A numerical study on propagation of premixed flames in small tubes

Combustion and Flame 146 (2006) 283–301www.elsevier.com/locate/combustflame

A numerical study on propagation of premixed flames insmall tubes

Nam Il Kim a,∗, Kaoru Maruta b

a School of Mechanical Engineering, Chung-Ang University, Heukseok, Dongjak, Seoul 156-756, Republic of Koreab Institute of Fluid Science, Tohoku University, 2-1-1, Katahira, Aoba, Sendai 980-8577, Japan

Received 20 September 2005; received in revised form 24 January 2006; accepted 15 March 2006

Available online 18 April 2006

Abstract

A premixed flame in a tube suffers strong variation in its shape and structure depending on boundary conditions.The effects of thermal boundary conditions and flow fields on flame propagation are numerically investigated.This study employs eight combinations of thermal and velocity boundary conditions. Navier–Stokes equationsand species equations are solved with a one-step irreversible global reaction model of methane–air mixtures.Finite volume method using an adaptive grid method is applied to investigate the flame structure. In the caseof an adiabatic wall, friction force on the wall significantly affected the flame structure while in the case of anisothermal wall, local quenching near the wall dominated flame shape and propagation. In both cases, variationsof flow fields occurred not only in the near field of the flame but also within the flame itself, which affectedpropagation velocities. Near the quenching conditions, strong similarity in the flame structure was found regardlessof the boundary velocity profiles due to self-induced velocity deformation. This study provides an overview of thecharacteristics of flames in small tubes at a steady state.© 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Propagation velocity; Flow redirection; Flame stabilization; Quenching

1. Introduction

Flame propagation in a tube has been studied asa fundamental issue in combustion research, despiteits simple geometry. A number of studies have beenconducted under various conditions of fuel composi-tion, tube size, pressure, etc. These works have beensummarized in many articles [1–4]. Recently, grow-ing interest in small-scale machines has motivatedthe development of small-scale combustors as energysources due to the higher energy density of fossil fu-

* Corresponding author. Fax: +82 2 814 9476.E-mail address: [email protected] (N.I. Kim).

0010-2180/$ – see front matter © 2006 The Combustion Institute.doi:10.1016/j.combustflame.2006.03.004

els in comparison with commercial batteries [5–7]. Indeveloping small-scale combustors, the most impor-tant issue is maintaining a flame in a small confinedspace. A flame in a tube constitutes one of the sim-plest models of such a small-scale combustor.

In principle, a flame can be stabilized in a tubeby adjusting the mean velocity to its propagation ve-locity if the tube diameter is larger than a criticalscale (quenching diameter). In practical experiments,however, stabilizing a flame in a small tube is muchmore difficult, mainly due to the great difference incharacteristic time scales between the flame and thetube [8]. Fig. 1a shows a stabilized flame under acondition close to quenching. Additionally, some ex-

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284 N.I. Kim, K. Maruta / Combustion and Flame 146 (2006) 283–301

Fig. 1. Photos of the flames: (a) stationary flame close to quenching conditions (propane/air mixture, equivalence ratio 1.45,pressure 0.07 atm, tube diameter is 51.6 mm), unpublished photo concerned with Kim et al. [8], (b) direct photographs of astable flame (b1) and a flame with repetitive extinction and reignition (b2), inner diameter of the tube was 2 mm, from Maruta etal. [10]. In all cases gravity force directed downward.

perimental studies have used enforced thermal bound-ary conditions to stabilize a flame in a small tube[9,10]. According to conditions, complex flame be-haviors were observed, especially near the flammablelimits, as shown in Fig. 1b [10].

The complex phenomena are attributed mainly tothe interactions of three mechanisms. The first is ther-mal interaction between the flame and the wall. One-dimensional analyses based on a simplified reactionand analytical methods have been successfully ap-plied to investigate this interaction [10]. The secondis chemical interaction between species in a flame.A one-dimensional computation method using a de-tailed reaction mechanism has been applied to explainthis interaction in small tubes [10]. The last is momen-tum interaction between a flow field and a flame. Theeffect of convective flow on a flame has been con-sidered an important issue following the pioneeringwork by Karlovitz et al. [11]. However, the momen-tum interaction in tubes requires further considera-tion in terms of its confinement and boundary con-ditions. While many experimental studies have beenconducted [12–15], most have focused on transientphenomena of flames in tubes. Only a limited num-ber of experimental studies investigated the effect offlow on flames independently by imposing additionalvelocity variations to the flames [16–20]. Neverthe-less, it is recognized that the momentum interaction,even under a steady state, has not been sufficientlyclarified, especially in small tubes.

In order to investigate the momentum interaction,three kinds of computational approaches have beenadopted. The first approach uses the assumption of afixed flow field to avoid a formidable computationalload [21–23]; however, it cannot evaluate the effectof a flame on a flow field. The second approach usesmatching jump conditions at the flame front with animposed or evaluated velocity field [18,24,25]. Even

though this approach addresses the parametric effectsdirectly, it is doubtful whether the employed assump-tion of constant flow rate through the flame is still use-ful in small tubes. The reason for skepticism over thisassumption is that the modification in the flame struc-ture becomes important if the characteristic length ofthe velocity wave is not sufficiently larger than theflame thickness [26,27]. Under such conditions, it isalso expected that the molecular transport of heat andspecies becomes important [28]. Therefore, we chosethe third approach, in which Navier–Stokes equationscoupled with species and energy equations employinga suitable reaction model and transport properties aresolved [29–33].

Lee and T’ien [29] showed flame structure un-der flashback conditions. Gonzalez et al. [30] simu-lated a propagating flame in a closed channel. Leeand Tsai [31] and Hackert et al. [32] found mul-tisolutions and bifurcation for a stationary flame ina cold tube. Recently, more detailed reaction mod-els have been applied for the flame in a tube byMichaelis and Rogg [33]. They reported the signifi-cance of flow redirection on the flame propagation fordifferent boundary conditions. However, it remains aformidable task to apply a detailed reaction model forvarious boundary conditions.

This paper aims to provide an overview of flamebehavior in small tubes. Navier–Stokes equationscoupled with energy and species equations are solvedwith a one-step irreversible global reaction modelof a methane–air mixture. Various characteristics offlames and flow fields in tubes, i.e., flame shape, prop-agation velocity, and variation of flow field, are inves-tigated. As a simple model of the flames in extremelysmall tubes, characteristics of the flame with slip-wall boundary conditions are investigated as well.Computational methods and the characteristics of anonstretched adiabatic flame are introduced in Sec-

N.I. Kim, K. Maruta / Combustion and Flame 146 (2006) 283–301 285

tion 2 and propagation velocities and flame shapesare compared for various boundary conditions in Sec-tion 3. Variation of the flow filed near the flame isdiscussed in Section 4. Discussion on similarity inflame structure near quenching is provided in Sec-tion 5.

2. Computation method

2.1. Governing equations

A finite volume method of the SIMPLE-R algo-rithm introduced by Patankar [34] was used to solvethe continuity and the momentum equations. Enthalpyand concentrations of species were solved together.Buoyancy force and radiant heat transfer were notconsidered. Governing equations could be presentedby the following equations.

Continuity equation:

(1)∂

∂tρ + ∇ · (ρu) = 0.

Momentum equation:

(2)∂

∂t(ρui) + ∇ · (ρuui) = ∇ · (μ∇ui) − ∂P

∂xi.

Enthalpy equation:

∂

∂t(ρh) + ∇ · (ρuh) = ∇ ·

(k

cp∇h

)−

5∑i

h0i ωiMi

(3)(ωi = γiωCH4 , i = 1,2, . . . ,5).

Species equation:

∂

∂t(ρYi) + ∇ · (ρuYi) = ∇ · (ρDi∇Yi) − ωiMi

(4)(i = 1,2, . . . ,5).

Unsteady formulations in cylindrical coordinateswere used for better relaxation to obtain a sensitivesteady solution. Time is denoted as t . Fluid velocityvector and its composition are respectively denotedas u and uj in Eqs. (1)–(4). Hereafter, displacementand velocity in axial direction are denoted as x and u,and those in the radial direction as r and v, respec-tively. Density, viscosity, thermal conductivity, andspecific heat of the fluid are denoted as ρ, μ, k, andcp , respectively. Heat of formation, reaction rate, mo-lar weight, stoichiometric ratio, and mass fraction ofa species i are respectively denoted as h0

i, ωi , Mi ,

γi , and Yi . The location of the maximum heat re-lease rate on the tube axis was defined as the flameposition, and the flame position was fixed at the ori-gin (x = 0) by adjusting the mean velocity Um. The

length of the computational domain was sufficientlylonger than the tube radius or the flame length.

In order to examine the effect of the flow fieldon the flame structure clearly, a simplified reactionmodel was used. This choice is reasonable because itis known that even a single-step reaction can presentthe effects of external perturbations, as explained byWilliams [1]. A stoichiometric mixture of methaneand air was chosen, since it is widely used and doesnot cause molar-number change by reaction. A single-step reaction model and fuel consumption rate ωCH4are given by Eqs. (5) and (6), respectively:

CH4 + 2(O2 + 3.76N2)

(5)→ CO2 + 2H2O + 7.52N2,

(6)ωCH4 = −A exp(−Ta/T )[CH4]a[O2]b,

where the activation temperature Ta is 24,358 K andparameters a and b are 0.2 and 1.3, respectively, asrecommended by Westbrook and Dryer [35]. It isknown that the preexponential factor A depends onthe properties and even on the computational gridsystem if it is not sufficiently small [32]. Hence theeffect of the grid size was examined to assure the va-lidity of the current computations below. Propertiesincluding density of the mixture were calculated bylinking CHEMKIN-II [36] and TRANFIT [37] basedon the local temperature and the composition of themixture at 1 atm. They were different from the pre-vious study by Westbrook and Dryer [35], and thusthe preexponential factor A in Eq. (6) was adjusted to2.29 × 1013 (g mol cm−3)−0.5 s−1 to match the lam-inar burning velocity S0

Lof 40 cm/s at stoichiometry.

2.2. Boundary and initial conditions

The propagation velocity of a premixed flame ina tube is generally defined as the relative mean ve-locity to the flame based on the cross-sectional areaof the tube. Thus, an infinite number of velocity pro-files can be defined for the same propagation velocityby changing the velocity profiles. In this study, fourkinds of velocity profiles having the same propaga-tion velocity were examined. First, three kinds of fullydeveloped velocity profiles, having different relativevelocities of the mixture to the wall, were chosen.Similar velocity profiles were employed as fixed flowfields by Cui et al. [38] in order to investigate thecorrelation between flame shape and differential dif-fusion effect. In this study, additionally, a uniformvelocity profile with a slip condition on the wall waschosen. Since none of these four flow fields vary with-out flames, the momentum interaction between flowand flame can be distinguished. Those velocity pro-files can be written as follows,

(7)u = 2(um − uw)[1 − (r/R)2] + uw,


Fig. 2. Velocity profiles having the same propagation velocity and initial flame shapes: (a) positive velocity profile, (b) uniformvelocity profile, (c) negative velocity profile, (a′) positive flame, (b′) flat flame, (c′) negative flame.

where um, uw , and R are the mean velocity of mix-ture, the wall velocity, and the radius of the tube,respectively. (1) Positive velocity profile (hereafterdenoted as “P”) is shown in Fig. 2a. The wall ve-locity was zero and the relative velocity (um − uw)

was positive. A flame in such a flow field correspondsto a stationary flame in a laboratory experiment [17–20]. (2) Uniform velocity profile (hereafter denoted as“U”) is shown in Fig. 2b. The mean velocity of themixture is the same as the wall velocity, and the ve-locity profile is flat. A flame in such a computationaldomain corresponds to a propagating flame within astatic mixture in a laboratory experiment [4]. (3) Neg-ative velocity profile (hereafter denoted as “N”) isshown in Fig. 2c. The relative velocity (um − uw)

is negative and the wall velocity is twice the meanvelocity; a fully developed velocity profile is then for-mulated toward the inverse direction. Similar velocityprofiles may arise when a flame propagates in a one-side closed tube. In this case, volume expansion inthe burned side generates flow of the mixture towardthe unburned side and then the relative velocity to theflame corresponds to the negative velocity profiles.(4) Slip condition (hereafter denoted as “slip”) is alsoemployed. Nonreactive flow has a uniform velocityprofile regardless of the wall velocity. It correspondsto a confined stream tube (or an ideal tube). Eventhough its velocity profiles correspond with the uni-form velocity profiles, the existence of a flame causessignificant variation.

Each case among these four velocity profiles is ex-amined for two kinds of thermal boundary conditions,i.e., an adiabatic (hereafter denoted as “A”) and anisothermal wall (hereafter denoted as “T”). Therefore,

a total of eight combinations of boundary conditionsare examined in this study with the same computationmethod, as summarized in Table 1. In all cases, thederivatives of mass-flow rate, enthalpy, and species inthe axial direction at the outlet boundary are set tozero (Neumann conditions).

In this study, three kinds of initial flame shapewere used, as shown in Figs. 2a′–2c′. Different ini-tial flame shapes were examined due to two reasons:a suitable choice of the initial flame can reduce com-putation time and a flame can exhibit multisolutionsunder particular conditions depending on the initialflame shapes, as reported by Lee and Tsai [31]. Theflame shape of Fig. 2a′ is typically observed in thecases of positive velocity profiles and thus is namedas a positive shape. Likewise, flat and negative shapesare shown in Figs. 2b′and 2c′, respectively. A normalvector on the flame surface is defined toward the un-burned side. Thus, the positive shape has a positivecurvature at the center.

2.3. Grid system

Laminar burning velocity of a flat flame was eval-uated in a sufficiently small tube with adiabatic andslip-wall boundaries. With sufficient relaxation formonotonic conversion, the estimated burning veloc-ities deviated depending on the grid size and on theinitially assumed velocities, as shown in Fig. 3. Thatis, a higher initial velocity resulted in a higher lami-nar burning velocity, while a lower value resulted in alower burning velocity. This shows that the grid sizenear the flame should be smaller than 50 µm in thestreamwise direction in order to estimate the prop-


Table 1Eight cases of boundary conditions depending on the ther-mal boundary conditions, velocity profiles, and slip or no-slip conditions on the wall

Case Boundary conditions

Temperature Velocityprofile

Slip/no-slip

AUslip Adiabatic Uniform SlipAU No-slipAP PositiveAN Negative

TUslip Isothermal Uniform SlipTU No-slipTP PositiveTN Negative

Fig. 3. Dependency of the estimated laminar burning veloc-ity on the grid system.

agation velocity with an error less than 1% and toexamine the thermal thickness δT , which is definedby the derivative of temperature as follows:

(8)δT ≡ Tad − Tin

dT/dx|max;

here Tad is the adiabatic flame temperature of the one-step reaction (2329 K) and Tin is the mixture temper-ature at the inlet (300 K).

Meanwhile, an investigation of the momentum in-teraction requires a large computation domain. Thisdiscrepancy between a large computational domainand a fine grid system near the flame necessitates awell-designed grid system. In this study, an adaptivegrid system was used and it consisted of initial andadapted grids near the flame. The initial grids werebuilt to have a grid scale not larger than 100 µm.The grid size was then reduced to be not larger than50 µm within a computational test section, as shownin Fig. 4a. Square grids were used within the test sec-

tion to eliminate the directivity of the flame propaga-tion.

Two steps consisting of grid elimination and gridadaptation were conducted consecutively near theflame during the simulation, as shown in Fig. 4b.At the first step, grid points of 25 µm were addednear the flame, including the diffusion layer, and thengrid points of 12.5 µm were added to the diffusionlayer including the reaction layer. Grid adaptation wasmade when a new node was required between twogrid points of the previous step. Inversely, added gridpoints were eliminated to prevent excessive genera-tion of grid points when they are no longer required.The criterion of grid adaptation was defined as themaximum deviation among three differences of nor-malized scalar values at two neighboring nodes: i.e.,the reaction rate of fuel ωCH4 , mass fractions of ox-idant YO2 , and fuel YCH4 . This can be presented asfollows,

�(1,2;n) = f (n)

(9)

× max

[c|�12ωCH4 |ω0

CH4|max

,|�12YO2 |

YO2,in,|�12YCH4 |

YCH4,in

],

where n is the step of grid adaptation, �12 is an oper-ator for the scalar difference between two neighboringpoints. The subscript “in” implies the value at the inletboundary, and the superscript “0” implies the value ofa nonstretched adiabatic flat flame. A constant c andan arbitrary function f (n) were suitably adjusted tohave sufficient grid points in the reaction and thermallayers, respectively. As a result, the computational de-viation in estimation of the laminar burning velocityusing this adaptive grid system was sufficiently small,as shown in Fig. 3.

2.4. Adiabatic flat flame

Characteristics of an adiabatic flat flame wereevaluated again with the consistent computationalmethod, and these data were used as references inthis study. An adiabatic flat flame is established in asufficiently narrow tube of an adiabatic wall and theslip-wall boundary conditions. Distributions of tem-perature, reaction rate, and mass-flow rate are shownin Fig. 5a, and the streamwise grid distribution isalso presented. Normalized temperature was definedas T = (T − Tin)/(Tad − Tin). Reaction rate was nor-malized by the maximum reaction rate of an adiabaticflat flame as ω = ωi/ω

0i,ad|max. Mass-flow rate was

normalized by the result of an adiabatic flat flameas mx = ρu/ρinS0

L. Thermal thickness was about

328 µm based on Eq. (8). This value can be com-pared with a characteristic length scale commonlyused in analysis. It is defined as � = αin/S0 , where

L


Fig. 4. Grid systems: (a) computational test section, (b) adapted grid near the flame front.

Fig. 5. Flame structure of the nonstretched adiabatic flat flame: (a) variation of the mass-flow rate, temperature, and reaction ratenear the flame front and the definition of the flame thickness, (b) distributions of the mass fraction of the species.

αin = kin/cp,inρin is the thermal diffusivity of an un-burned mixture. The characteristic length scale was50 µm and the thermal thickness in this paper wasabout 6.6 times larger than that (δT ∼ 6.6�).

Mass-flow rate through the flame was constant, asshown in Fig. 5a. Even though this is not a new find-ing in an adiabatic flat flame, it is worth noting atthis state because significant variations of mass-flowrate near the flame will be introduced and discussedin later sections. Mass fractions of five species, con-sidered in this study, are shown in Fig. 5b.

3. Propagation velocities and flame shapes

A comparison between flames in adiabatic tubesand those in isothermal tubes can distinguish the ef-fects of the thermal interaction and the momentuminteractions.

3.1. Propagation velocities and flame shapes inadiabatic tubes

Propagation velocities of the flame in adiabatictubes are shown in Fig. 6 for various velocity profiles(described in Table 1; ANn and ANp are explainedbelow). Contours of reaction rate and temperatureare shown in Fig. 7 under representative conditions.When the tube size is smaller than the flame thick-ness, the flame structure will likely be more signif-icantly affected by the reaction mechanisms. Hence,we limited our discussion to cases when the tube di-ameter is larger than the flame thickness.

Before discussing each velocity profile case, theeffect of grid size on the propagation velocity is ex-amined again for the case of two-dimensional com-putations. Three kinds of grid systems were comparedfor the case of AP, as shown in Fig. 6 (i.e., an adap-tive grid system and two uniform grid systems of 50


Fig. 6. Effects of velocity profiles on propagation velocitiesof the flames in adiabatic tubes (×, AP with uniform grid of50 µm; +, AP with uniform grid of 25 µm).

and 25 µm). When the tube diameter was 0.4 mm,the calculated propagation velocities of three grid sys-tems overlapped and their deviations were negligible.When the tube diameter was 2.0 mm, meanwhile, thedeviations in propagation velocity increased. Sinceflames in large tubes have long flames and large cur-vatures, they require finer grids systems. As the gridsize decreases, estimated propagation velocity alsodecreases. The propagation velocity using an adap-tive grid was the smallest but still very close to theresult of the uniform grid of 25 µm; thus the resultswith the adaptive grid were the most reliable. There-fore, all the results in this study were obtained usingthe same method, i.e., the adaptive grid.

The case of the adiabatic slip wall (AUslip) cor-responds to a flame in a confined stream tube. Itis known that even in the adiabatic and slip wall,the stabilization of a flat flame is hindered by theDarrieus–Landau instability [39,40], which gives riseto wrinkled flames. While experimental studies on theDarrieus–Landau instability suffer from various ex-perimental disturbances [41], computational studiescan show the effect of the Darrieus–Landau instabil-ity more clearly [30,42]. Fig. 7a shows that wrinkledflames exist only when the tubes are larger than a crit-ical size (larger than 2 mm), as expected from otherstudies. However, one notable aspect is that the in-crease of the propagation velocity is smaller than thatof other cases, as shown in Fig. 6. This implies thatthe Darrieus–Landau instability is not a major causeof the variation of the propagation velocity, especiallyfor a flame in a tube.

The case of the uniform velocity profiles with adi-abatic no-slip wall (AU) can be compared with theresults of the slip condition (AUslip). As shown in

Fig. 6, the propagation velocity was almost the samewith the laminar burning velocity in a sufficientlysmall tube, while it increased when the tube diameterwas larger than a critical size (about 1 mm). Increaseof the propagation velocity was much larger than theresult of the slip conditions (AUslip). In addition, theflame shapes shown in Fig. 7b were more wrinkledthan those of the AUslip case shown in Fig. 7a. Theseresults imply that even though the Darrieus–Landauinstability may initially generate the wrinkled flameshape, the no-slip condition on the wall is a muchmore important mechanism in terms of affecting thepropagation velocity and the flame shape.

The case of the positive velocity profiles with adi-abatic no-slip wall (AP) corresponds to a stationaryflame in an adiabatic tube. When the tube diameterwas larger than a critical size, the flame became muchlonger than the other cases and had positive shapes,as shown in Fig. 7c. The slope of the propagation ve-locity to the tube diameter in Fig. 6 is much steeper(η ∼ 385 s−1) compared with that of the AU case(η ∼ 108 s−1).

The case of the negative velocity profiles with adi-abatic no-slip wall (AN) had two steady solutionsdepending on the initial conditions. When the initialflame was a flat or a positive shape, steady solutionshaving positive flame shapes were obtained under suf-ficient computational relaxation, as shown in Fig. 7d.This case was denoted as ANp case, and its propaga-tion velocity was close to that of the AUslip case, asshown in Fig. 6. On the other hand, when the initialflame shape was negative or the velocity disturbanceduring the computation was not sufficiently small,a negative flame shape could be obtained, as shownin Fig. 7e. This case was denoted as ANn case, and itspropagation velocity was much faster than that of theANp case, as shown in Fig. 6. Such complex propaga-tion velocities are related to the variation of the flowfield, and are discussed in Section 5.

3.2. Propagation velocities and flame shapes inisothermal tubes

Propagation velocities with four kinds of veloc-ity profiles in isothermal tubes are shown in Fig. 8.The shapes of the reaction layer and the distributionof temperature are shown in Fig. 9.

In the case of TP (isothermal wall, positive ve-locity profile, no-slip wall), propagation velocity in-creased with the tube diameter, as shown in Fig. 8.Within a tube diameter range of 4 to 6 mm, thepropagation velocities were almost constant and wereslightly smaller than the laminar burning velocity. Thequenching diameter was 2.47 mm. Fig. 9a shows thatthe flame shapes vary from hemispherical to positivewith the increase of tube diameter. An experimental


Fig. 7. Flame shapes in adiabatic tubes (upper half tube depicts normalized reaction rate and lower half tube depicts normalizedtemperature; contour line shows values of 0.1, 0.3, 0.5, 0.7, and 0.9; left-hand side is unburned mixture).

study by Kim et al. [8] reported similar results in awide range of equivalence ratios and pressures.

In the case of TU (isothermal wall, uniform veloc-ity profile, no-slip wall), propagation velocities hadtwo solutions based on their propagation velocitiesand shapes. That having a negative flame shape wastermed “TUn” (see Fig. 9b) and another having a pos-itive flame shape was termed “TUp” (see Fig. 9c).In the case of TUn, propagation velocity decreasedsteeply with a decrease in the tube diameter. Theflame shape became hemispherical and the flame wasquenched when the tube diameter was 2.23 mm. Themaximum propagation velocity of the TUn case was

about 1.7 times that of S0L

. On the other hand, varia-tion of the propagation velocity of the TUp case wasrelatively smaller than that of the TUn case. Thesetwo solutions could coexist when the tube diameterswere in the range of 6.8 to 9.0 mm.

Previous numerical studies reported the existenceof two solutions as such in tubes [31] and betweenparallel flat channels [32], although their magnitudesare not in agreement with the results of this study.Lee and Tsai [31] showed that these two solutionsare related to two key factors which determine theflame shapes; the shapes of the initial flames and ex-ternal forces such as buoyancy force. Discontinuity


Fig. 8. Effects of velocity profiles on propagation velocitiesof the flames in isothermal tubes.

in propagation velocity was also found in an exper-imental study by Kim et al. [20]. However, to theauthors’ knowledge, the effect of friction force onsuch hysteresis has not been discussed directly, de-spite that friction is one of the representative sourcesof irreversible work. Thus, flame propagation in a slipisothermal wall was conducted.

In the case of TUslip (isothermal wall, uniform ve-locity profile, slip wall), two aspects are notable. Thefirst is that the propagation velocity of the TUslip fol-lows a trend similar to that of the TU (see Fig. 8).Further, flame shapes are similar to those of the TUcase (see Figs. 9b, 9c, and 9d). The existence of alocally quenched space near the wall, namely “deadspace,” suppresses sudden change of the mass-flowrate induced by thermal expansion and friction force.The second is that the TUslip case has only one propa-gation velocity for the same tube diameter regardlessof the initial flame shapes, even though a transitionstill takes place. Steady solutions are found on thetransition branch connecting two different branches ofupper and lower solutions. Such solutions on the tran-sition branch are not obtained in the case of the no-slip condition (TU). These observations imply that thefriction force plays a significant role in the hysteresisin the propagation velocity and the flame shape. Thequenching diameter in this case was 2.12 mm, whichwas slightly smaller than that of the TU.

In the case of TN (isothermal wall, negative veloc-ity profile, no-slip wall), the flame shapes were nega-tive for all cases, as shown in Fig. 9e. Heat loss nearthe wall suppressed the possibility of positive shape,which was observed in the adiabatic case (ANp case;see Fig. 7e). Flame length and propagation veloci-ties were much larger than those of the other cases.Flame shapes in large tubes were similar to the “finger

flame” in studies by Clanet and Searby [15] and Kimet al. [20]. The reason for such increase in the propa-gation velocity requires discussions on the flow redi-rection coupled with the flame structure, which willbe discussed later. Under conditions close to quench-ing, meanwhile, the flame shape became hemispher-ical, similar to the other cases, and the quenchingdiameter was the smallest (1.98 mm). Propagationvelocity under the quenching conditions was slightlysmaller than the laminar burning velocity of the adia-batic flat flame.

3.3. Comparison between adiabatic and isothermaltubes

Comparisons of the propagation velocities of theflames in adiabatic and isothermal tubes are presentedin Fig. 10. It is notable that the isothermal and adia-batic cases are extreme cases of heat transfer for agiven velocity boundary condition.

The cases of the positive velocity profiles (AP andTP) are compared in Fig. 10a. Propagation velocitiesof the AP case are much larger than those of the TPcase. In the AP case, the mass-flow rate across theflame front near the wall steeply decreases due to vol-ume expansion and friction force. This will be shownin Fig. 12d and discussed later. It causes a decreaseof the flow rate in front of the flame as well, and thusa larger flow rate is required to stabilize the flame.However, such a decrease of the mass flow near thewall was not significant in the case of TP due to thedead space (see Fig. 9a). This large deviation betweenAP and TP cases implies that the propagation velocityis most sensitive to the heat transfer conditions underthe positive velocity profiles, even though it is ther-mally stable, i.e., less heat loss corresponds to fasterpropagation velocity.

The cases of uniform velocity profiles (AU andTU) are shown in Fig. 10b. Compared with the pos-itive velocity case (Fig. 10a), the difference in thepropagation velocities between adiabatic and isother-mal tubes was much smaller. This implies that alaboratory-scale flame propagating in a static mixtureis weakly affected by the thermal conditions. Despitetheir similar propagation velocities, the flame shapesof AU (see Fig. 7b) and TUn were directed in oppo-site at the center (see Fig. 9b).

The cases of the slip conditions (AUslip and TU-slip) are compared in Fig. 10c so as to distinguishthe thermal effect from the friction force. The prop-agation velocities of both cases were very close toeach other in large tubes, since the difference in ther-mal boundary conditions becomes minor with the in-crease of the tube diameter. When the tube diameterswere between 2.4 and 7 mm, however, the propa-gation velocities of the flames were larger in cold


Fig. 9. Flame shapes in isothermal tubes (upper half tube depicts normalized reaction rate and lower half tube depicts normalizedtemperature; contour line shows values of 0.1, 0.3, 0.5, 0.7, and 0.9; left-hand side is unburned mixture).

tubes (TUslip) than in adiabatic tubes (AUslip). Thissurprising result implies that the one-dimensional ap-proach assuming the propagation velocity as a func-tion of temperature will not be applicable under such

conditions. In addition, flame stabilization may notbe easily achieved by simple control of its thermalboundary. Although more studies are required, thisphenomenon can be another cause of flame instability


Fig. 10. Comparison of the propagation velocities of the flame in adiabatic and isothermal tubes.

in a small tube, which has been observed in a previ-ous experimental study [10]. This phenomenon mightalso be important under the unsteady state, becausewhen the gas phase undergoes velocity variation dur-ing a short period (or with a higher frequency) thegrowth of a viscous layer near the wall may becomenegligible. Such a sudden variation in velocity mayoccur not only by forced convective flow but also byvolume expansion accompanied by combustion itself[19,20].

Inverse dependency of the propagation velocity onthe heat loss was recently reported by Kurdyumov andFernández [22]. Their computation was conductedemploying a fixed velocity field similar to the posi-tive velocity profile in this study. They suspected asufficiently small Lewis number as the cause of suchinverse dependency of propagation velocity on heatloss. On the contrary, our results show the effect ofthe flow redirection on the propagation velocity ratherthan the Lewis number effect.

The cases of negative velocity profiles (AN andTN) are shown in Fig. 10d. Both adiabatic and isother-mal flames tended to show negative flame shapesrather than positive ones. In this situation the flamefront at the center was weakly affected by the differ-ence in thermal boundary conditions. Fast flow near

the wall also suppressed the growth of the thermalboundary layer. Thus, the propagation velocities ofthe AN and TN cases were very close.

From these results, we can expect that the effectof the velocity field on the flame structure would beminimized if a flame changes the velocity profilesnear the flames to locally negative ones. Such veloc-ity variation may also occur near the quenching, andthis can account for the similarity in the flame shapesand structures near the quenching despite the differentpropagation velocities. More discussion on the flamecharacteristics near the quenching is presented in Sec-tion 5.

4. Flow redirection near the flame

Some experimental studies [2,4] have introducedthe variation of relative velocity to the flame. The ve-locity variations generated by the flames are calledflow redirection, and this can cause significant varia-tion of the propagation velocity coupled with the vari-ation of the flame shape. Flow redirection in adiabaticand isothermal tubes is discussed in the following sec-tions.


Fig. 11. Flow redirection near the flame: (a) contour shows the normalized velocity in radial direction and in axial direction,(b) contour shows the normalized mass-flow rate in radial direction and in axial direction.

4.1. Flow redirection in adiabatic tubes

For the AU case of a 3-mm tube, distributions ofnormalized velocity in the axial direction u = u/S0

L

and in the radial direction v = v/S0L

are shown inFig. 11a. Two notable features were detected. First,the velocity variation in the axial direction is not al-ways monotonic. Second, the magnitude of the veloc-ity variation across the flame is much larger than thatexplainable by volume expansion alone. Thus, nor-malized mass-flow rates, defined as mr = ρv/ρinS0

L

in the radial direction and mx = ρu/ρinS0L

in theaxial direction, were used to distinguish the momen-tum change from the thermal expansion, as shownin Fig. 11b. Contours of the mass-flow rates showgrowth of a shear layer near the wall or variation ofthe streamlines more clearly.

Even though the assumption of constant mass fluxthrough the flame has been widely used in many com-bustion problems, real flames in tubes suffer strongvariation in mass-flow rate through the flame. In orderto investigate the variation of the mass-flow rate, twostraight streamlines were observed, one on the walland another on the tube axis. Note that the mass-flowrate on the wall is not trivial only when the slip-wallcondition is used (AUslip), as shown in Fig. 12a. Forthe other cases, the mass-flow rates on the tube axesare shown in Figs. 12b–12f. Mass-flow rate throughthe centerline is affected not only by the local charac-teristics but also by the global structure of the flame,since the total mass-flow rate in a tube is conserved.Thus, inversely, the variation of mass-flow rate alongthe tube axis reflects the overall variation of the flowfield.

In the AUslip case, variations of the mass-flowrate both on the wall and on the axis were smallerthan those of the other cases. When the tube diameterwas larger than the critical diameter (above 2 mm),flames near the wall become slightly convex towardupstream. Thus, streamlines slightly diverged in frontof the flame near the wall, and the mass-flow rateon the wall decreased in front of the flame along the

streamline (as shown in Fig. 12a), where the origin(x = 0) was defined as the location of the maximumheat release rate on the tube axis identically. On theother hand, the mass-flow rate on the tube axis resultsin opposite trends, as shown in Fig. 12b. The ampli-tude of the velocity variation was much larger at thetube axis. This is due to the configuration of the tube;i.e., even a small disturbance near the wall has signifi-cant effects on the flame structure at the center. Underthe slip conditions, the mass-flow rates both far up-stream and far downstream were the same.

The AU case (shown in Fig. 12c) demonstratesthat the no-slip condition on the wall significantlycontributes to the increase of the mass-flow rate at thecenter. Mass-flow rates far upstream were the sameas the propagation velocities. As the flow approachesthe flame front, the mass-flow rate increases due tothe flame shapes as shown in Fig. 7b. After sufferingadditional flow redirection in the thermal layer, thevelocity field deformed by the flame develops again.

A newly developed velocity far downstream canbe estimated from Eq. (7) by multiplying the meanvelocity by the ratio of volume expansion, and is pre-sented as Eq. (10). The mass-flow rate at the tubecenter (far downstream) can then be normalized bythe averaged value of the mass-flow rate at the inletboundary as Eq. (11):

(10)

ux=∞ = 2

(ρin

ρoutum − uw

)[1 −

(r

R

)2]

+ uw,

(11)mx=∞ρinum

= ρout

ρin

(2

ρin

ρout− uw

um

),

where the volume expansion ratio ρin/ρout can bepredicted using the temperature ratio based on theideal gas law, and is 7.76 in this study. In the caseof AU, the velocity ratio uw/um is unity. Thus, themass-flow rate became 1.87 times that at far up-stream; Fig. 12c shows the same result.

The AP case showed the longest flame in the axialdirection and the largest curvature at the center among


Fig. 12. Mass-flow rate on the wall and along the tube axis: (a) AUslip case (on wall), (b) AUslip case (tube axis), (c) AU case,(d) AP case, (e) ANp case, (f) ANn case.

the four cases of velocity profiles (see Fig. 7). Thiscaused the largest variation in the mass-flow rate, asshown in Fig. 12d. As can be expected from Eq. (11),when the wall velocity was zero, the mass-flow ratesof both far upstream and far downstream were twotimes the mean mass-flow rate, due to parabolic ve-locity profiles. The mass-flow rate immediately be-hind the reaction layer slightly increased in proportionto the propagation velocity.

The AN case had double solutions of differentpropagation velocities and flame shapes (see Figs. 6,7d, and 7e). Their mass-flow rates along the tube axis

are shown in Figs. 12e and 12f. If the flame shape ispositive (ANp, see Fig. 12e), the characteristics of themass-flow rates are similar to the results of most cases(AU, AUslip, and AP). On the contrary, when theflame has a negative shape (ANn case), the mass-flowrate increases monotonically as shown in Fig. 12f. Inmost cases, continuous incoming flow from the re-tarded flame surface near the wall increased the mass-flow rate along the tube axis. Finally, the mass-flowrate far downstream can be estimated from Eq. (11)by substituting uw/um ∼ 2, and was 1.74 times themean mass-flow rate at far upstream.


Fig. 13. Temperature variation in axial direction in the caseof TUslip.

4.2. Flow redirection in isothermal tubes

To examine the flow redirection in isothermaltubes, the temperature variation should be discussedfirst since the heat loss from the mixture is the mainissue in the comparison with the adiabatic tube. Tem-perature variation along the tube axis for the TUslipcase is shown in Fig. 13. The burned gas temperaturein isothermal tubes eventually decreases to the tubetemperature at the far downstream. As the tube di-ameter decreases, the propagation velocity and heatrelease from the flame decrease as well. Under thequenching condition, the gas temperature decreasedto the wall temperature in only a few millimeters (thesame order as the tube diameter). Such trends are sim-ilar regardless of the velocity profiles, as can be ex-pected from the temperature contours shown in Fig. 9.

The flow field near the flame experiences morecomplex variation coupled with temperature varia-tion. Flow redirection in the isothermal tube is shownin Fig. 14 with normalized mass-flow rates in the ra-dial direction mr and in the axial direction mx . TheTUn and TUp cases at the same tube diameter, 8 mm,correspond to Figs. 14a and 14b, respectively. Flamefronts near the isothermal walls are retarded due toheat loss. Thus, flow redirection starts further up-stream, and the mass-flow rate increases near the wall.The mass-flow rate near the wall of the TUn case wasmuch larger than that of the TUp case in the vicinityof the wall.

The mass-flow rates along the tube axis are shownfor different velocity profiles in Fig. 15. In general,variations of the mass-flow rates were more compli-cated compared with the adiabatic cases (see Fig. 12).As previously noted, when the tube diameter in-creases infinitely, the difference between adiabaticand isothermal tubes will become negligible. Thus,variations of the mass-flow rate in small tubes needto be highlighted in the cases of isothermal tubes.

In the case of TU (see Fig. 15a), mass-flow rates insmall tubes have minimum values at the reaction lay-ers. Decrease of the mass-flow rates in the upstreamis due to the flow redirections coupled with thermalexpansion while the increase of the mass-flow rates atthe downstream is due to the development of the flowfield. This flow development is affected by frictionforce on the wall, internal shear force, and heat trans-fer in the mixture. Mass-flow rates of the TUslip case(see Fig. 15b) were similar to those of the TU caseat upstream, while they were different at downstreamsince the complicated temperature distribution at thedownstream causes additional expansion or contrac-tion in volume.

The case of the TP (see Fig. 15c) shows trendssimilar to that of the TU case. One notable result wasfound in a tube size (6 mm); specifically, the mass-flow rate suffers both decrease and increase along thestream. The smooth decrease of the mass-flow rate atthe far upstream can be explained by the flow throughthe dead space near the wall, while a small peak infront of the flame was generated due to the locallypositive curvature of the flame. Finally, the case ofthe TN (see Fig. 15d) showed that the negative cur-vature and the dead space did not directly cause thedecrease of the mass-flow rates at the center. That is,even though all flame shapes were negative in the TNcase, the mass-flow rates across the flame increasedmonotonically. It is notable that the flame front atthe center having a finite mass consumption rate can-not be stabilized without the increase of the flow ratecaused by the flow redirection. And the flow redirec-tion in front of the flame at the center is suppressedwith the increase of the flame curvature, which is en-larged with the mean velocity in the negative velocityprofiles. As a result, much larger mean velocities arenecessary to adjust the mass consumption rate of theflame.

5. Characteristics of flame quenching

As the tube diameter decreases, the dead space in-creases and eventually the whole flame is quenchedat a critical scale, namely the “quenching diameter.”Close to quenching conditions, some amount of un-burned mixture can pass through the dead space. Eventhough such quenching phenomena demand more de-tailed chemical reaction models to investigate chem-ical effects including radical quenching, we are in-terested here in the flame–flow interaction mainlycaused by thermal expansion. Thus, a simple one-stepreaction model was used to distinguish the thermal ef-fects from chemical effects.


Fig. 14. Variation of the mass-flow rate near the flames of two solutions obtained in the same isothermal tube diameter of 8 mm:(a) TUn case and (b) TUp case.

Fig. 15. Variation of the mass-flow rates along the tube axis in isothermal tubes.

5.1. Dead space and unburned ratio

The role of dead space becomes important as thetube diameter decreases. The dead spaces for differentvelocity profiles are shown in Fig. 16. Here, the deadspace was scaled by the minimum distance from thewall to the peak of the reaction rate in the radial direc-

tion. When tube diameters were sufficiently large, thedead spaces were similar to the thermal thickness ofthe adiabatic flat flame. Dead space of the TP casewas the largest and decreased in the order of TU,TUslip, and TN case. However, the dead spaces un-der the quenching conditions were proportional to thetube diameter, and the ratio of dead space to the tube


Fig. 16. Dead spaces of the flames in isothermal tubes de-pending on the velocity profiles and tube diameters.

diameter was about a quarter regardless of the veloc-ity profiles. This result agrees with the experimentalstudy reported by Kim et al. [8].

These observations imply that the flame under thequenching conditions occupies only about a quarterof the cross-sectional area, regardless of the velocityprofiles. A significant amount of unburned mixtureflow through the dead space is expected. Due to therapid decrease of the temperature under the quench-ing conditions, as shown in Fig. 13, such unburnedmixture may flow downstream without burning. To-tal fuel flow rate at the outlet was normalized by thefuel flow rate at the inlet, and this ratio was termedthe unburned ratio. Fig. 17 shows the unburned ratiosagainst different velocity profiles. Near the quenchingthe unburned ratio steeply increased. In the case of theslip condition (TUslip) it increased to 30% or larger,and in the other cases it increased to approximately20%.

5.2. Similarity in flow field near quenching

The mass-flow rates at four positions in the axialdirection under the conditions closest to quenchingare shown in Fig. 18. Those positions are x = −1D

(D: tube diameter), x = −0.5 mm, x = 0, and x =2D, which respectively correspond to far upstream,the leading edge of the thermal layer, the center ofthe reaction layer, and far downstream. In most casesvelocity variations occurred near the flame and theywere negligible farther upstream (x � −1D). Mass-flow rate at x = −0.5 mm and at x = 0 had neg-ative velocity profiles at the tube axis, even in thecases of positive velocity profiles at the inlet bound-ary.

At this state, profiles of the mass-flow rate at thereaction layer (x = 0) are of interest and comparedin Fig. 19 since they dominate mass consumption or

Fig. 17. Unburned ratio of the mixture depending on the ve-locity profiles and tube diameters.

combustion at the flame fronts in most cases. Imposedvelocity profiles have been redirected by the flame it-self before they reach the reaction layer. Edges of thedead spaces of the TP and TN cases showed radii ofthe flame region. The mass-flow rates in the flame re-gion were less than 0.5, and strong similarity in theprofiles was found in the flame region regardless ofthe imposed velocity profiles far upstream. These re-sults support the existence of similar flame structuresunder the quenching conditions.

Flame thicknesses are compared in Fig. 20. Thecases of TUslip and TU showed a sudden jump offlame thickness under the same conditions, at whichthe transition in propagation velocities and flameshapes occurred as illustrated in Figs. 8 and 9. In allcases, flame thicknesses drastically increased near thequenching conditions. They are shown in more detailin a small graph in Fig. 20. The averaged flame thick-ness under the quenching conditions was 0.427 mm.The error bar in the small graph shows the variation ofthe flame thickness for a disturbance of 10 µm in tubediameter. This implies that the flame thickness underthe slip condition is less sensitive to the variation oftube diameter.

Quenching conditions are compared using thequenching Peclet numbers, which was defined as theaspect ratio of tube diameter to the characteristiclength scale of the flame. In this study, as a whole,quenching Peclet numbers based on characteristiclength, Pe = Dq/� = DqS0

L/αu, were 40 to 50 as

shown in Table 2. These values are comparable withprevious analytical or experimental results; Pe ∼ 64[4], Pe ∼ 50 [43], Pe ∼ 40 [44], Pe ∼ 38 [45], andPe ∼ 59 [46].

Additionally, this study shows the dependency ofthe quenching Peclet numbers on the velocity pro-files. Quenching Peclet number decreased in the orderof TP, TU, TUslip, and TN. The quenching Peclet


Fig. 18. Variation of the mass-flow rate due to the flame under conditions closest to quenching.

Fig. 19. Comparison of the mass flow at the position of thereaction layer (x = 0). Each case corresponds to the quench-ing conditions of each velocity profile.

number of TP was approximately 25% larger thanthat of the TN case. This result implicates the re-sult by Hackert et al. [32]. They predicted that thequenching Peclet number of the stationary flame (cor-responds to the TP case) was slightly larger thanthat of the propagating flame (corresponds to the TUcase). Meanwhile, other quenching Peclet numbers

Fig. 20. Flame thickness depending on tube diameters andvelocity profiles. Magnified figure shows the ultimate flamethickness under the quenching conditions and the sensitivityof the flame thickness to the tube diameters.

based on flame thickness, Pef = D/δT , were 4.7to 5.9. Even though their values are much smallerthan the former Peclet numbers (∼10%), quite sim-ilar trends are maintained due to the strong similar-ity in the flame thickness under quenching condi-tions.


Table 2Characteristic length scales and Peclet numbers under thedifferent velocity profiles

Case Dq

[mm]δT[mm]

PeD/�

Pef

D/δT

TP 2.470 0.417 49.4 5.92TU 2.230 0.434 44.6 5.14TUslip 2.120 0.430 42.4 4.93TN 1.984 0.426 39.68 4.66

6. Conclusions

Structures and propagation velocities of flames intubes were investigated numerically. Two kinds ofthermal boundary conditions and four kinds of veloc-ity profiles were employed. Different velocity profilesinduced significant variations in propagation veloc-ities and the flame structures coupled with thermalconditions and tube sizes.

In adiabatic tubes, Darrieus–Landau instabilitycan be the origin of flame disturbance while variationof the propagation velocity is mostly caused by fric-tion force on the wall. The confined structure of theflame caused significant variation in the flow field, notonly near the flame but also within the flame, even inthe case of a slip-wall condition.

In isothermal tubes, the existence of dead spacenear the wall dominated the flame structure and theflow redirection near the flame. It was found thatthe existence of multisolutions is related to the fric-tion force. From a comparison of the results betweenadiabatic and isothermal walls, the sensitivity of thepropagation velocity on the thermal boundary couldbe estimated. In the slip tube, it was predicted thatthe propagation velocity of the flame in an isothermaltube is faster than that of the flame in an adiabatictubes. Such inverse dependency of the propagationvelocity on the heat loss presented the limitation ofone-dimensional approaches and the importance ofthe flow redirection near the flame.

Flow redirection near the flame was investigatedbased on the variation of local mass-flow rates. Mass-flow rate varied significantly not only near the flamebut also in the flame itself. Furthermore, flow redi-rection near the flame can generate a similar flamestructure under the quenching conditions; i.e., flamethickness under quenching conditions has a similarvalue regardless of the velocity profiles and the devi-ation in the quenching Peclet numbers was less than aquarter, regardless of the velocity profiles and veloc-ity boundary condition at the wall.

Acknowledgments

We express our gratitude for discussions with Mr.T. Kataoka in Mitsubishi Heavy Industry (in Japan).

This study was partly supported by the CombustionEngineering Research Center in the Korea AdvancedInstitute of Science and Technology.

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A numerical study on propagation of premixed flames in small tubes

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