Flame acceleration in the early stages of burning in tubes

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Flame acceleration in the early stages of burning in tubes
  Combustion and Flame 150 (2007) 263–276www.elsevier.com/locate/combustflame Flame acceleration in the early stages of burning in tubes Vitaly Bychkov a , ∗ , V’yacheslav Akkerman a , b , Gordon Fru a ,Arkady Petchenko a , Lars-Erik Eriksson c a  Institute of Physics, Umeå University, S-901 87 Umeå, Sweden b  Nuclear Safety Institute (IBRAE) of Russian Academy of Sciences, B. Tulskaya 52, 115191 Moscow, Russia c  Department of Applied Mechanics, Chalmers University of Technology, 412 96 Göteborg, Sweden Received 20 March 2006; received in revised form 13 December 2006; accepted 18 January 2007 Abstract Acceleration of premixed laminar flames in the early stages of burning in long tubes is considered. The accel-eration mechanism was suggested earlier by Clanet and Searby [Combust. Flame 105 (1996) 225]. Accelerationhappens due to the initial ignition geometry at the tube axis when a flame develops to a finger-shaped front, withsurface area growing exponentially in time. Flame surface area grows quite fast but only for a short time. Theanalytical theory of flame acceleration is developed, which determines the growth rate, the total acceleration time,and the maximal increase of the flame surface area. Direct numerical simulations of the process are performed forthe complete set of combustion equations. The simulations results and the theory are in good agreement with theprevious experiments. The numerical simulations also demonstrate flame deceleration, which follows acceleration,and the so-called “tulip flames.” © 2007 Published by Elsevier Inc. on behalf of The Combustion Institute. Keywords:  Premixed flames; Flame acceleration; Tulip flames; Direct numerical simulations 1. Introduction When speaking about flame acceleration, resear-chers typically mean detonation-to-deflagration tran-sition (DDT) according to the Shelkin scenario; see[1–6]. However, flame may accelerate not only in thescope of the DDT problem. For example, flame ve-locity increases due to external turbulence or intrin-sic flame instabilities. We do not discuss these ef-fects in the present paper because of limited space;one can see the following reviews and papers on * Corresponding author. Fax: +46 90 786 66 73.  E-mail address:  vitaliy.bychkov@physics.umu.se(V. Bychkov). turbulent burning [7–14] and on flame instabilities[15–19].Another interesting example of flame accelerationhas been suggested and studied experimentally byClanet and Searby [20]. Let us consider a flame prop- agating in a cylindrical tube of radius  R  with ideallyslip adiabatic walls as shown in Fig. 1. One end of the tube is closed; the flame is ignited at the symmetryaxis at the closed end. In that case the flame front de-velops from a hemispherical shape at the beginning toa “finger”-shape; see Fig. 1. In the process of burning, the volume of the burnt gas increases as(1) dV dt  = ΘS  w U  f  , where  U  f   is the planar flame velocity,  S  w  is the to-tal surface area of the flame front, and  Θ  = ρ f  /ρ b 0010-2180/$ – see front matter  © 2007 Published by Elsevier Inc. on behalf of The Combustion Institute.doi:10.1016/j.combustflame.2007.01.004  264  V. Bychkov et al. / Combustion and Flame 150 (2007) 263–276  Fig. 1. Geometry of a flame accelerating due to the initialconditions. is the expansion factor, defined as the density ratioof the fuel mixture and the burnt matter. The majorcontribution to the flame surface area  S  w  in Fig. 1comes from the flame “skirt,”  S  w ≈ 2 πRZ tip , where Z tip  is the coordinate of the flame tip, and the vol-ume of the burnt matter may be roughly evaluated as V   ≈ πR 2 Z tip . Then Eq. (1) reduces to(2) dZ tip dt  = 2 ΘU  f  RZ tip , which leads to the acceleration of the flame tip as(3) Z tip ∝ exp ( 2 ΘU  f  t/R). It is interesting to compare this effect to the Shelkinacceleration mechanism. The growth rate 2 ΘU  f  /R is rather large in comparison with that obtained ac-cording to the Shelkin scenario for laminar flames[5,6]. However, the Shelkin mechanism is not limitedin time; it takes place until the detonation is trig-gered. In contrast, the acceleration mechanism [20]works in a short time interval, and it is unlikely toproduce a DDT under normal conditions. The accel-eration starts when the flame evolves from a hemi-spherical kernel to the finger-shaped front of  Fig. 1( t >t  sph ); the acceleration stops when the flame skirttouches the wall ( t < t  wall ). According to the experi-mental measurements [20], one has  t  sph ≈ 0 . 1 R/U  f  , t  wall ≈ 0 . 26 R/U  f  , which leaves only a short time in-terval  t  wall  − t  sph  ≈ 0 . 16 R/U  f   for the acceleration.It is interesting how strongly the flame surface areamay increase during such a short time interval, butRef. [20] did not address this question. The purposeof the present paper is to answer this question as wellas to clarify other aspects of the flame accelerationobtained experimentally by Clanet and Searby. Thecurrent study is also conceptually close to the work of Zeldovich [21].In the present paper we consider flame accelera-tion in the early stages of burning in tubes with slipwalls, as proposed by Clanet and Searby [20]. We de-velop the analytical theory of flame acceleration. Wederive the formulas for the acceleration time interval, t  sph  <t <t  wall , and for the growth rate. We show thatthe flame surface area increases because of accelera-tion approximately by a factor of 2 Θ  in comparisonwith the tube cross section. We also perform directnumerical simulations of the flame acceleration. Theanalytical and numerical results agree with the ex-perimental data [20]. The simulations also clarify the effect of the so-called “tulip” flame.The paper is organized as follows. In Section 2we develop the analytical theory of flame accelerationin the early stages of burning. Details of the directnumerical simulations are presented in Section 3. InSection 4 we compare the theory and the simulationresultstotheexperimentaldata[20].Thepaperiscon- cluded by a brief summary. 2. The analytical theory of flame acceleration We consider a flame propagating in a cylindricaltube of radius  R  with ideally slip adiabatic walls andwith one end closed. The flame is ignited at the sym-metry axis at the closed end. In the analytical theorywe will use the dimensionless coordinates  (η ; ξ) = (r ; z)/R , velocities  (w ; v) = (u r ; u z )/U  f  , and time τ   = U  f  t/R . We employ the standard model of an in-finitesimally thin flame front, which propagates nor-mally with the velocity  U  f   (or unity in the dimension-less variables). At the beginning, the flame is ignitedat the tube axis at the closed end  (η ; ξ) = ( 0 ; 0 ) . Ini-tially, the front is hemispherical, but the flame shapechanges as the flame skirt  η f   moves along the tubeend wall ( ξ   = 0) from the axis  η = 0 to the side wall η = 1 as shown in Fig. 2. We stress that Fig. 2 and the calculations below consider only the flow infini-tesimally close to the wall, at  ξ   → 0. In that limit theflamefronttouching the wallmaybe treated aslocallycylindrical not only in the case of a finger shape, buteven for the hemispherical front. The flame separatesthe flow into two regions of the fresh fuel mixture andthe burntmatter. The incompressible flowis described Fig. 2. Flow close to the tube end wall.  V. Bychkov et al. / Combustion and Flame 150 (2007) 263–276   265 by the continuity equation(4)1 η∂∂η(ηw) + ∂v∂ξ  = 0 . The boundary condition at the end-wall  ξ   =  0 is v = 0. We are interested in the flow along the wall,in the limit  ξ   → 0. In the fuel mixture (labeled “1”)the flow is potential; we assume(5) v 1 = A 1 ξ, where the factor  A 1  may depend on time. The as-sumption is consistent with the “porous piston” effectas the flame approaches the tube wall. In the hemi-spherical regime of flame propagation it works onlysufficiently close to the wall as the leading term in  ξ  for  ξ   → 0. Then the radial velocity in the fuel mixtureis calculated from Eq. (4) as(6) w 1 = A 1 2  1 η − η  . In Eq. (6) we have taken into account the bound-ary condition at the side-wall of the tube  w  = 0 at η = 1. The velocity distribution in the burnt matter(labeled “2”) takes the form(7) v 2 = A 2 ξ, (8) w 2 =− A 2 2  η. In general, the flow in the burnt matter of  Fig. 1 is ro-tational because of the curved flame shape. However,close to the wall, the flame front is locally cylindri-cal and the flow (7) and (8) is potentialy similar to (5) and (6). In Eq. (8) we have also taken into account the boundary condition at the tube axis  w = 0 at  η = 0.To complete the solution we consider the matchingconditions at the flame front  η = η f  ,(9) dη f  dτ  − w 1 = 1 , (10) w 1 − w 2 = Θ − 1 , (11) v 1 = v 2 . The condition (9) specifies the fixed propagation ve-locity  U  f   of the flame front with respect to the fuelmixture (which is unity in scaled variables). The con-ditions (10) and (11) describe the jump of the normalvelocity and continuity of the tangential velocity atthe front. We stress that Eq. (11) applies only at theflame skirt close to the wall and it follows from theirrotational assumption combined with the cylindricalflame shape, i.e., no shear across the flame front. Sub-stituting Eqs. (5)–(8) into Eqs. (9)–(11), we obtain (12) A 1 = A 2 = 2 (Θ − 1 )η f  , and the equation for the flame front(13) dη f  dτ  − (Θ − 1 )  1 − η 2f   = 1 . According to Eq. (13), we can separate two oppositeregimes of flame propagation: when the flame skirtis close to the axis  η f  ≪ 1, and when it is close to thewall 1 − η f  ≪ 1.Inthelimitof  η f  ≪ 1,theflameprop-agates with the velocity  dη f  /dτ   = Θ  (or  ˙ R f  = ΘU  f  ).The same velocity takes place for an expanding hemi-spherical flame front. In the other case of 1 − η f   ≪ 1, the flame propagation velocity is  dη f  /dτ   = 1 (or ˙ R f  = U  f  ). In that limit a locally cylindrical flame skirtapproaches the wall; the radial velocity of the freshfuel mixture tends to zero, and the flame skirt prop-agates with the planar flame velocity with respect tothe tube end wall. Integrating Eq. (13) with the initialcondition  η f  = 0 at  τ   = 0, we find(14) τ   = 12 α ln  Θ + αη f  Θ − αη f   , or(15) η f  = Θα exp ( 2 ατ) − 1exp ( 2 ατ) + 1  = Θα tanh (ατ), where(16) α =   Θ(Θ − 1 ). According to Eq. (15), the flame velocity is equal to the velocity of a hemispherical front close to theaxis, when 2 ατ   ≪ 1 and  η f   = Θτ   (or  R f   = ΘU  f  t  ).Respectively, one should expect transition to the“finger”-shape at the characteristic time(17) τ  sph ≈ 1 / 2 α, when the flame skirt is at  η f  ≈ 0 . 46 Θ/α . Of course,there is no exact mathematical definition of the tran-sition time. Still, the characteristic time, Eq. (17),comes as a parameter into Eq. (15) and may be com-pared to the experiments [20]. For the expansion fac- tors  Θ  =  6–8, typical for propane flames, we find α ≈ 5 . 5–7 . 5 and 0 . 06  < τ  sph  <  0 . 09. The theoreticalevaluation (17) agrees quite well with the experimen-tal estimates  τ  sph  ≈ 0 . 1 [20] taking into account arather vague definition of the value. Transition to the“finger” shape takes place approximately when theflame skirt has moved halfway to the tube side wall. Itis much easier to determine the time instant when theflame skirt touches the tube wall. Substituting  η f  = 1into Eq. (14), we obtain (18) τ  wall = 12 α ln  Θ + αΘ − α  . For the expansion factors  Θ  = 6–8 we find 0 . 23  <τ  wall  <  0 . 28, which is in very good agreement withthe experimental evaluation  τ  wall ≈ 0 . 26 [20]. Thus,  266  V. Bychkov et al. / Combustion and Flame 150 (2007) 263–276  Fig. 3. The time limits of flame acceleration,  τ  sph  and  τ  wall ,versus the expansion factor  Θ . the flame surface area grows almost exponentially intime during the interval  τ  sph  <τ <τ  wall . The time in-stant  τ  sph  defined by Eq. (17) and the time  τ  wall  whenthe flame skirt touches the wall, Eq. (18), are shownin Fig. 3 versus the expansion factor  Θ .We can also find evolution of the flame tip. If weconsider the flow along the axis  η = 0, as shown inFig. 4, then the equation for  ξ  tip  becomes(19) dξ  tip dτ  − v 2 = Θ. Equation (19) is the condition of a fixed propagationvelocity of a planar flame front written with respect tothe burnt matter. Similarly to Fig. 2, the flame shape islocally planar close to the axis, at  η → 0. In that limitthe flow may be described as a potential one, with theaxial velocity component  v  determined by a functionsimilar to Eq. (7). Besides, the solution for  v  alongthe axis has to coincide with Eq. (7) at  η → 0,  ξ   → 0.Thus weobtain the sameformula for v 2  along the axisas in Eqs. (7) and (12). We stress that such reasoning does not hold away from the axis in the burnt gas,where the flow is rotational. Still, in the present cal-culations we have to know only the gas velocity alongthe axis. As a result, we come to the differential equa-tion for the flame tip,(20) dξ  tip dτ  − 2 (Θ − 1 )η f  (τ)ξ  tip = Θ, with the initial condition  ξ  tip ( 0 ) = 0 and with the so-lution ξ  tip = Θ 4 α  exp ( 2 ατ) − exp ( − 2 ατ)  (21) = Θ 2 α sinh ( 2 ατ). Just after ignition, 2 ατ   ≪ 1, the flame tip moves inthe same way as the flame skirt,  ξ  tip = η f  = Θτ  ; seeEqs. (15) and (21), which are similar to an expand- ing hemispherical flame front. When the flame skirttouches the wall,  τ   = τ  wall , the position of the flametip is(22) ξ  wall = ξ  tip (τ  wall ) = Θ 2 α sinh ( 2 ατ  wall ) = Θ, Fig. 4. Flow close to the tube axis. or Z wall = ΘR  inthedimensionalunits.Bytheendof the acceleration we can neglect the decaying term inEq. (21). Then the flame tip accelerates exponentially, ξ  tip ∝ exp (στ) , with the growth rate(23) σ   = 2 α = 2   Θ(Θ − 1 ), which is a little different from the model estimate 2 Θ of Eq. (3); see [20]. This difference is small, since α/Θ = 0 . 91–0 . 94 for propane burning with  Θ = 6–8.It is also interesting to evaluate total increase of the flame surface area during the flame acceleration.By the end of the acceleration, the flame shape is al-most self-similar, with  η f   ≈ 1,  ξ  tip  ≫ 1, and  ξ  tip  ∝ exp ( 2 ατ) . We look for the flame shape  ξ  f  = f(η,τ) in the form(24) f(η,τ) = ϕ(η) exp ( 2 ατ). The accuracy of such an approximation is  Θ − 1 ≪ 1,which is acceptable for the propane flames used in theexperiments [20]. Then the equation of flame evolu- tion may be written with respect to the burnt matter as ∂f ∂τ  + w 2 ∂f ∂η − v 2 = Θ  1 +  ∂f ∂η  2  1 / 2 (25) ≈− Θ∂f ∂η. Of course, the approximation | ∂f/∂η |≫ 1 holds onlynear the wall, certainly not near the axis. Still, theregion near the wall contributes most to the flame sur-face area. Taking into account the exponential regimeof flame acceleration (24) and the velocity distribu-tion (7) and (8) we reduce Eq. (25) to (26)2 (α − Θ + 1 )ϕ =  (Θ − 1 )η − Θ  dϕdη. Integrating Eq. (26), we find(27) ϕ = C  Θ − (Θ − 1 )η  χ , where  C  is the constant of integration and(28) χ = 2   αΘ − 1  − 1  = 2    ΘΘ − 1  − 1  . In the solution (27) and (28) we have to keep onlythe terms of the principal order in  Θ − 1 ≪ 1, and takeinto account that  f( 0 ) = ξ  wall  = Θ , Eq. (22). Then  V. Bychkov et al. / Combustion and Flame 150 (2007) 263–276   267 Eqs. (24), (27), and (28) reduce to(29) f   = Θ( 1 − η) 1 /Θ . We stress that the solution (29) does not satisfy theboundary condition at the axis  ∂f/∂η = 0 at  η = 0.This is, of course, a consequence of the approxima-tion(30)  1 +  ∂f ∂η  2  1 / 2 ≈− ∂f ∂η used in Eq. (25) in the limit of   Θ − 1 ≪ 1. Still, theprincipal contribution to the flame surface area comesfrom the flame skirt. Another shortcoming of the eval-uation (29) is that we adopted the potential flow (7) and (8) everywhere in the burnt matter, while in re-ality it holds only close to the tube end wall and tothe tube axis. Outside of these regions, the flow maybe rotational because of the curved flame shape. Us-ing the evaluation (29), the scaled surface area of the curved flame front is calculated as S  w /πR 2 = 2 1   0 η   1 +  ∂f ∂η  2 dη (31) ≈− 2 1   0 η∂f ∂ηdη = 2  Θ 2 Θ + 1 . Thus, for stoichiometric propane flames with  Θ = 8,we should expect a maximal flame surface areaas large as  S  w /πR 2 ≈  14 . 2. Of course, order-of-magnitude evaluations may be obtained within aneven simpler and cruder approximation of a cylindri-cal shape of radius  R  and height  ΘR  with  S  w /πR 2 ≈ 2 Θ . Relative difference between this approximationand Eq. (31) is about 1 /Θ . 3. Details of the numerical simulations To assess accuracy of the analytical theory de-veloped in the previous section, we performed directnumerical simulations of the hydrodynamic and com-bustion equations including chemical kinetics andtransport processes. In this section we are using di-mensional variables. We investigated the case of anaxisymmetric flow described by the equations(32) ∂ρ∂t  + 1 r∂∂r(rρu r ) + ∂∂z(ρu z ) = 0 ,∂∂t (ρu z ) + ∂∂z  ρu 2 z − γ  zz  (33) + 1 r∂∂r  r(ρu z u r  − γ  zr )  =− ∂P ∂z,∂∂t (ρu r ) + ∂∂z(ρu z u r  − γ  zr ) (34) + 1 r∂∂r  r  ρu 2 r  − γ  rr  =− ∂P ∂r − 1 rγ  θθ  ,∂ε∂t  + ∂∂z  (ε + P)u z − γ  zz u z − γ  zr u r  + q z  + 1 r∂∂r  r  (ε + P)u r  − γ  rr u r  − γ  zr u z + q r  (35) = 0 ,∂∂t (ρY) + 1 r∂∂r  ρru r Y   − rµ Sc ∂Y ∂r  + ∂∂z  ρu z Y   − µ Sc ∂Y ∂z  (36) =− ρY τ  R exp ( − E a /R p T), where(37) ε = ρ(QY   + C V T) + ρ 2  u 2 z + u 2 r  is the total energy per unit volume,  Y   is the mass frac-tion of the fuel mixture,  Q  is the energy release in thereaction, and  C V  is the heat capacity per unit massat constant volume. We consider a single irreversiblereaction of the first order. Temperature dependenceof the reaction rate obeys the Arrhenius law with theactivation energy  E a  and the constant of time dimen-sion  τ  R . The stress tensor  γ  ij   is(38) γ  zz = µ  43 ∂u z ∂z − 23 ∂u r ∂r − 23 u r r  , (39) γ  rr  = µ  43 ∂u r ∂r − 23 ∂u z ∂z − 23 u r r  , (40) γ  θθ   = µ  43 u r r − 23 ∂u z ∂z − 23 ∂u r ∂r  , (41) γ  zr  = µ  ∂u z ∂r + ∂u r ∂z  , and the heat diffusion vector  q i  is given by(42) q z =− µ  C P Pr ∂T ∂z + Q Sc ∂Y ∂z  , (43) q r  =− µ  C P Pr ∂T ∂r + Q Sc ∂Y ∂r  . Here  µ = ρν  is the dynamic viscosity,  C P  is the heatcapacityperunitmassatconstantpressure,andPrandSc are the Prandtl and Schmidt numbers, respectively.The fuel mixture is assumed to be a perfect gas of constant molecular weight  m = 2 . 9 × 10 − 2 kg / mol,with  C P = 7 R p / 2 m  and  C V = 5 R p / 2 m , where  R p = 8 . 31 J /( molK )  is the perfect gas constant. The equa-tion of state is(44) P   = ρmR p T.
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