 3, the time moments to which the explosion photos in Figure 1 correspond are marked.It can be seen that the maximums of the apparent flame velocity correspond to the maximums of the explosive pressure, indicating the explosive pressure's dependence on the apparent flame velocity in a deflagration explosion.
In addition, the satisfactory agreement between the experimental data and the results of calculating dynamic parameters of explosive pressure obtained from the experimental kinematic characteristics of the flame front confirm the unambiguous dependence of explosive pressure on the apparent flame velocity in a deflagration explosion.
It follows from the previous material that to provide a reliable prediction of explosive loads; it is necessary to know the apparent flame velocity parameters of a deflagration explosion.
Various empirical relationships are used to calculate the apparent flame velocity [6][7][8][9][10][11].In one of the first papers on the subject, it is shown that the apparent flame speed increases with the size of the fireball.An empirical formula for determining the maximum value of the apparent flame velocity is given [6,7]: , where where U n is the normal combustion rate of the gas-air mixture (GAM), m/s; ε is the coefficient of expansion of the combustion products; R 0 is the radius of the spherical cloud, m; v is the coefficient of kinematic viscosity, m 2 /s.The empirical relationship (1) was derived from experimental data obtained at the test site for hydrogen-air mixture explosions.
In [2], a relation linking the apparent flame velocity (when the flame propagates on a sphere in unbounded space) to the path traveled by the flame (by acceleration) R p and the size of detonation cells ∆ of the mixture was obtained, which is similar in structure to formula (1): (2) where W is the maximum value of the apparent flame velocity in free space.
In [8], the propagation velocity of the flame front is calculated using a formula similar to (2), only it includes an empirical coefficient A that depends on the type of combustible substance involved in the explosion: In [9][10][11], it is considered that the maximum propagation velocity of the flame front in deflagration explosions depends on the size of the detonation cells of the mixture or the mass of the substance involved in the explosion.The relation for determining the maximum propagation velocity of the flame front in [9][10][11] is as follows: , where K is the empirical coefficient (K= 26-43).Formula (4) is reduced to the ratio , where R 0 is the radius of a  =  •  0 1/2 spherical cloud.Formula (4) is, in a certain sense, similar to formulas (1)(2)(3).
In [12], an attempt was made to theoretically determine the maximum propagation velocity of the flame front in free space.
We consider the hydrodynamic aspects of the deflagration combustion of a mixture to determine the numerical value of the apparent flame velocity in a deflagration explosion.
First, it is necessary to consider the physical meaning of the following phenomena: hydrodynamic friction, diffusion, and diffusivity, characterized respectively by kinematic viscosity coefficient -ν, diffusion coefficient -D, and diffusivity coefficient of mediumα.Firstly, the mentioned coefficients have the same dimensionality -m 2 /s, which, to some extent, indicates that they belong to a common group of physical phenomena.Secondly, they characterize the phenomena representing the manifestation of molecular (in laminar flows) or fragmentary (in turbulent flows) structure of a continuous medium, i.e., they describe the same phenomenon but concerning different influences: force (describing tangential stresses), entropy (characterizing the measure of chaos in the mixture) or energy (describing the distribution of heat in the medium).The kinematic viscosity coefficient indicates the flow of momentum transferred by chaotically moving molecules or fragments of the medium from a region with one flow velocity (time-averaged) to an area with another velocity.The change in the momentum of the medium is interpreted as the appearance of tangential stresses in a moving continuous medium.The diffusion coefficient characterizes the flow of matter transferred by chaotically moving molecules or fragments of the medium from a region with one concentration to an area with another concentration.A change in the mass of a given substance in a volume results in a change in the concentration of that substance in the medium.The diffusivity coefficient characterizes the flow of high-speed chaotically moving molecules or fragments of the medium from the region with their increased content, characterized by an increased medium temperature, to the area with a decreased temperature.A change in the number of high-velocity molecules or fragments of a substance leads to a change in the substance's temperature at the point of the medium under consideration.Thus, all the coefficients listed above generally describe the same phenomenon but concerning different medium properties.Therefore, the change in the values of these coefficients depending on some parameters should be approximately the same.The most studied group of coefficients is the coefficient of hydraulic (hydrodynamic) friction, which was determined by practical necessity.
Let us analyze the methods of its determination or calculation.Historically, all sorts of instruments and equipment for hydraulic (mainly water) structures were created first, and then theory emerged.Taking into account this circumstance, at one time, it was accepted that pressure losses in channels and pipelines, as well as the pressure of resistance to the movement of bodies in a continuous medium, should be attributed to the value of velocity head.For example, the following formula is used to determine the pressure drop in a pipe: where ΔP is the pressure drop on the pipe section, L is the pipe length, d is the pipe diameter, ρ is the density of the moving medium, U is the average flow velocity along the pipe section, λ -hydraulic resistance coefficient, which is a measure of pressure drop (in fractions of the velocity head) for a specific (in pipe diameters) flow section.The use of the numerical value of the velocity head of a flow as the unit of pressure drop or resistance pressure is due to the fact that man deals with or is surrounded by mainly turbulent flows, the resistance of which is proportional to the velocity head.In this case, we speak about the quadratic law of resistance, in which the coefficient of hydraulic resistance -λ or drag coefficient -С Х do not depend on the velocity of the oncoming flow but are determined by the surface roughness or shape of the body, i.e., λ=const and С Х =const.
In laminar flows, the drag coefficient depends on the flow velocity to the first degree, so the numerical value of the drag coefficient in formula (5) must rely on the flow velocity.
Indeed, for laminar flows, an analytical expression for determining the value of the drag coefficient is obtained, which has the form: In this case, the viscosity coefficient of the medium is taken as its value obtained from the molecular exchange of impulses between neighboring layers of the liquid or from the condition of laminarity (layering) of the flow.In this case (in laminar flow), as stated earlier, the resistance will be proportional to the first degree of velocity rather than to the square of velocity, as is the case in turbulent flows.However, at turbulent flows and at large Reynolds numbers, the hydraulic resistance coefficient λ does not depend on the velocity of the impinging flow and, consequently, on the Reynolds number.
Therefore, the coefficient λ must be a functional dependence on the flow velocity or Reynolds number, or it is required for the kinematic viscosity coefficient of the medium to depend on the flow velocity.Historically, the viscosity coefficient of the medium is assumed to be a constant (ν=ν 0 ), which corresponds to the conditions of the flow laminarity or the conditions of exchange between neighboring layers of liquid molecules, not fragments of liquid, as it takes place in reality in turbulent flows.As a result, it is assumed that the variable is the hydraulic resistance coefficient λ and not the kinematic viscosity coefficient of the medium ν, i.e., it is assumed that ν=ν 0 .Therefore, various empirical relations are introduced in the practice of calculating the hydraulic resistance coefficient λ, e.g., [13]: where K E is the equivalent uniform absolute roughness.The above relation for hydraulically smooth surfaces (K E =0) transforms into the well-known Blasius formula [13]: .
The above formulas, assuming the constancy of the kinematic viscosity coefficient for all flow regimes of the medium, can be transformed in such a way that they would have the structure of an analytical solution, namely , but with a variable kinematic viscosity λ = 64  coefficient of the medium depending on K E and the number Re 0 (in the currently accepted understanding, i.e., for v =v 0).This approach to the problem of determining the hydraulic resistance coefficient λ makes it possible to estimate the change of other coefficients related to the same physical phenomenon (diffusion coefficient -D and medium diffusivity coefficient -α), depending on the medium flow conditions.
Let us consider how the kinematic viscosity coefficient of the medium ν should change during the transition from laminar flow to turbulent flow in pipes with hydraulically smooth surfaces.In laminar flows, the value of the kinematic viscosity coefficient obtained from the molecular exchange of momentum between neighboring fluid layers is ν 0 .For example, under standard conditions, for air, ν 0 =0.15 cm 2 /s, and for water, ν 0 =0.01 cm 2 /s.
When the Reynolds number of the flow is less than the critical value Re 0 <Re 0KR at laminar flows, the drag coefficient is determined by the formula (6): . For laminar λ = 64 flows, the viscosity coefficient has a constant value, which is ν=ν 0 .
At turbulent flows (Re 0 >Re 0KR ) in pipes with hydraulically smooth surfaces, the hydraulic resistance coefficient λ is determined by formula (8): .It is obvious that "stitching" of drag coefficient values at laminar flows with drag coefficient values at turbulent flows should be performed, i.e., .Then, for hydraulically smooth surfaces and turbulent flows, we obtain the following relation for the drag coefficient: , which corresponds to the analytical formula for laminar flows, but with the viscosity coefficient increased by a factor of .

( ) ¾
As a result, we obtain that the turbulent kinematic viscosity coefficient ν for hydraulically smooth surfaces (K E =0) exceeds the laminar kinematic viscosity coefficient ν 0 by a factor of or we have: The same situation should be observed for other coefficients similar in physical meaning to the kinematic viscosity coefficient.Therefore, for the turbulent diffusion coefficient and the diffusivity of gases, the following dependencies should take place: and . ( The flame propagation velocity in deflagration explosions is determined by the diffusivity [14] and is described by the formula: where α is the coefficient of thermal conductivity; τ = 1/k is the characteristic reaction time; k is the rate constant of the chemical reaction. The relation (11) is obtained under the condition that the Lewis number, characterizing the ratio of the diffusion coefficient to the diffusivity, is approximately equal to unity [14].
Taking into account relations (10), we obtain an expression for the flame propagation velocity under turbulent regimes corresponding to the motion of the gas medium in front of the flame front with hydraulically smooth surfaces: , (12) where R is the current radius of the combustion region, and R KR is the critical radius of explosive combustion, at which the transition of laminar gas motion at the flame front into the turbulent regime occurs.From the last equation, we get: where U is the maximum value of the apparent flame propagation velocity at the radius of the fireball R; ε is the degree of expansion of combustion products; U H is the normal flame propagation velocity in laminar combustion.
Consequently, to determine the apparent velocity in a deflagration explosion in free space, it is necessary to determine the critical radius of the flame front at which the transition of laminar gas motion at the flame front into the turbulent flow regime occurs [15][16][17].
For this purpose, we conducted the following experimental studies.Parameters of deflagration explosions of propane-air and acetylene-air mixtures of stoichiometric composition were carried out and recorded.Several series of experiments were performed.
In the first series of experiments, explosions were conducted in a cubic volume with an internal dimension of 1m and a fully open top face.Its upper edge was covered with a thin film during the chamber filling with gas.The mixture was ignited by a spark in the cube's center.The explosion process was recorded with a high-speed movie camera at a pull rate of 240 frames per second (4.17 ms between frames).Figure 4 shows photographs of some of the moments of the propane-air mixture explosion.The first photo captures the initial moment of the explosion (the moment of realization of the spark), the second photo captures the moment of the explosion 62.5 ms after the spark, and the third photo corresponds to the 125th millisecond of the explosion.
From the above photos, it can be seen that at first, laminar combustion of the mixture occurs, so the flame front is flat and smooth, but at a certain distance from the ignition source, there is turbulization of the mixture moving in front of the flame front, and the explosive combustion process becomes turbulent.This is due to hydrodynamic phenomena in the mixture moving in front of the flame front, which leads to an increase in the heat transfer coefficient and, consequently, to an increase in the apparent flame velocity.A more detailed analysis of photographs of the explosion process showed that the critical radius of the explosion cloud at which the transition of the laminar combustion process into turbulent is R KP =0.45 m.This numerical value of the critical radius corresponds to the situation when the mixture of stoichiometric composition is calm and stationary.
Similar experiments were performed using acetylene as a propellant.Figure 5 shows photos of some of the moments of the acetylene-air mixture explosion.The first photo captures the initial moment of the explosion (the moment of realization of the spark), the second photo captures the moment of the explosion 12.5 ms after the spark, and the third photo corresponds to the 25th millisecond of the explosion.Detailed analysis of photographs of the explosion process showed that the critical radius of the explosion cloud at which the transition of the laminar process of combustion of acetylene-air mixture into turbulent is R KP =0.24 m.
Following relations ( 12) and ( 13), it can be stated that the acceleration of the explosive combustion process in free space occurs faster in acetylene-air mixtures than in propane-air mixtures.At the same time, the rate of increase of visible flame speed in acetylene-air mixtures is about 1.5 times faster (more precisely, by 46%) than in propane-air mixtures.
It was previously stated that previous publications on this topic [2,[6][7][8][9][10][11][12] indicated the detonation cell size as the main parameter responsible for flame acceleration in a deflagration explosion.The experiments showed that, for the two mixtures under study, the critical blast radius and detonation cell size are related by the following relationship: .( 14) ( ) The physical picture explaining the structure of formula ( 14) is not very clear, but it is quite suitable for the two mixtures under study.

Discussion
The second series of experiments was conducted using the same substances (propane and acetylene).The experiments involved the explosion of a compact cloud of a mixture of stoichiometric composition.The volume of the cloud resting on the ground was 410 liters or 0.41 m 3 .A cube-shaped box with a 74.5 cm edge was used to form this cloud.After filling the box with the required volume of combustible gas, it was removed without significant effect on the formed gas-air cloud.Ignition of the mixture was carried out at the center point of the lower edge of the cloud.Let us consider the results of an experimental explosion of a propane-air mixture (hereinafter referred to as experiment 1), which was recorded with a high-speed movie camera at a rate of 240 frames per second.Figure 6 shows photos of the experimental propane-air mixture explosion process.The time intervals between photos are 50 ms.Experimental materials were processed similar to the previous experiments to determine the apparent flame velocity of the deflagration explosion.The horizontal dimensions of the fireball for each time point were measured.After that, an interpolation of the experimental points describing the dependence of the flame front position on time R(t) was carried out.The interpolation was performed by a polynomial of degree 8. Figure 7 shows the derivative of the interpolation relationship that describes the apparent flame speed.In addition, in Figure 7, the dots indicate the numerical values of the flame velocity obtained by formal calculation of the time derivative obtained from the experimental values of the flame front position.
From the graph in Figure 7 of the dependence of the apparent flame velocity on the radius of the cloud of explosion products, we can see that starting from about R=0.40-0.45m, which can be considered critical (R KR ), there is some acceleration of the flame front.The obtained distance of the beginning of flame acceleration corresponds to the previously obtained critical radius (R KP =0.45 m) of the transition of the laminar combustion process of the propane-air mixture into a turbulent one.Photo 3 of Figure 6 corresponds to this moment.This photo shows the moment of the explosion at the 150th millisecond.
In addition, it follows from the graphs of Figure 7 that at R = R C = 0.60 m, which corresponds to 0.75 * R FB (R FB is the radius of the fireball or the final radius of the explosion products), the growth rate of the apparent flame velocity first decreases, and then the flame velocity begins to decrease at R = R FB =0.78 m explosive combustion ceases.This is because the movement of the mixture near the explosive cloud boundary, caused by the expansion of combustion products during central ignition of the mixture, leads to additional saturation of the boundary mixture with air.Saturation of the boundary mixture with air occurs due to diffusion, which leads to its "depletion" and significantly reduces the rate of explosive combustion of the mixture, which initially had a stoichiometric concentration.Let us consider the results of an experimental explosion of an acetylene-air mixture (hereinafter referred to as experiment 2), which was recorded with a high-speed movie camera at 240 frames per second.Figure 8 shows photographs of the process of the experimental explosion of acetylene-air mixture.The time intervals between photos are 25 ms.The experimental materials were processed using the procedure described above and used in processing the materials of the previous experiments to determine the apparent flame velocity of a deflagration explosion of an acetylene-air mixture.Experimental points describing the time dependence of the flame front position would be determined from the filming data.A curve was constructed using a 12th-degree polynomial interpolating the experimental points obtained.From the graph in Figure 9 of the dependence of the apparent flame velocity on the radius of the cloud of explosion products, we can see that starting from about R=0.15-0.20 m, there is some acceleration of the flame front.The obtained distance of the beginning of flame acceleration corresponds to the previously obtained critical radius (R KP =0.24 m) of the transition of the laminar combustion process of the acetylene-air mixture into a turbulent one.
The dependences of the apparent flame velocity on time and coordinate shown in Figure 9 suggest the following.An increase in the apparent flame velocity of an acetylene-air mixture compared to the flame velocity of a propane-air mixture results in a faster saturation of the mixture surrounding the combustion region with air.This is due to the increase in the diffusion coefficient due to the increase in the velocity of the mixture.Therefore, the depletion process of the mixture surrounding the combustion region is accelerated, and there is a decrease in the radius of the flame front, at which the apparent flame speed begins to decrease.The apparent flame velocity decrease begins to appear at R C =0.35 m at the 25th millisecond of the explosion.The second photo shown in Figure 8 corresponds to this moment in time.This photo corresponds to an explosion at the 25th millisecond.Significant air saturation of the mixture surrounding the combustion area is confirmed by the readings of temperature sensors located on the floor at a distance of 0.2 m from each other (see Photo 1 of Figure 8).
Figure 10 shows the time dependencies of temperature change at the indicated points.The graph in Figure 10 shows that the temperature of the explosion products corresponds to the value of the temperature at full combustion of fuel only at distances up to R=0.6 m.At greater distances, the temperature of the explosion products decreases, indicating a "depletion" of the mixture.

Temperature, K
The results of a series of experimental explosions of a propane-air mixture in a sufficiently large volume (2.25*2.25*2.0m 3 ) confirmed the general ideology describing the acceleration process of explosive combustion in free space [18][19][20].

Conclusion
The conducted experiments and performed theoretical studies have shown that hydrodynamic processes determine flame acceleration in explosive combustion.The numerical value of the apparent flame velocity in deflagration explosions determines the levels of explosive loads and, consequently, accidental damage.Therefore, issues related to hydrodynamic phenomena accompanying accidental explosions need considerable attention.
The characteristic dimensions at which laminar explosive combustion transitions to turbulent combustion have been determined experimentally.The dependence determining the flame acceleration in an accidental explosion is given.
It is experimentally shown that there is no rapid cessation of explosive combustion and flame arrest at the central initiation of a spherical explosive cloud of stoichiometric concentration.The visible flame velocity near the cloud boundary begins to decrease gradually, and when the size of the fireball (R=R FB ) is reached, explosive combustion ceases.This is due to the movement of the mixture near the explosive cloud boundary, which is caused by the expansion of the combustion products when the mixture is centrally ignited.Hydrodynamic flows lead to additional saturation of the boundary mixture with air.Saturation of the boundary mixture with air occurs due to diffusion, which leads to its "depletion" and significantly reduces the rate of explosive combustion of the mixture, which initially had a stoichiometric concentration.
The analysis of flame kinematic characteristics showed that the intensification of the explosive combustion process caused by turbulators on the flame path increases the apparent flame velocity and explosive pressure.An increase in the apparent flame velocity leads to an acceleration of the saturation process (due to diffusion) with air of the mixture surrounding the combustion region.This is due to the increase in the diffusion coefficient caused by the increase in the velocity of the mixture.Accordingly, the process of "depletion" of the mixture surrounding the combustion area is accelerated.The consequence is a decrease in the radius of the fireball (R FB ) and a decrease in the radius of the flame front, where the apparent flame speed decreases.

E3S 2023 ,
Web of Conferences 457, 02039 (2023) https://doi.org/10.1051/e3sconf/202345702039FCInumber, U is the flow velocity, d is the characteristic linear = • ν 0 dimension of the region under consideration, and v 0 is the kinematic viscosity coefficient based on the molecular structure of the medium.

Figure 9
shows the derivative of the interpolation dependence, which describes the apparent flame speed, and the numerical values of the flame speed obtained by formal 11 E3S Web of Conferences 457, 02039 (2023) https://doi.org/10.1051/e3sconf/202345702039FCI-2023 calculation of the time derivative obtained from the experimental values of the flame front position.