Unburned mixture fingers in premixed turbulent flames

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Proceedings of the Combustion Institute 35 (2015) 1401–1408

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Unburned mixture fingers in premixed turbulent flames Andrei N. Lipatnikov a,⇑, Jerzy Chomiak a, Vladimir A. Sabelnikov b, Shinnosuke Nishiki c, Tatsuya Hasegawa d a

Department of Applied Mechanics, Chalmers University of Technology, Go¨teborg 412 96, Sweden b ONERA - The French Aerospace Lab., F-91761 Palaiseau, France c Department of Mechanical Engineering, Kagoshima University, Kagoshima 890-0065, Japan d EcoTopia Science Institute, Nagoya University, Nagoya 464-8603, Japan Available online 9 July 2014

Abstract Data obtained in 3D direct numerical simulations of statistically planar, 1D premixed turbulent flames indicate that the global burning velocity, flame surface area, and the mean flame brush thickness exhibit significant large-scale oscillations with time. Analysis of the data shows that the oscillations are caused by origin, growth, and subsequent disappearance of elongated channels filled by unburned gas. The growth of such an unburned mixture finger (UMF), which deeply intrudes into combustion products, is controlled by a physical mechanism of flame-flow interaction that has not yet been highlighted in the turbulent combustion literature, to the best of the present authors knowledge. More specifically, the fingers grow due to strong axial acceleration of unburned gas by local pressure gradient induced by heat release in surrounding flamelets. Under conditions of the present DNS, this physical mechanism plays an important role by producing at least as much flame surface area as turbulence does when the density ratio is equal to 7.5. Although, similarly to the Darrieus-Landau (DL) instability, the highlighted physical mechanism results from the interaction between a premixed flame and pressure field, it is argued that the UMF and the DL instability are different manifestations of the aforementioned interaction. Disappearance of an UMF is mainly controlled by the high-speed self-propagation of strongly inclined flame fronts (cusps) to the leading edge of the flame brush, but significant local increase in displacement speed due to large negative curvature of the front plays an important role also. Ó 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Premixed turbulent flame; Physical mechanisms; Darrieus-Landau instability; DNS; Modeling

1. Introduction In recent 3D Direct Numerical Simulation (DNS) studies [1–4] of statistically planar, 1D, premixed turbulent flames, the following phenom⇑ Corresponding author. Fax: +46 31 18 09 76.

E-mail address: [email protected] (A.N. Lipatnikov).

enon was documented. Both turbulent burning velocity U t and mean flame brush thickness dt , evaluated by averaging the DNS data over yz-planes, which are parallel to the mean flame surface, exhibit significant large-scale oscillations with time, see Fig. 1 in Section 3. Bell et al. [2] and Poludnenko and Oran [3,4] associated these oscillations with formation of a cusp [5–7], i.e. a highly curved tip of a conical flame surface with

http://dx.doi.org/10.1016/j.proci.2014.06.081 1540-7489/Ó 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

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A.N. Lipatnikov et al. / Proceedings of the Combustion Institute 35 (2015) 1401–1408

a small angle, with unburned gas being inside the cone. For instance, Bell et al. [2] attributed the oscillations of U t ðtÞ to periodical growth of cusps into “elongated channels” filled by unburned gas. Similar structures were documented in experimental studies of premixed turbulent flames, see Fig. 2 in Ref. [8], Fig. 1 in Ref. [9], Figs. 9 and 10 in Ref. [10], Fig. 10 in Ref. [11], Fig. 4 in Ref. [12], Figs. 3, 7 and 12 in Ref. [13], Fig. 4d in Ref. [14], Fig. 4 in Ref. [15], Figs. 2 and 4 in Ref. [16], or Figs. 4a and 10 in Ref. [17]. To the best of the present authors’ knowledge, physical mechanisms that control the initiation and growth of the elongated channels have not yet been analyzed. As far as the disappearance of the channels is concerned, Bell et al. [2] have noted “period of apparent rapid movement when the sides of the channel close upon each other and the cusp returns to a more typical position relative to the rest of the flame”. Poludnenko and Oran [4] highlighted the rapid propagation of a cusp along its axis by exploiting the classical theory by Zel’dovich [7] and by numerically simulating a cusp formed due to collision of two planar laminar premixed flames that move in almost opposite directions. Formation of isolated pockets of unburned gas, documented in a 2D DNS study by Kollmann et al. [18,19], can also be relevant to the disappearance of the channels, as noted by Bell et al. [2]. The goal of the present work is to analyze DNS data in order to gain further insight into physical mechanisms that control the growth and disappearance of the aforementioned elongated channels. In the next section, the attributes of the DNS database are summarized. Obtained results are discussed in the third section followed by conclusions. 2. DNS database Because DNSs addressed here were discussed in detail elsewhere [1,20,21] and the computed data were used in a couple of papers [22–27], we restrict ourselves to a brief summary of the simulations. The DNSs dealt with statistically planar, 1D, adiabatic premixed flames modeled by unsteady 3D continuity, Navier–Stokes, and energy equations, as well as the ideal gas state equation. The Lewis and Prandtl numbers were equal to 1.0 and 0.7, respectively. The dependence of the molecular transfer coefficients on the temperature T was taken into account, e.g. the kinematic viscosity m ¼ m0 ðT =T 0 Þ0:7 . Combustion chemistry was reduced to a single reaction. The computational domain was a rectangular Kx  Ky  Kz , with Kx ¼ 8 mm, Ky ¼ Kz ¼ 4 mm, and was resolved using a uniform mesh of 512  128  128 points. Homogeneous isotropic

turbulence (rms velocity u0 ¼ 0:53 m/s, integral length scale L ¼ 3:5 mm, Kolmogorov scale g ¼ 0:14 mm, and Ret ¼ u0 L=mu ¼ 96) was generated at the inlet boundary, entered the computational domain with a mean velocity U, and decayed along the direction x of the mean flow. The flow was periodic in y and z directions. At an initial instant, a planar laminar flame was embedded into statistically the same turbulence assigned for the velocity field in the entire computational domain. Subsequently, the inflow velocity was increased at two instants, i.e. U ð0 6 t < t1 Þ ¼ S L < U ðt1 6 t < t2 Þ < U ðt2 6 tÞ, in order to keep the flame in the computational domain till the end t3 of simulations. Three cases characterized by three different density ratios r ¼ qu =qb were investigated. Here, subscripts u and b designate unburned and burned gas, respectively. Characteristics of these flames are listed in Table 1, where S L is the laminar flame speed and dL ¼ ðT b  T u Þ= max jdT =dxj is the laminar flame thickness. We will place the focus of discussion on results obtained in cases H and L, characterized by the highest and lowest density ratio, respectively. Results reported later were obtained for t P t2 , with the instants t2 being different in cases H, M, and L. Because the mean inlet velocity was constant at t P t2 , no external acceleration affected results obtained in the laboratory coordinate framework. Both time-dependent mean quantities qðtÞ averaged over transverse yz-planes and mean quantities hqi averaged also over time interval t2 6 t 6 t3 , with t3  t2  1:5L=u0 , will be discussed in the next section. Conditioned quantities, e.g. hqjn1 < c < n2 i, and Probability Density Functions (PDFs), e.g. P ðq; n1 < c < n2 Þ, reported in the following, were obtained using joint PDFs P ðc; q; xÞ, which were computed by processing the DNS data saved for a plane x ¼const at various instants within the time interval t2 6 t 6 t3 . Here, c ¼ ðT  T u Þ=ðT b  T u Þ is the combustion progress variable, q is an arbitrary quantity, while hqjn1 < c < n2 i is a mean value of q averaged by considering only points x and instants t such that cðx; tÞ is within the interval ðn1 ; n2 Þ. The statistical convergence of the DNS data obtained at t2 6 t 6 t3 was shown in earlier papers [20–27]. Table 1 Flame characteristics. r ¼ qu =qb S L , m/s dL , mm hU t ðtÞi, m/s 0 hU 2t i1=2 =hU t i hdt ðtÞi, mm 0 hd2t i1=2 =hdt i

Case H

Case M

Case L

7.53 0.600 0.217 1.13 0.10 1.23 0.16

5.0 0.523 0.191 1.00 0.17 1.41 0.29

2.5 0.416 0.158 0.74 0.11 1.35 0.25

A.N. Lipatnikov et al. / Proceedings of the Combustion Institute 35 (2015) 1401–1408

3. Results and discussion

(a)

4

normalized using S L , see solid curves, and (ii) integrated Flame Surface Density jrcj (FSD) Z x2 Z K y Z K z jrcjðx; tÞdxdydz Aðc1 ; c2 Þ  ; ð2Þ Ky Kz x1 0 0

integrated Flame Surface Density

H

L

H

Ut FSD, cR<0.5 FSD, cR>0.5 FSD

0.5 0

10

15

flame brush thickness, m

20

time, ms

30

25

(b)

5

max gradient 0.01
4

H

3

2

4

6

8

x, mm

(b)

4

18.71 ms 18.81 ms 18.92 ms 19.02 ms 19.12 ms 19.22 ms

3

2

1

0 0

2

4

6

8

x, mm

L

1.5 1

0 0

Fig. 2. Iso-lines cðx; y; z; tÞ ¼ 0:85 on xz-planes associated with the furthest advancement of the reaction zone to the burned gas. Time is specified in legends. (a) Growth of an unburned-mixture finger. (b) Disappearance of the finger.

(a) 2

2

1

z, mm

see dotted-dashed, dashed, and dotted curves, associated with Að0; 0:5Þ; Að0:5; 1Þ, and Að0; 1Þ, respectively. The distance xk is such that cðxk ; tÞ ¼ ck , where ck is a reference value of the mean combustion progress variable, and xk ¼ 0 or Kx if ck ¼ 0 or 1, respectively. The oscillations of U t ðtÞ are controlled by the oscillations of the total FSD Að0; 1Þ, cf. solid and dotted curves. Figure 1b shows oscillations of a mean flame brush thickness dt , which was either determined using the maximum gradient method, i.e. dt ¼ 1= max j@c=@xj, see solid curves, or equal to the axial distance between planes characterized by cðx; tÞ ¼ 0:01 and 0.5 (dotted curves), cðx; tÞ ¼ 0:5 and 0.99 (dotted-dashed

17.73 ms 17.93 ms 18.14 ms 18.35 ms 18.55 ms 18.66 ms

3

z, mm

Figure 1a shows significant oscillations of (i) turbulent burning velocity Z Kx Z Ky Z Kz W c ðx; tÞdxdydz U t ðtÞ ¼ ð1Þ qu Ky Kz 0 0 0

1403

L

curves), and cðx; tÞ ¼ 0:01 and 0.99 (dashed curves). Mean and rms values of U t and dt ¼ 1= max j@c=@xj are reported in Table 1. In all cases, a peak burning velocity U t ðtÞ is accompanied by formation of an UnburnedMixture Finger (UMF) that deeply intrudes into burned gas. The growth and disappearance of such a finger, associated with the third peak of U t ðtÞ in case H, is shown in Fig. 2a1 and 2b, respectively (the value of c ¼ 0:85 is chosen, because the local mass rate of product creation W c ðc ¼ 0:85Þ is close to its maximum value). Similar results were obtained when investigating other peaks of U t ðtÞ and in other cases. 3.1. Growth of unburned mixture fingers

2

Figure 2a shows that growth of an UMF results in increasing flame brush thickness and surface

1 0

10

15

20

time, ms

25

30

Fig. 1. Oscillations of (a) normalized burning velocity (solid line), integrated FSD (broken lines), and (b) mean flame brush thicknesses, obtained in cases H (thick lines, 9:3 < t < 20:7 ms) and L (thin lines, 20:6 < t < 31 ms).

1 Although the iso-line cðx; tÞ ¼ 0:85 on the xz-plane at t ¼ 17:73 ms, see dotted-dashed curve in Fig. 2a, is weakly wrinkled, the same iso-line plotted on the perpendicular xy-plane is wrinkled so that Að0; 1Þ  1:8 at this instant.

A.N. Lipatnikov et al. / Proceedings of the Combustion Institute 35 (2015) 1401–1408

H

conditioned PDF

=0.1 =0.5 =0.9

0.4 0.3 0.2 0.1 0

0

4

8

-2

12

16

enstrophy, ms

Fig. 3. Conditioned PDF P ðX; c < 0:005Þ for enstrophy, obtained for various hci in case H.

L

0

-4

-8 10

15

20

time, ms

30

25

(b)

5 4 3

H

2

L

1 0

10

15

20

time, ms

30

25

Fig. 4. (a) Speeds dxf =dt of flamelet elements that advance the farthest into unburned (broken curves) or burned (solid curves) gas and (b) axial flow velocities uðxf Þ in the vicinity of these elements, obtained in cases H (thick lines, 9:3 < t < 20:7 ms) and L (thin lines, 20:6 < t < 31 ms). Here, xf ðtÞ is the minimum (for various y and z) or maximum value, respectively, of xðy; z; tÞ such that cðx  hx ; y; z; tÞ < 0:01 6 cðx; y; z; tÞ, where hx is the grid resolution in the axial direction.

Shown in Fig. 5 are conditioned PDFs P ðT l ; c < nu Þ for various terms T l in the balance equation for the axial momentum @u @u @u @u 1 @p 1 @sxk þu þv þ w ¼ þ ; @t |fflffl{zfflffl} @x @y @z q @x q @xk |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflffl{zfflfflffl} |fflffl{zfflffl} T1

0.5

(a)

4

dxf/dt, m/s

area, thus, increasing burning velocity. Such an explanation of the large-scale oscillations of U t ðtÞ and dt ðtÞ was already suggested by Bell et al. [2] and by Poludnenko and Oran [3,4], but a physical mechanism that controls the growth of long UMFs has not yet been clarified, to the best of the present authors’ knowledge. Accordingly, the goal of this subsection is to reveal such a mechanism and to assess its importance. Turbulence could trigger the origin of an UMF, but does not seem to control its growth, because we did not observe intensification of turbulence within UMFs. For instance, shown in Fig. 3 is conditioned PDF P ðX; c < nu Þ for enstrophy X ¼ 0:5ðr  uÞ  ðr  uÞ at various mean hci specified in legends and nu ¼ 0:005. The PDF becomes narrower and its peak moves to lower X when hci is increased, thus, indicating that fluctuations in vorticity r  u are weaker within UMFs when compared to the leading edge or the middle part of the flame brush. It is worth noting that, due to deep intrusions of UMFs into the burned gas, conditioned quantities hqjc < nu iðxÞ are solely averaged over the UMFs provided that hciðxÞ is sufficiently close to unity. Accordingly, conditioned PDF P ðq; c < nu ; hciÞ characterizes the statistics of a quantity q within the UMFs if 1  hci  1. For instance, hci ¼ 0:995 at x ¼ 4:9 and 4.3 mm in cases H and L, respectively, and unburned gas arrives at these points only with UMFs, see Fig. 2. A physical mechanism that controls the growth of UMFs can be guessed by considering Fig. 4, which indicates high observed speeds dxf =dt of flamelet elements that advance the farthest into products (solid curves in Fig. 4a) and large axial velocity uðxf Þ of unburned gas (c  0:01) in front of these elements (solid curves in Fig. 4b). It is worth stressing that, in case H, the peak values of uðxf Þ are an order of magnitude higher than u0 , and, therefore, these peak values are unlikely to be controlled by the turbulence.

axial velocity, m/s

1404

T2

T3

ð3Þ

T4

obtained at various hci in case H. Here,   @ui @uj 2 @uk sij ¼ qm þ  dij @xj @xi 3 @xk

ð4Þ

is the viscous stress tensor, dij is the Kronecker delta, and the summation convention applies for the repeated index k. These results indicate that (i) the magnitudes of all terms T l are comparable at the leading edge of the flame brush, see Fig. 5a, (ii) the most probable magnitudes of the axialconvection term T 1 (solid curves) and the pressure-gradient term T 3 (dotted-dashed curves) are increased with an increase in hci, cf. Figs. 5 a and b, and, (iii) at large hci, the magnitudes of these two terms are much higher than the magnitudes of transverse-convection and viscous terms,

A.N. Lipatnikov et al. / Proceedings of the Combustion Institute 35 (2015) 1401–1408

(a) =0.1

conditioned PDF

0.002

1 2 3 4

0.0015

0.001

0.0005

0

-4000

-2000

0

2

2000

4000

terms, m/s

(b) =0.5 0.0025

1 2 3 4

conditioned PDF

0.002

0.0015

0.001

0.0005

0

-1000

0

1000

2000

2

3000

4000

terms, m/s

(c) =0.995 0.006

1 2 3 4

conditioned PDF

0.005 0.004 0.003 0.002 0.001 0

0

2000

2

4000

terms, m/s

Fig. 5. Conditioned PDFs P ðT l ; c < 0:005; hciÞ for terms T l in Eq. 3, with l being specified in legends. Case H.

the unburned flow acceleration at large hci, as shown by the DNS data reported in Fig. 5c. The fact that pressure gradient and flow convergence are induced in unburned gas enveloped by a flame front is well known, e.g. see [28]. In order to argue that such a pressure gradient can strongly accelerate the unburned gas, let us invoke the following rough order-of-magnitude estimate. If we neglect turbulent velocity fluctuations within an UMF, then, integration of the Euler equation yields [28] @u=@t þ p=qu þ u  u=2 ¼ f ðtÞ, where uðx; tÞ is a potential, i.e. u ¼ ru, and f ðtÞ is an arbitrary function of time. If we (i) extend this relation to hci ! 0, (ii) disregard the term @u=@t, and (iii) assume that axial velocity uuf of unburned gas near the tip of an UMF is much higher than the local transverse velocity, then, we arrive at u2uf  u20 þ 2ðp0  puf Þ=qu , where subscript 0 designates quantities at hci ! 0. The maximum uuf is associated with an instant when the UMF tip reaches the trailing edge hci ! 1 of the flame brush. At that instant, the pressure difference p0  puf can be estimated to be of the order of the difference between the pressure drop Dpt ¼ qu U 2t ðr  1Þ across a statistically planar, 1D premixed turbulent flame and the pressure drop DpL ¼ qu S 2L ðr  1Þ across an unperturbed laminar flame (DNS data support the estimate of the latter pressure drop in spite of high curvature of the local flame tip). Using the instantaneous U t ðtÞ ¼ 1:34 m/s and u0 ¼ U ¼ 1:15 m/s, this estimate yields maxfuuf g  4:5 m/s, in line with Fig. 4b. It is worth stressing that the UMFs contribute significantly to the global flame surface area Að0; 1Þ and U t . Curves in Fig. 6 show a ratio of Að0; hciÞ to Að0; 1Þ vs. hci, whereas symbols plotted on the curves show the lowest hci (cb in the following) and Að0; cb Þ=Að0;R1Þ such that the 1 probability Pðw P b; hciÞ ¼ b P ðw; c < nu ; hciÞdw is 50% or higher. Here, w is a ratio of minfjT 1 j; jT 3 jg to maxfjT 2 j; jT 4 jg and b is a number, e.g. five circles plotted in Fig. 6 show 1 0.8

integrated FSD

T 2 (dashed curves) and T 4 (dotted curves), respectively, see Fig. 5c. Therefore, at large hci, the high magnitude of the axial flow acceleration (term T 1 ) is controlled by the local axial pressure gradient (term T 3 ), with the difference between the PDFs P ðT 1 ; c < nu ; hci ¼ 0:995Þ and P ðT 3 ; c < nu ; hci ¼ 0:995Þ, cf. solid and dotteddashed curves in Fig. 5c, being associated with transient effects. Contributions by terms T 2 and T 4 to the acceleration of the axial flow of unburned gas are negligible within UMFs (large hci). It is the axial pressure gradient that is the driving force of the acceleration discussed and pushes local flamelet elements farther into products, thus, creating a long UMF, whereas the turbulence (terms T 2 and T 4 ) contributes weakly to

1405

case H case H case L case L

0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Reynolds-averaged combustion progress variable Fig. 6. Relative contribution (curves) Að0; hciÞ=Að0; 1Þ to the production of the global flame surface area vs. hci and the probability Pðw P b; hciÞ for b ¼ 2; 3; 4; 5 and 10 (symbols from left to right, respectively).

A.N. Lipatnikov et al. / Proceedings of the Combustion Institute 35 (2015) 1401–1408

cb obtained for b ¼ 2, 3, 4, 5, and 10 in case H. Thus, in this case characterized by the density ratio typical for flames under room conditions, about a half of the global flame surface area is created at hci larger than cb ðb ¼ 10Þ  0:67 (right circle), i.e. this half of the area is mainly created due to the axial flow of unburned gas induced by the negative pressure gradient, whereas the turbulence plays a minor role (the DNS data show that the ratio of minfjT 1 j; jT 3 jg to maxfjT 2 j; jT 4 jg is larger than 10 with probability 50%). If b ¼ 2 (left circle), cb  0:22 and more than 86 % of flame surface area is created in the region where Pðw P 2Þ P 0:5. Separation between PDFs P ðT 1 ; c < nu Þ and P ðT 3 ; c < nu Þ, which peak at large positive T 1 and T 3 if hci is close to unity, and P ðT 2 ; c < nu Þ and P ðT 4 ; c < nu Þ, which peak at low jT 2 j and jT 4 j and vanish for large T 2 and T 4 , is also observed in case L (not shown). However, the effect is less pronounced, because the magnitude of the pressure gradient is lower due to a lower density ratio when compared to case H. Accordingly, cb  0:65 if b ¼ 2 or cb  0:92 if b ¼ 4, see left and right triangles, respectively, and only about 40 or 10 % of the global flame surface area (dashed curve) is created in zones characterized by Pðw P 2Þ P 0:5 or Pðw P 4Þ P 0:5, respectively. The probability Pðw P bÞ does not reach 50 % if b P 5 in case L. The physical mechanism highlighted above (acceleration of unburned gas by pressure gradient induced by heat release in surrounding flamelets) is somehow similar to the classical Darrieus-Landau (DL) instability discussed in many textbooks, e.g. [28]. However, the DL instability and the formation of UMFs seem to be different manifestations of the interaction between a premixed flame and pressure field. First, Robin et al. [26] have noted that, under conditions of the present DNS, “the dimensions of the computational domain are not large enough to permit the development” of the DL instability even if the density ratio is as high as r ¼ 7:53. Second, experiments by Truffaut and Searby [29] have shown that the maximum growth rate of the DL instability is about 0.3 ms1 in C3H8air flames, whereas the rate a1 dxuf =dt of growth of an UMF tip reaches 2 ms1 in case H even if the largest possible estimate ðxuf  xle Þ=2 of perturbation amplitude a is used. Here, xuf and xle are the maximum and minimum, respectively, values of x such that cðx  hx ; y; z; tÞ < 0:005 and cðx; y; z; tÞ > 0:005, while hx is the grid resolution in x-direction. Third, the DL instability not only pushes some elements of the flame front to products, but also pulls other elements of the flame front into unburned gas. However, we have not yet found a pronounced correlation between U t ðtÞ and qle ðtÞ,

qle is a flow characteristic averaged over where  an yz-plane associated with cðtÞ  1. Figure 1 shows that oscillations of the FSD averaged over the leading half of flame brush (dotted-dashed curves in Fig. 1a) or of its thickness (dotted curves in Fig. 1b) are much less pronounced than oscillations of the counterpart quantities associated with the trailing half of the flame brush. Figure 4 shows that, in case H, the observed speed dxf =dt of flamelet element that advances the farthest into unburned gas and the local axial flow velocity uðxf Þ (broken curves) are much less than the counterpart quantities associated with flamelet element that advances the farthest into products (solid curves). Figure 7 indicates that, in case H, flamelet elements that advance the farthest into unburned (solid curves) and burned (dashed curves) gas move mainly in the same direction with respect to cðx; tÞ ¼ 0:5, whereas such elements move in opposite directions with the framework of the DL theory. Fourth, as discussed in detail elsewhere [30], interaction of flames with the local pressure field manifests itself in a number of ways in combustion science. For instance, as far as governing physical mechanisms of premixed turbulent combustion are concerned, the rapid propagation of an inherently laminar instantaneous flame front along the axis of a vortex tube should be borne in mind [31]. Although the DL instability and the rapid flame propagation in a vortex tube are both manifestations of the influence of combustion-induced pressure perturbations on the constant-density flow, these two phenomena are widely recognized to be different physical mechanisms of premixed turbulent combustion. Similarly, two other manifestations of the aforementioned influence, i.e. the DL instability and UMFs, appear to be different physical mechanisms of an increase in premixed flame surface area in a turbulent flow. Although acceleration of unburned gas by pressure gradient induced due to heat release in enveloping flame front occurs during growth of the DL instability of a laminar premixed flame, this 4 3

distance, mm

1406

2 1

H

0

leading flamelet trailing flamelet

L

-1 -2

10

15

20

time, ms

25

30

Fig. 7. Distance between plane cðx; tÞ ¼ 0:5 and xf ðtÞ.

A.N. Lipatnikov et al. / Proceedings of the Combustion Institute 35 (2015) 1401–1408 1

normalized strain rate

1.2

probability

0.8 0.6 0.4 0.2 0 0.9

1407

0.92

0.94

0.96

0.98

1 0.8 0.6 0.4 0.2 0 -3

1

-2

mechanism appears to be weak for small perturbations and can be overwhelmed by flame stabilization due to cusp formation [7]. Under conditions of the present DNS, turbulent eddies appear to create perturbations of flame front that are sufficiently strong in order to result in high pressure-driven acceleration of the unburned gas and the growth of elongated UMFs. Further study of the birth of an UMF is definitely required. 3.2. Disappearance of unburned mixture fingers In the present DNS, collisions of sides of UMFs were not observed and the fingers disappeared due to the motion of the enveloping flame front to the left, see Fig. 2b. In spite of high local axial flow velocity of unburned gas, such motion was possible when an angle between the local flame front and the x-axis was small so that the speed S d =jnx j of the self-propagation of the highly inclined flame front in the x-direction was larger than the local u. Here, S d ¼ ½r  ðqDrcÞþ W c =ðqu jrcjÞ is the local displacement speed, n ¼ rc=jrcj is the unit vector normal to the front and pointed at the unburned gas, and D is molecular diffusivity. For instance, Fig. 8 shows that probability of jn1 x j > 5, evaluated under conditions of 0:015 < c < 0:045 and V f  S d þ u  n > 0, is higher than 60 % at large hci in case H (dashed curve). Very large values of jn1 x j cause high negative dxf =dt shown in Fig. 4a, see solid curves. A crucial role played by high jn1 x j in the disappearance of UMFs was already highlighted by Poludnenko and Oran [3,4]. The present data reveal another important physical mechanism, i.e. substantial increase in local displacement speed S d due to high negative curvature2 of the

2 Because Le ¼ 1 in the present DNS, variations in the local consumption velocity were weak.

0

1

Fig. 9. Normalized strain dL at =S L vs. normalized flame curvature dL r  n computed in case H in grid points such that 0:05 < cðx; tÞ < 0:1 and 0 6 xf ðtÞ  x 6 0:5 mm at t ¼ 18:76 ms. xf ðtÞ is the axial position of the finger tip.

2.5

conditioned PDF

Fig. 8. Conditioned (0:015 < c < 0:045 and V f > 0) probabilities of n1 x < 5 (dashed curves), S d > S L (solid curves), and S d > 2S L (dotted-dashed curves) computed in cases H (curves) and L (curves with crosses).

-1

normalized curvature

Reynolds-averaged combustion progress variable

=0.5, case H =0.999, case H =0.5, case L =0.999, case L

2 1.5 1 0.5 0 0

1

2

3

4

5

6

7

8

9

Sd/SL Fig. 10. Conditioned P ðsd ; 0:05 < c < 0:1; V f > 0Þ.

enveloping flame front, e.g. see Fig. 9. Figure 10 shows that conditioned PDF for the normalized displacement speed sd  S d =S L does not vanish in a wide range of sd > 1 at hci ¼ 0:999, whereas the conditioned PDF computed at hci ¼ 0:5 is narrow and peaks at sd ¼ 1. The former PDF (solid or dotted curve) indicates also that S d > S L when the tip of an UMF moves to unburned gas, i.e. V f > 0. The importance of the increase in S d is further supported in Fig. 8, which shows that the conditioned probability of sd > 1 reaches unity at large hci and the conditioned probability of sd > 2 is strongly increased by large hci. Thus, although the last term in S d =jnx j ¼ S L sd ð1=jnx jÞ statistically larger than sd , the constraint of sd > 1 is also of importance in order for UMFs to disappear. It is worth noting that, while the axial velocity at UMF tip is substantially lower in case L than in case H, cf. thin and thick solid curves in Fig. 4b, the former u½xf ðtÞ is still significantly (by a factor of about five) higher than S L . Therefore, in order for an UMF tip to move to left, 1=jnx j should be large, i.e. the finger should be long enough. Accordingly, a L-finger grows during a longer

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time when compared to a H-finger so that a decrease in the density ratio by a factor of three reduces UMF length only from 3.5 to 2 mm, cf. thick and thin dashed curves in Fig. 7. Figure 9 shows that flame enveloping an UMF is characterized by large negative curvature j ¼ r  n and moderate positive strain rate at ¼ @uk =@xk  nj nk @uj =@xk , whereas perturbed laminar flames that are commonly considered to model the influence of turbulent stretching on local burning rate [32] are associated with either jat ¼ 0 (twin planar counterflow flames or a collapsing spherical flame) or jat > 0 (an expanding spherical flame). Bearing in mind an important role played by UMFs under conditions of the present and other [2–4] DNSs, the influence of negative curvature and positive strain rate on local burning rate is worth studying. The tip of a laminar Bunsen flame is a proper model problem. 4. Conclusions The above analysis of DNS data implies that acceleration of unburned gas by local pressure gradient induced by heat release in surrounding flamelets can create unburned mixture fingers that deeply intrude into combustion products, thus, increasing flame surface area, turbulent burning velocity, and mean flame brush thickness. Under conditions of the present DNS, this physical mechanism plays an important role by producing at least as much flame surface area as turbulence does in case H. Similarly to the DL instability, this physical mechanism results from the interaction between a premixed flame and pressure field but does not seem to be reduced to the DL instability. Unburned mixture fingers disappear mainly due to the propagation of highly inclined flame fronts to the leading edge of the flame brush, but significant local increase in displacement speed due to large negative curvature of the front plays an important role also.

Acknowledgments AL gratefully acknowledges the financial support by the Chalmers e-Science Centre, Swedish Energy Agency, and Combustion Engine Research Center. VS gratefully acknowledges the financial support by ONERA, France.

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