PHYSICS OF FLUIDS 30, 106101 (2018)

Moderation of flame acceleration in obstructed cylindrical pipesdue to gas compression

V’yacheslav Akkerman1,a) and Damir Valiev21Center for Innovation in Gas Research and Utilization (CIGRU),Center for Alternative Fuels, Engines and Emission (CAFEE),Computational Fluid Dynamics and Applied Multi-Physics Center,Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown,West Virginia 26506-6106, USA2Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineeringof the Ministry of Education, Department of Energy and Power Engineering, Tsinghua University,Beijing 100084, China

(Received 25 July 2018; accepted 11 September 2018; published online 2 October 2018)

The role of gas compression in the process of extremely fast flame acceleration in obstructed cylin-drical tubes is studied analytically and validated by computational simulations. This acceleration isassociated with a powerful jet-flow produced by delayed burning in spaces between the obstacles.The acceleration mechanism is Reynolds-independent and conceptually laminar, with turbulenceplaying only a supplementary role. In this work, the incompressible formulation is extended toaccount for small but finite initial flame Mach number up to the first-order terms. As a result,it is shown that flame-generated compression mitigates the acceleration process noticeably. Beingqualitatively the same as in two-dimensional channels, quantitatively, the effect is much stronger incylindrical pipes, which reduces the validity domain of the incompressible cylindrical-axisymmetricformulation as compared to the incompressible two-dimensional theory. The latter is also testedwhen the theory is validated by the computational simulations. Published by AIP Publishing.https://doi.org/10.1063/1.5049736

I. INTRODUCTION

Sporadic acceleration of an initially slow flame (deflagra-tion) front commands both fundamental and practical inter-ests,1,2 especially when flame acceleration (FA) is followedby a deflagration-to-detonation transition (DDT) event. On theone hand, prevention of FA and DDT is a critical need fromthe fire safety viewpoint. On the other hand, promotion of FAand fruitful usage of the subsequent DDT can improve thenovel combustion technologies, for instance, providing a keyelement in design and operation of pulse-detonation engines.1

While spontaneous FA in pipes with smooth walls is oftentimesattributed to wall friction,3–8 such acceleration is however rel-atively slow as compared to that occurring in the presenceof obstacles.9–16 While the role of obstacles was typicallyassumed to be in the generation of turbulence that augments theburning rate,9–12 Bychkov et al.13–18 have revealed that obsta-cles can provide a qualitatively new physical mechanism of FA.Specifically, the flame spreading through a “tooth-brush”-likearray of tightly packed obstacles in a pipe creates the pock-ets of the fresh fuel mixture trapped between the obstaclessuch that delayed burning in the pockets produces a powerfuljet-flow in the unobstructed segment of the pipe. Such a jet-flow renders a flame tip to move much faster, which producesnew trapped pockets and, consequently, generates a positiveflame-flow feedback, thereby leading to extremely strong FA

a)Author to whom correspondence should be addressed: Vyacheslav.Akkerman@mail.wvu.edu

and, subsequently, to an explosion of the fuel mixture evolv-ing into detonation. It is noted that while an optimal obsta-cles’ design was actively debated,1,9,12,14 such a “tooth-brush”configuration of Fig. 1 yields fastest FA due to continuousfeedback between the flame tip velocity and new pockets’ignition.14 Moreover, tightly packed obstacles provide a pre-dictable acceleration mechanism, supported by the analyticalmodels,13–15 which is opposed to the case of widely spacedobstacles, when a chaotic (turbulent) motion leads to higheruncertainty of the flame behavior. Since more predictableflame behavior is better for practical applications,17,18 a “tooth-brush” array of tightly packed obstacles is nowadays widelyemployed in the studies,13–18 including the present work.

To be more specific, the study13 quantified FA in two-dimensional (2D) channels, with the acceleration rate appear-ing independent of a Reynolds number associated with flamepropagation; Ref. 14 extended the theory13 from a 2D channelto a cylindrical-axisymmetric geometry. Both the formula-tions13,14 employed the approximation of an incompressibleflow, which holds well at the initial (quasi-isobaric) stage of FAfor relatively slow hydrocarbon-air flames. The theories havebeen validated by extensive numerical simulations13,14 as wellas by comparison to the experiments on propane-air flames.11

It is noted that the simulations showed a strong decrease inthe acceleration rate for faster flames with an increase in theinitial Mach number associated with flame propagation. Fur-thermore, moderation of FA was observed as soon as the flamevelocity in the laboratory reference frame became compara-ble to the speed of sound. Both of these effects indicate that

1070-6631/2018/30(10)/106101/11/$30.00 30, 106101-1 Published by AIP Publishing.

106101-2 V. Akkerman and D. Valiev Phys. Fluids 30, 106101 (2018)

FIG. 1. Schematic of flame propagation in an obstructed pipe.

compressibility of the burning gas could substantially influ-ence FA, which is especially important at the developedstages of acceleration. Obviously, the incompressible formu-lations13,14 were not able to explain the finite Mach numbereffects, and therefore there was/is a critical need to analyze therole of compressibility in the process of FA (and, presumably,subsequent DDT) in obstructed pipes. This problem has beenaddressed in Ref. 15, where the impact of gas compression hasbeen quantified in 2D obstructed channels. However, while a2D geometry is a fruitful simplifying approach, conventionallyadopted in numerous theoretical and computational studies,obstructed conduits are intrinsically three-dimensional (3D)in practice, resembling, in particular, cylindrical tubes or thatwith rectangular cross sections.10 Consequently, a compari-son with the future and ongoing experiments will presumablyrequire a 3D formulation rather than a 2D one.15 It is noted,in this respect, that an analytical, computational, and exper-imental study19 showed a significant difference between FAin 2D and 3D/cylindrical-axisymmetric unobstructed pipes,with the acceleration rate being substantially larger in thecylindrical tubes than that in the 2D channels. May we expectthe same effect for obstructed conduits?

This question is addressed in the present work, forobstructed cylindrical pipes. Specifically, the analytical for-mulation considers cylindrical-axisymmetric geometry, and itaccounts for small but finite gas compressibility by employingexpansion in small Mach numbers up to the first-order terms.It is shown that flame-generated compression mitigates theacceleration process noticeably, which explains moderationof acceleration observed in the simulations.14 Being qualita-tively the same as in 2D channels, quantitatively, the effect ismuch stronger in cylindrical pipes, which reduces the validitydomain of the incompressible cylindrical-axisymmetric for-mulation14 as compared to the incompressible 2D theory.13

The latter is also tested when the theory of Sec. II is validatedby the computational simulations in Sec. III.

II. ANALYTICAL FORMULATION

Figure 1 is a schematic of the problem, considering thecylindrical axisymmetric geometry. A premixed flame front

propagates from the closed end of a pipe of radius R, withideally slip and adiabatic wall and with a fraction αR blockedby the tightly spaced obstacles (∆z� R), while the inner frac-tion, i.e., a cylinder of radius (1 − α)R, α < 1, remainsunobstructed. As shown in Refs. 13 and 14, the flame frontmoves extremely fast along the central unobstructed segmentof the pipe leaving behind (in the deep narrow “pockets”between the obstacles) the fresh mixture that will be burnedlater, after the flame tip passes. According to Ref. 16, the block-age ratio should be, approximately, in the range 1/9 ≤ α ≤ 2/3to provide such obstacles-based FA. Similar to the original,incompressible theory,14 the spacing between the obstacles ∆zis not a parameter of the analytical formulation. It is neverthe-less assumed that the obstacles are “tightly packed” such thatthe spacing between them is much smaller than the obstaclelength, ∆z� αR < R; otherwise, the geometry would actuallynot be “tooth-brush”-like, and the Bychkov mechanism ofFA due to delayed burning in the pockets between the obsta-cles could be significantly modified or even suppressed byvorticity generated between the obstacles.18 Since ∆z� R,a flame is assumed to propagate mainly in the radial direc-tion in the pockets. As such, a standard approach of aninfinitesimally thin flame front moving normal to itself with anunstretched laminar velocity SL is adopted, with pressure- andtemperature-dependences of SL providing only negligible cor-rections in the first-order expansion in small Mach numbers.20

A characteristic Mach number associated with flame propa-gation is conventionally defined as Ma ≡ SL/c0, where c0

is the speed of sound in the fresh gas. In this work, weextend an incompressible (Ma→ 0) cylindrical-axisymmetricformulation14 to study FA in the compressible burning gasusing the expansion in small Ma up to the first order. In thismanner, the theory14 obviously provides the zeroth-order termsin the expansion. Overall, the main assumptions of the presentformulation (and the respective imposed limitations) are: (i)tight spacing between the obstacles (∆z� R), (ii) an unlim-ited length L of the pipe (R � L; L → ∞), and (iii) theideally slip and adiabatic surfaces of the pipe wall and obsta-cles. The appropriateness of such idealized surface condi-tions in the Bychkov model has been recently justified inour computational work.16 The major quantities characteriz-ing the flame dynamics in the present theory and modellingare the instantaneous flame tip position Z f (t) and its scaledcounterpart Z f /R. It is also expected that the derivative ofthis quantity, dZ f /dt, will correlate with the flame propa-gation velocity in the laboratory reference frame because aflame propagates mainly in the unobstructed segment of thepipe.

From the side of the burnt gas, the flame tip velocity in thelaboratory reference frame is the sum of the flame tip velocitywith respect to the burnt gas plus the flow velocity of the burntgas. Then the evolution equation for the flame tip reads

dZf /dt = uz,bf + ϑSL, (1)

where uz ,b(z, t) is the flow velocity of the gas in the z-direction,with uz ,bf = uz ,b(Z f , t) being the velocity just behind theflame front, and the instantaneous thermal expansion ratioϑ ≡ ρa/ρbf is determined as the ratio of the densities justahead (index “a”) and behind (index “bf ”) the flame front.

106101-3 V. Akkerman and D. Valiev Phys. Fluids 30, 106101 (2018)

We further designate the initial value of the expansion factoras Θ ≡ ϑ0 at t = 0.

We start with the limit of an incompressible flow, Ma→ 0.Thenϑ → Θ and the continuity equation simply reads∇·u = 0.The fresh mixture trapped in all pockets between the obstaclesis at rest, wp = 0, where w is the radial velocity componentand the subscript “p” stands for the fuel mixture in the pocket.Then the planar flame propagating in a pocket produces a flow,with the boundary at the border between the obstructed andunobstructed segments of the pipe being wb = −(Θ − 1)SL.Consequently, the velocity distribution in the burnt gas in theunobstructed segment of the pipe is14

ur,b = −Θ − 11 − α

SL

Rr = −σ0

r2

, uz,b = 2Θ − 11 − α

SL

Rr = σ0z,

σ0 ≡ 2Θ − 11 − α

SL

R.

(2)

Then, with Eq. (1), we obtain the differential equation for theflame tip,

dZf /dt = ΘSL + σ0Zf , (3)

with the solution

Zf =ΘSL

σ0

[exp(σ0t) − 1

],

orZf

(1 − α)R=

Θ

2(Θ − 1)[exp(σ0t) − 1

]. (4)

The flow velocity in the fuel just ahead of the flame frontfollows from the matching relation

ua = (dZf /dt) − SL = σ0Zf + (Θ − 1)SL. (5)

Obviously, ua → σ0Zf asymptotically in time as the flameaccelerates.

While Eqs. (2)–(5) reproduce the incompressible theory,14

we next extend the analysis to attain the accuracy of the firstorder in Ma by employing the method.15,19–21 Specifically, aslong as Ma is sufficiently smaller than unity, the flow aheadof the flame can be treated as isentropic such that the den-sity, pressure, and temperature of the fuel mixture obey thefollowing relations:20

ρa/ρ0 =

[1 +

(γ − 1)2

ua

c0

]2/(γ−1)

≈ 1 +ua

c0≈ 1 + Maσ0

Zf

SL+ Ma(Θ − 1), (6)

Pa/P0 =

[1 +

(γ − 1)2

ua

c0

]2γ/(γ−1)

≈ 1 + γua

c0≈ 1 + Maγσ0

Zf

SL+ Maγ(Θ − 1), (7)

Ta/T0 =

[1 +

(γ − 1)2

ua

c0

]2

≈ 1 + (γ − 1)ua

c0

≈ 1 + Ma(γ − 1)σ0Zf

SL+ Ma(γ − 1)(Θ − 1), (8)

with ρ0, P0, and T0 being the initial parameters in the fuel mix-ture, and the specific heat ratio (adiabatic index) γ = cP/cV . Inthis respect, the rigorous mathematical limitation of validityfor the present formulation reads ua/c0� 1, i.e.,

ua/c0 = Ma(Θ − 1){1 + Zf /(1 − α)R

}� 1. (9)

The matching relations at the flame front for the density,pressure, and temperature are15

ρa(Zf − uz,a) = ρbf (Zf − uz,bf ), (10)

Pa + ρa(Zf − uz.a)2 = Pbf + ρbf (Zf − uz,bf )2, (11)

Ta +QcP

+(Zf − uz,a)

2

2cP= Tbf +

(Zf − uz,bf )2

2cP, (12)

respectively, where Q ≡ (Θ−1)cPT0 is the specific heat releasein the reaction and cP is the specific heat at constant pressure.Neglecting higher-order terms in Ma, we reduce Eqs. (11) and(12) to Pa = Pbf and Ta + (Θ − 1)T0 = Tbf , respectively, suchthat Eq. (10) and the ideal gas law, ρaTa = ρbf Tbf , yield in thefirst-order approximation in Ma� 1,

ϑ ≡ ρa/ρbf = 1 + (Θ − 1)(T0/Ta) ≈ Θ −Ma(γ − 1)(Θ − 1)2

−Ma(γ − 1)(Θ − 1)σ0Zf /SL. (13)

To find the pressure in the burnt gas, Pb, we next employ theEuler equation,

∂ uz,b

∂ t+ uz,b

∂uz,b

∂z= −

1ρb

∂Pb

∂z. (14)

With the first-order accuracy in Ma � 1, Eq. (14) actuallyyields quasi-uniform pressure in the burnt gas, i.e., Pb(t) ≈ Pbf .However, being uniform in space, this pressure grows in time,thus increasing the temperature and density of the burnt gasdue to adiabatic compression.

We next consider flame propagation in a single pocketbetween the obstacles, Fig. 1. Within the same accuracy ofMa, the pressure of the fuel mixture in the pocket is the sameas that of the burnt gas, namely, Pb = Pb(t) = Pbf = Pa

given by Eq. (11). Then we have adiabatic pre-compressionof the fuel mixture in the pocket from the initial density ρ0

to ρp. It is noted that the density and temperature of the fuelmixture in the pocket are the same as those of the fuel mixturejust ahead of the flame front in the unobstructed segment ofthe pipe, ρp = ρa and Tp = Ta, since in both cases, we haveadiabatic compression and the same final pressure, Pb(t) = Pbf

= Pa. Then the compressible continuity equation for the fuelmixture in the pocket, Rf < r < R, reads

1r∂

∂r

(ur,pr

)+∂uz,p

∂z= −

1ρp

dρp

dt= −

1γPp

dPp

dt= −

1γPa

dPa

dt,

(15)

with the solution obeying the matching relation at the wall,r = R,

ur,p =1

2γPa

dPa

dt

(R2

r− r

), ur,pa =

12γPa

dPa

dt

(R2

Rf− Rf

),

(16)

where the second term in Eq. (15) is neglected, and the sub-script “pa” denotes the flow velocity in the pocket just aheadof the flame, at r = Rf . Using the matching relations at theflame, we find the velocity in the burnt gas just at the flamefront in the pocket, subscripted by “pbf,” as

106101-4 V. Akkerman and D. Valiev Phys. Fluids 30, 106101 (2018)

ur,pbf = ur,pa − (ϑ− 1)SL =1

2γPa

dPa

dt

(R2

Rf−Rf

)− (ϑ− 1)SL.

(17)

In a similar manner, we subsequently solve the continuityequation in the burnt gas in the pocket, with relation (17) beingthe boundary condition at the front, r = Rf . The solution reads

ur,pb =1

2γPa

dPa

dt

(R2

r− r

)− (ϑ − 1)SL

Rf

r. (18)

Then the gas velocity at the interface between the obstructedand unobstructed segments of the pipe, i.e., at the radiusr = (1 − α)R, is given by

ur,pb =1

2γPa

dPa

dt

(R2

(1 − α)R− (1 − α)R

)− (ϑ − 1)SL

Rf

r

=1γPa

dPa

dtα(2 − α)2(1 − α)

R −(ϑ − 1)SL

(1 − α)RRf . (19)

In the limit of weak gas compressibility, the continuityequation in the unobstructed segment of the pipe takes theform

1r∂

∂r(ur,br

)+∂uz,b

∂z= −

1ρ

dρdt= −

1γPa

dPa

dt, (20)

with the solution at r = (1 − α)R being

ur,b = −(ϑ − 1)

(1 − α)2

SLRf

R2r +

1γPa

dPa

dtα(2 − α)

2(1 − α)2r, (21)

uz,b = 2(ϑ − 1)

(1 − α)2

SLRf

R2z −

1γPa

dPa

dtz

(1 − α)2. (22)

As a result, with a further approximation Rf (Ztip) ≈ (1 − α)R,the evolution equation for the flame tip, uz ,b = uz ,bf at z = Z f ,Eq. (1), becomes

dZf /dt = ϑSL + uz,bf

= ϑSL +

[2

(ϑ − 1)(1 − α)

SL

R−

1γPa

dPa

dt1

(1 − α)2

]Zf . (23)

Finally, substituting Eqs. (11) and (13) into Eq. (23), omittingthe second- and higher-order terms in Ma, and accounting forthe zeroth-order approximation (3), we arrive at the differentialequation

dZf /dt = ψSL + σ1Zf −Maχσ20Z2

f /SL, (24)

with σ0 of Eq. (2) and other designations being

χ = γ − 1 + (1 − α)−2, ψ/Θ = 1 −Ma(γ − 1)(Θ − 1)2/Θ,

(25)

σ1/σ0 = 1 −Ma{2(γ − 1)(Θ − 1) + Θ(1 − α)−2

}. (26)

In the incompressible limit, Ma = 0, we have σ1 = σ0, ψ = Θ,and Eq. (24) reproduces Eq. (3). However, accounting forgas compressibility, we obtain moderation of acceleration inEq. (24), which is provided by two terms: the linear term withσ1Z f and the nonlinear one Maχσ2

0Z2f /SL. The linear term

does not change the exponential state of FA, although it reducesthe acceleration rate to σ1 as compared to the incompressiblevalue σ0. In other words, at the initial stage, Z f → 0, acceler-ation is moderated by the linear term only such that Eq. (24)reduces to

dZf /dt ≈ ψSL + σ1Zf , (27)

with the solution

Zf =ψSL

σ1

[exp(σ1t) − 1

]. (28)

Obviously, Eqs. (27) and (28) have the same qualitative formas Eqs. (3) and (4), respectively, with minor quantitative cor-rections incorporated. Nevertheless, the nonlinear term ofEq. (24), Maχσ2

0Z2f /SL, becomes important quite fast and

modifies the exponential state of acceleration to a slower one.The general analytical solution to Eq. (24) takes the form

Zf =2ψSL

[exp(σ2t) − 1

](σ2 − σ1) exp(σ2t) + (σ2 + σ1)

≈ψSL

σ2×

exp(σ2t) − 11 + Maχψ

[exp(σ2t) − 1

] , (29)

where

σ2 =

√σ2

1 + 4Maχψσ20 ≈

√σ2

1 + 4MaχΘσ20

≈ σ1(1 + 2Maχψ) ≈ σ1(1 + 2MaχΘ). (30)

While the result (29) resembles (4) and (28) while the Ma-termin the denominator is small, this term becomes important verysoon such that Eq. (29) eventually saturates to the limit

Zf ,sat =SL

Maχσ2≈

SL

Maχσ0=

(1 − α)2(Θ − 1)

RMaχ

. (31)

Equation (31) provides one more limitation on the present for-mulation because a flame should propagate through a largeenough number of obstacles, i.e., Zf ,sat � R or, from Eq. (31),

Maχ� SL/σ2R ∼ SL/σ0R ≈(1 − α)

2(Θ − 1),

or

Ma�(1 − α)

2(Θ − 1)χ≈

(1 − α)3

2(Θ − 1). (32)

For instance, for Θ = 8 and α = 1/3, Eq. (32) yieldsMa � 0.02; for Θ = 8 and α = 1/2, we have Ma � 0.01;finally, for Θ = 8 and α = 2/3, this yields Ma� 0.0025.

It is noted that Eqs. (24)–(30) remarkably resemble the2D formulation,15 except for the 2D factor 1 − α, which isreplaced by (1 − α)2 in Eqs. (25) and (26) for χ and σ1. Thuswe predict a 2D channel and a cylindrical pipe, both obstructed,to be equivalent in terms of the flame tip acceleration rate if(1 − α2D) = (1 − αcyl)2, i.e.,

α2D = αcyl(2 − αcyl) or αcyl = 1 −√

1 − α2D. (33)

Therefore, obstructed cylindrical pipes are expected to yieldstronger acceleration as compared to that in 2D channels pro-vided that α is the same in both cases. In other words, forexample, α2D = 1/3, 1/2, and 2/3 provide the same flametip acceleration rate as αcyl = 0.18, 0.29, and 0.42, respec-tively, whereas αcyl = 1/3, 1/2, and 2/3 are equivalent interms of FA to α2D = 5/9, 3/4, and 8/9.

III. DESCRIPTION OF THE COMPUTATIONALSIMULATIONS

Equations (24)–(26) with the solution (29) present arevised theory of FA in obstructed cylindrical pipes with oneextreme closed. Unlike the original incompressible theory,14

106101-5 V. Akkerman and D. Valiev Phys. Fluids 30, 106101 (2018)

Eqs. (2)–(5), the present formulation accounts for small butfinite Ma up to the first order terms. While Eq. (9) provides arough evaluation for a limitation of the approach, the formu-lation is next validated and its validity domain is quantifiedby performing the computational simulations of the hydrody-namic and combustion equations including transport processes(thermal conduction, diffusion, and viscosity) and one-stepArrhenius chemistry. Similar to the formulation of Sec. II,a cylindrical-axisymmetric configuration is employed in thesimulations, with the basic equations in the form

∂ρ

∂t+

1r∂

∂r(rρur) +

∂

∂z(ρuz) = 0, (34)

∂

∂t(ρur) +

∂

∂z(ρuzur − ζzr) +

1r∂

∂r

[r(ρu2

r − ζrr

)]+∂P∂r

+1rζθθ = 0, (35)

∂

∂t(ρuz) +

∂

∂z

(ρu2

z − ζzz

)+

1r∂

∂r[r(ρuzur − ζzr)

]+∂P∂z= 0,

(36)∂ε

∂t+∂

∂z[(ε + P)uz − ζzzuz − ζzrur + qz

]+

1r∂

∂r[r((ε + P)ur − ζrrur − ζzruz + qr)

]= 0, (37)

∂

∂t(ρY ) +

1r∂

∂r

(rρurY − r

µ

Sc∂Y∂r

)+∂

∂z

(ρuzY −

µ

Sc∂Y∂z

)= −

ρYτR

exp(−Ea/RuT ), (38)

where ε = ρ(QY + cPT ) + ρ(u2z + u2

r )/2 is the total energyper unit volume and Y is the progress variable showing themass fraction of the fuel mixture. The ideal gas model isemployed such that P = ρRuT/m and cP − cV = Ru, whereRu = 8.31 J/(mol K) is the universal gas constant; boththe fresh and the burnt gases have the same molar massesm = 2.9 × 10−2 kg/mol; cV = 5Ru/2m and cP = 7Ru/2m arethe specific heats, with γ = cP/cV = 1.4. The stress tensor ζαβis given by

ζrr = µ

(43∂ur

∂ r−

23∂uz

∂z−

23

ur

r

),

ζzz = µ

(43∂uz

∂z−

23∂ur

∂r−

23

ur

r

),

(39)

ζθθ = µ

(43

ur

r−

23∂uz

∂z−

23∂ur

∂r

), ζrz = µ

(∂uz

∂r+∂ur

∂z

),

(40)

and the energy diffusion vector qα takes the form

qr = −µ

(cP

Pr∂T∂r

+QSc

∂Y∂r

), qz = −µ

(cP

Pr∂T∂z

+QSc

∂Y∂z

),

(41)

with the dynamic viscosity µ, the Prandtl number Pr, and theSchmidt number Sc. To avoid the diffusional-thermal instabil-ity, we take the unit Lewis number Le ≡ Sc/Pr = 1 with Pr =Sc = 1 and µ = 1.7 × 10−3 N s/m2. The initial density, pres-sure, and temperature are ρ0 = 1.16 kg/m3, P0 = 105 Pa, and

T0 = 300 K, respectively, with the initial thermal expansionratio being Θ ≡ ϑ0 ≡ ρa0/ρbf 0 = 8. Equations (34)–(37)represent the conservation of mass, momentum, and energy,whereas Eq. (38) describes a single irreversible Arrhenius reac-tion of the first order, with an activation energy Ea and a char-acteristic constant of time dimension τR, which is estimatedto obtain a particular value of the unstretched laminar burningvelocity SL by solving the associated eigenvalue problem. Theburning zone is conventionally described by the parameter oflength dimension Lf ≡ µ/Prρ0SL, which is oftentimes identi-fied as a characteristic flame thickness and specifies the appro-priate numerical mesh. It is nevertheless noted that the quantityLf is just a thermal-chemical parameter of length dimension,while the burning zone may be an order of magnitude wider inreality.2 To have a better resolution of the reaction zone, in oursimulations, we employed a relatively small activation energy,Ea/RuT0 = 32. To investigate the role of obstacles, we havevaried the blockage ratio in the range α = 1/3, 1/2, and 2/3.It is noted, in this respect, that if the blockage ratio was toosmall, α� 1, the pockets between the obstacles would dis-appear and the obstacles-based acceleration scenario wouldreduce to the finger-flame acceleration scenario.19 A rangeof applicable blockage ratios has been investigated in ourrecent work,16 which identified the minimal threshold block-age ratio providing the obstacles-based acceleration mecha-nism to beαc ∼ 1/Θ = 1/8 (an analytical prediction) orαc = 1/9(a computational result). Similarly, the Bychkov accelerationmechanism will stop working in the opposite limit of too largeblockage ratio, (1 − α)� 1, because the central unobstructedpipe segment will disappear in that case. According to themodelling,16 the obstacles-based acceleration mechanism stillworks as long as α ≤ 2/3. Consequently, all the blockageratios employed in the present work is within such an “appli-cability” domain 1/8 ≤ α ≤ 2/3. To identify the role of gascompression, we used various initial flame propagation Machnumbers in the range Ma ≡ SL/c0 = 0.001 ∼ 0.01, where thelower value, 0.001, corresponds to the typical methane andpropane flames. The theory of Sec. II does not involve theReynolds number, thus yielding a minor dependence on thepipe radius as long as R � Lf . Therefore, to reduce the com-putational costs, we used a relatively narrow pipe, R = 30 Lf ,with a Reynolds number characterizing the flame propaga-tion in the form Ref ≡ RSL/ν = R/Lf = 30. Obviously, aReynolds number associated with a flow, Re = 〈uz〉R/ν, mayexceed Ref by several orders of magnitude, due to FA andthermal expansion of the burning gas, which thereby allowsa possibility of turbulization of the flame and the flow. Whilethe theory of Sec. II does not involve a spacing between theobstacles∆z, this is certainly an inherent parameter in the com-putational simulations. In this particular work, the obstacles’spacing was chosen as small as ∆z = 0.25R. Our previoussimulations14,16 demonstrated that the impact of ∆z is minorindeed as long as the obstacles’ spacing does not exceed theobstacles’ length. In contrast, for the large spacing such as∆z = 2R, the Bychkov mechanism stops working and theimpact of ∆z becomes profound, in particular, due to vorticityemerging in large pockets.16 The Bychkov mechanism of FAis also expected to stop working in the opposite limit of very-very tightly packed obstacles. We believe this may occur when

106101-6 V. Akkerman and D. Valiev Phys. Fluids 30, 106101 (2018)

∆z becomes comparable to the flame thickness,∆z ∼ Lf . How-ever, the event of ∆z ∼ Lf is not the case in our present simu-lations, where we have ∆z = 0.25 × 30 Lf = 7.5 Lf . Overall,a range of obstacles’ spacing providing the Bychkov mech-anism of FA in obstructed pipes can be evaluated, approx-imately, as Lf � ∆z� αR < R. To avoid impacts of wallfriction and heat losses, the slip (n · u = 0) and adiabatic(n · ∇T = 0) surfaces of the pipe walls and obstacles areemployed, where n is the normal unit vector to the surface.Our recent computational study16 demonstrated that variationsof the wall and obstacles’ surface conditions provide a minoreffect on flame propagation in obstructed pipes, which therebyjustifies the choice of idealized boundary conditions employedin the present theory and modelling. It is noted that this situa-tion is conceptually different from unobstructed pipes, wherewall boundary conditions influence FA substantially.3–8 Inaddition, the non-reflecting boundary conditions at the openend of the pipe are used. A hemispherical premixed flameembryo of radius rf , rf � (1 − α)R, ignited at the pipe center-line, at the closed end of the pipe, is employed as the initialcondition for the simulations, with the internal flame struc-ture imitated by the classical Zeldovich-Frank-Kamenetskii(ZFK) analytical solution.22 In a hemispherical geometry, theZFK solution reads

T = Tf + (Tb − Tf ) exp(−√

r2 + z2/Lf

), if z2 + x2 < r2

f ,

T = ΘTf , if z2 + x2 > r2f ,

Y = (Tb − T )/(Tb − Tf ), P = Pf , ux = 0, uz = 0.

(42)

We use a cylindrical-axisymmetric Navier-Stokes codewhose core was originally developed in Volvo Aero. The solverwas subsequently widely tested and utilized successfully innumerous combustion problems6,7,13–19 as well as in studies ofaero-acoustics and transonic radiative jets,23–25 including themodeling of flame instabilities, FA and DDT. The code is there-fore robust and accurate, and it was subsequently upgraded14

to incorporate an array of obstacles as in Fig. 1. More detailsof the numerical approach and the basic elements of the solverare presented, for instance, in Refs. 14 and 19. In particular, thenumerical approach is based on a cell-centered, finite-volumenumerical scheme, which is of the second-order accuracy intime, of fourth-order in space for the convective terms, and ofsecond-order in space for the diffusive terms.

The computational mesh in the solver is rectangular, withthe cell walls parallel to the radial and axial directions. Inthis particular work, a uniform grid with the quadratic cellsof size 0.2Lf was used to ensure a possibility of a curvedflame to propagate freely in both the radial and the axial direc-tions. The longitudinal size of the calculation domain changesdynamically, following the leading pressure wave. Here, thesplines of the third order are used for re-interpolation of theflow variables during periodic grid reconstruction to preservesecond-order accuracy of the numerical scheme. Such a meshhas been successfully tested (in this particular geometry) inRef. 14. Specifically, the resolution test has compared the flametip acceleration rates for the grids of sizes 0.125Lf , 0.25Lf , and0.5Lf and the computational inaccuracy did not exceed 6%.

IV. RESULTS AND DISCUSSION

We start the discussion of the results obtained with the 3Dcolor snapshots of Fig. 2, where the evolution of a flame frontin a cylindrical-axisymmetric obstructed tube is representedby the four consecutive instantaneous snapshots [Figs. 2(a)–2(d), respectively]. It is noted that the “boxes” are shown toillustrate the relative flame size and do not represent the com-bustion chamber geometry, which is cylindrical-axisymmetric.This particular simulation run is related to Ma = 10−2 andα = 2/3. It is clearly seen that while the flame expands almosthemi-spherically in the unobstructed part of the channel at theinitial stage of burning, Fig. 2(a), the pockets of the fresh gasare formed as soon as the flame front penetrates the obstructed

FIG. 2. A 3D representation of the time evolution ofan axisymmetric cylindrical flame for Ma = 10−2 andα = 2/3. The four consecutive time instants are shown,namely, τ ≡ SL t/R = 0.05 (a), 0.1 (b), 0.15 (c), and0.2 (d). The “boxes” are shown to illustrate the relativeflame size and do not represent the combustion chambergeometry, which is cylindrical-axisymmetric.

106101-7 V. Akkerman and D. Valiev Phys. Fluids 30, 106101 (2018)

segment, Fig. 2(b). Subsequently, as expected, delayed burn-ing in these pockets generates a jet flow that promptly pushesthe flame tip towards the fuel mixture, thereby generating newpockets behind the flame tip; see Figs. 2(c) and 2(d). Conse-quently, there is a positive feedback between the jet and thenew pockets, which provides extremely fast FA.

Figures 3(a)–3(d) validate the analytical formulation ofSec. II by the computational simulations described above.Specifically, the analytical and computational time evolutionsof the scaled flame tip position, Z f /R, are compared for variousblockage ratios α and flame propagation Mach numbers Ma.In all four figures, the incompressible model,14 Eq. (4), its lin-ear modification, Eq. (28), and the general solution, Eq. (29),are presented by the dotted, dashed, and solid lines, respec-tively, with the simulations presented by the markers. WhileFigs. 3(a)–3(c) are related to the fixed Ma = 10−3 at vari-ous α = 1/3 (a), 1/2 (b), and 2/3 (c), Fig. 3(d) correspondsto a different Mach number, Ma = 3.5 × 10−3, with α = 1/3.It is seen in all four figures that the difference between the“linear” formulations (3) and (28) is minor, but they both dif-fer, substantially, from Eq. (29). This thereby certifies the roleof the last (nonlinear) term in Eq. (24). Comparison of thesolid and dotted curves justifies that gas compression mod-erates FA and quantifies such moderation. The effect is pro-moted with the Mach number and the blockage ratio, but thesame concerns the inaccuracy of the model. Indeed, Eq. (29)agrees with the simulations much better than the incompress-ible theory,14 Eq. (4), but this tendency breaks as soon aswe go beyond the limitation of the present theory in termsof α, Ma, and time. Namely, while agreement is quantita-tively good in Figs. 3(a) and 3(b) (for small α and Ma), itis rather qualitative than quantitative in Figs. 3(c) and 3(d).

Moreover, even initially good agreement as that observedat Figs. 3(a) and 3(b) starts breaking relatively prompt atτ ≡ SLt/R ≈ 0.15 ∼ 0.2.

The observation of Figs. 3(a)–3(d) can be explained asfollows. While the modeling has been performed for a rela-tively wide range of parameters, Ma = 0.001, 0.0035, 0.005,0.0075, and 0.01 andα = 1/3, 1/2, and 2/3, in fact, only fewof the pairs {Ma; α} obey the limitation imposed by Eq. (32)and provide best agreement between the theory and modeling,namely, Ma = 0.001 and 0.0035 for α = 1/3 and Ma = 0.001for α = 1/2, whereas all the results for α = 2/3 appear beyondthe criterion (32) such that only qualitative (if any) rather thanquantitative agreement between the simulations and Eq. (29)could be expected for α = 2/3. Overall, all the results of Fig. 3are qualitatively the same as those in a 2D geometry.15 How-ever, in a cylindrical pipe, the effect of gas compression ismuch stronger than that in a 2D channel, as was explained,theoretically, in Sec. II by means of the extra factor (1 − α)in the expressions for χ, Eq. (25), and σ1, Eq. (26). Conse-quently, the validity domain of the present theory is smallerthan that of 2D.15

We next investigate the rest of the computational results.Even being beyond the limit of the formulation of Sec. II, wewould nevertheless like to describe these results, with a relationto our theory, at least qualitatively. The whole set of simula-tions is presented in Fig. 4(a), where the time evolution of Z f /Ris plotted for all α and Ma. Within this cloud of markers, differ-ent types of symbols correspond to different blockage ratios,with triangles, squares, and circles representing α = 1/3, 1/2,and 2/3, respectively. The different Ma are shown by differentcolors. As expected, it is seen in Fig. 4(a) that FA is promotedwith α but diminishes with Ma.

FIG. 3. The scaled flame tip position Z f /R versus the scaled time SL t/R for Θ = 8 and Ma = 10−3, α = 1/3 (a); Ma = 10−3, α = 1/2 (b); Ma = 10−3, α = 2/3(c); and Ma = 3.5 × 10−3, α = 1/3 (d). The dotted lines represent the incompressible flows, Eq. (4), the dashed lines show the linear compressible approach,Eq. (28), and the solid lines are devoted to the complete formulation, Eq. (29). The numerical simulations are shown by the markers.

106101-8 V. Akkerman and D. Valiev Phys. Fluids 30, 106101 (2018)

FIG. 4. (a) The scaled flame tip position Z f /R versus thescaled time SL t/R for the thermal expansion ratio Θ = 8,various Mach numbers associated with flame propaga-tion Ma = 0.001, 0.0035, 0.005, 0.0075, and 0.01, shownby different colors, and various blockage ratios α = 1/3,1/2, and 2/3 shown by the triangles, squares, and circles,respectively. (b) The quantity Zfσnum/ΘSL versusσnumtforΘ = 8, various Ma = 0.001, 0.0035, 0.005, 0.0075, and0.01 (different colors), and variousα = 1/3 (triangles), 1/2(squares), and 2/3 (circles).

We next try to find a unified formulation such that theentire cloud of markers in Fig. 4(a) is aimed to collapse intoa single (preferably) or at least few curves. First of all, all thesimulation runs of Fig. 4(a) demonstrate various exponentialtrends. We thereby extract the fitting exponential growth ratesfor each run, σnum. To justify the exponential state of FA, inFig. 4(b), Zfσnum/ΘSL is plotted versus σnumt for all cases ofFig. 4(a). It is clearly seen that the entire set of markers col-lapses now into a unified curve, which in turn fits the exponentshown by the solid line.

In addition to the identification of the exponential trend,undertaken by means of Fig. 4(b), we also would like to relatethis computational trend to the theoretical prediction. For thispurpose, the theoretical and numerical acceleration rates σare compared in Figs. 5(a) and 5(b). Specifically, Fig. 5(a)presents σ versus Ma for various α = 1/3, 1/2, and 2/3, shownby different colors. Here, the horizontal dashed lines show the

Ma-independent quantity σ0, Eq. (2), while σ1, Eq. (26), isshown by the solid lines, and the numerical results are pre-sented by the markers. It is seen that both the incompressible,Eq. (2), and compressible, Eq. (26), predictions generallyexceed the simulation results. However, the general trend isthe same: σnum decreases with Ma and grows with α. It shouldalso be noted that there is no reason to compare σnum with σ2

because σ2 > σ0 and would therefore yield less agreement.In fact, this demonstrates one more time that Eq. (29) doesnot simply modify the exponential state of acceleration, fromσ1 to σ2. Instead, Eq. (29) yields another trend, slower thanthe exponential one, due to a Ma-term in the denominator ofEq. (29).

Figure 5(b) presents the computational acceleration ratescaled by the theoretical one, σnum/σtheor , versus Ma, withsquares, triangles, and circles related to α = 1/3, 1/2, and 2/3,respectively. Here, the two scalings by σ0, Eq. (2), and σ1,

106101-9 V. Akkerman and D. Valiev Phys. Fluids 30, 106101 (2018)

FIG. 5. The scaled exponential acceleration rate σR/SL(a) and σnum/σtheor (b) versus the Mach number asso-ciated with flame propagation Ma for various blockageratios:α = 1/3 (triangles), 1/2 (squares), and 2/3 (circles).The quantities σ0 and σ1 are given by Eqs. (2) and (26),respectively, while the numerical results are presented bythe markers.

Eq. (26), are employed. Namely, the empty markers showσnum/σ0, while the filled ones are related to σnum/σ1. Itis interesting to see that both ratios σnum/σ0 and σnum/σ1

show a weak dependence on α, at least as long as α and Maare small. In fact, this computational finding agrees with theanalytical prediction of the present work. Indeed, while thequantity α is immersed into σ0, Eq. (2), the correction factorσ1/σ0 shows only a weak α-dependence as long as Ma � 1;see Eq. (26). The simulations show the same trend as thetheory: the numerical value σnum tends to reproduce a theoret-ical σ, with α immersed into σ0 and a minor α-dependenceof the value σnum/σtheor as long as Ma � 1. Consequently,with this scaling, we may remove such an important param-eter as the blockage ratio, which is a notable step towardsa unified formulation. Moreover, while the function σnum/σ0

depends on Ma (it decreases with Ma) as expected, the functionσnum/σ1 practically does not for α = 1/3; its Ma-dependence

is very weak for α = 1/2 and even for α = 2/3 as long asMa ≤ 0.005.

In this light, even beyond the validity domain of thetheory of Sec. II, an empirical formula σ = Cσ1 can beproposed, with σ1 = σ1(Θ, α, Ma) of Eq. (26) and a phe-nomenological constant C that may depend, slightly, on theblockage ratio. For example, the present modelling yieldsC ≈ 0.6, 0.63, and 0.73 for α = 1/3, 1/2, and 2/3, respec-tively. To justify such an idea, Fig. 6 presents a counterpartof Fig. 4(b), but for the quantity σ1 instead of σnum. Namely,the quantity Zfσ1/ΘSL is plotted versus σ1t. Now the diver-sity is less than that in Fig. 4(a), but the coincidence is stillnot as good as in Fig. 4(b). Nevertheless, the Ma-dependenceidentified by different colors is clearly seen in Fig. 6. Further-more, all squares and triangles fall into a unified curve quitewell. While these markers appear not too close to the purelytheoretical curve (solid line), they however agree well with

106101-10 V. Akkerman and D. Valiev Phys. Fluids 30, 106101 (2018)

FIG. 6. The quantity Zfσ1/ΘSL versus σ1t for Θ = 8,various Mach numbers associated with flame propagationMa = 0.001, 0.0035, 0.005, 0.0075, and 0.01 (differentcolors), and various blockage ratios α = 1/3 (triangles),1/2 (squares), and 2/3 (circles). The solid line is relatedto the exponential acceleration rate σ = σ1 of Eq. (26),while the dashed-dotted lines are related to the phe-nomenological dependenceσ = Cσ1, with C = 0.6, 0.63,and 0.73 found for α = 1/3, 1/2, and 2/3, respectively.

the modified curves, with σ = Cσ1, for C ≈ 0.6, 0.63. Con-sequently, the simulation runs for α = 1/3 and α = 1/2 obeythe trend exp(σt) with σ = Cσ1 very well. Even for α = 2/3(circles), this agreement is reasonable, though not as good.

Finally, it is noted that since the aim of the present workwas to extend the cylindrical theory,14 by accounting for gascompression, and to validate the new formulation by means ofcomputational simulations; in the present theory and model-ing, we employed the same simplifying assumptions (such asthe unity Lewis number, some identical properties of the burntand unburnt gas, idealized boundary conditions at the walls,etc.) as in the previous work.14 Otherwise, a comparison ofthese two formulations would not be self-consistent and theimpact of gas compressibility could not be distinguished andanalyzed accurately. It is nevertheless recognized that prac-tical reality may differ noticeably from the assumptions wemade. In particular, premixed combustion is oftentimes non-equidiffusive such that Le , 1. The impact of Le on FA isexpected to be profound. Namely, Le < 1 flames are expectedto accelerate faster than the equidiffusive (Le = 1) ones becauseof the diffusional-thermal instability. In contrast, Le > 1flames may propagate slower because of the flame thicken-ing (a thicker flame front is harder to be corrugated than athinner one). Anyway, a detailed analysis of an impact of Leon FA in obstructed pipes requires a separate study, which willbe presented elsewhere.

V. CONCLUSION

In this investigation, we have successfully demonstrated,analytically and computationally, that gas compression isresponsible for moderation of FA in cylindrical obstructedtubes observed in the simulations.14 Specifically, the analyti-cal formulation of Sec. II considers a cylindrical-axisymmetricgeometry, and it accounts for small but finite gas compress-ibility by employing expansion in small flame Mach numbers,Ma ≡ SL/c0�1, up to the first-order terms. In this respect,the theoretical analysis resembles a 2D formulation15 in terms

that moderation of FA is described as a combination of a lin-ear plus and a nonlinear effect. The linear effect reduces therate of exponential FA with the initial Mach number at the ini-tial stage of the process, whereas the nonlinear effect becomesimportant as the flame moves away from the closed tube end.It is shown that flame-generated compression mitigates theacceleration process strongly, modifying the acceleration statefrom an exponential to a slower one. Being qualitativelythe same as in 2D channels, quantitatively, the effect ismuch stronger in cylindrical pipes, which reduces the validitydomain of the incompressible cylindrical-axisymmetric for-mulation14 as compared to the incompressible 2D theory.13

The analytical formulation of Sec. II is subsequently validatedby the computational simulations.

ACKNOWLEDGMENTS

V’yacheslav Akkerman is supported by the U.S. NationalScience Foundation (NSF), through the CAREER Award No.1554254, as well as by the West Virginia Higher EducationPolicy Commission, through the Grant No. HEPC.dsr.18.7.Damir Valiev is supported by the National Science Foundationof China (NSFC), through the Grant No. 51750110503, as wellas by the Thousand Young Talents Plan program.

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