Acceleration of a flame by flame- vortex interactions

C O M B U S T I O N A N D F L A M E 82: 111-125 (1990)

111

Acceleration of a Flame by Flame-Vortex Interactions P A M E L A K. B A R R Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94551-0969

The acceleration of a premixed flame propagating in a planar channel past stationary obstacles is investigated using a computer model based on combining the discrete vortex method with a flame interface algorithm. Results presented in this article show that the initial acceleration of the flame is caused by the sudden contraction of the flow due to the presence of the obstacles. Because the burning speed SL remains constant, it is the increase in surface area that is responsible for the apparent acceleration. This combustion-generated flow also causes turbulent recirculation regions to form downstream of the obstacles. When the flame interacts with these turbulent eddies, the burning rate increases, producing a stronger velocity field in the channel that further accelerates the flame. In geometries containing a series of obstacles, the higher flow velocities caused by the distorted flame result in stronger turbulent eddies behind subsequent obstacles. Numerical results for these geometries demonstrate the positive feedback mechanism of acceleration as the flame encounters these increasingly stronger turbulent regions. In this article, results are presented for different locations of the series of obstacles, as well as a clustering of obstacles near the ignition source. The effects of blockage ratio, ignition shape, and fuel concentration are also investigated with the computer model.

NOMENCLATURE

V x, y x0, Y0

cell volume Cartesian coordinates vortex location

BR F H i, j

blockage ratio (obstacle height/H) volume fraction of burned gas channel height cell indices

k rnb m n rcore s

discrete vortex index burned mass number of vortices time level index vortex core size volume source to account for gas expan-

F At Pb pu ~b

sion laminar burning speed time velocity field (U = U p + U r) velocity used in vortex shedding component of U r that accounts for presence o f walls potential, or irrotational, velocity field from gas expansion rotational velocity field from discrete vorticies

INTRODUCTION

SL t U Ue UBC Up Ur

Copyright (~) 1990 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010

Greek Symbols discrete vortex circulation strength time step size burned gas density unburned gas density velocity potential

As a flat flame travels away from a wall it induces a flow in the unburned gas because the expansion of the freshly burned gas pushes the surrounding gas away. The flow of this gas past an obstacle creates a region of vorticity downstream of the obstacle. As the flame reaches this vortex the flame distorts, causing its surface area and burning rate to increase, and producing larger convective velocities. For the case of flame propa-

0010-2180/90/$3.50

112 gation in a tube or channel containing a series of obstacles, a positive feedback can be established that results in a continuous acceleration of the flame. That is, the larger velocities that were created in the channel when the first vortex distorted the flame produces stronger recirculation regions behind the succeeding obstacles. As the flame burns into the next (stronger) vortex, it will be distorted faster than before, increasing the rate of burning, and thus producing yet stronger velocities in the channel. Eventually the flow velocities may become sonic and shock waves could form in the combustion-generated flow field. The present analysis only considers the processes of early acceleration where compressibility is not important, This acceleration is caused by the increase in the flame surface area, and not by an increase in the local burning velocity SL, because SL is constant in the model, Accelerating flames pose serious problems because they can produce significant pressure rises, Further, these accelerating flames may result in a large energy release rate, enabling the flame to become a detonation wave. Thus, it may be possible to produce a detonation wave in a lean combustible mixture even though no available ignition source is strong enough to cause direct initiation. Examples of accident damage caused by these blast waves are shown in Baker et al. [1]. These examples show that an accident producing a combustible gas mixture creates a dangerous situation if it occurs within a confined or semiconfined volume with obstacles such as large equipment or doorways between rooms, or if the combustible mixture can enter confining geometries such as ventilation ducts or elevator shafts. The presence of these obstacles can lead to an acceleration of the flame, resulting in overpressures that may exceed the design limit of the confining structure and possibly creating detonation waves that can produce massive localized damage, Lee et al. [2] have studied the acceleration of a flame propagating in a long tube containing a series of evenly spaced obstacles. They showed that for a variety of fuel-air mixtures the leading edge of the flame reaches a steady propagation speed, except in the regime where flame quench dominates (i.e., lean mixtures and small orifice diam-

P . K . BARR eters). They categorize the accelerated flames as either fast flames traveling just under the sound speed of the combustion products or as detonation or quasi-detonation waves. A combination of flame stretch, gas-dynamic choking, and the relative sizes of the orifice diameter and the detonation cell sizes determines the final steady propagation speed. Moen et al. [3] performed similar experiments in large tubes (2.5 m diameter). They observed a significant increase in the overall propagation rate of the flame when a series of obstacles with large internal diameters (2.3 m) were present in the tube. Although these obstacles did not cause much restriction to the flow, they created overpressures ten times that produced without obstacles. The dynamic coupling between the combustiongenerated flow field and the flame distortion is illustrated by Moen et al. [4] to explain results from their study on the effect of obstacles on a freely expanding cylindrical flame. By placing spirals of tubing in the path of the flame, they were able to increase the propagation rate by up to 24 times that observed without the turbulence-producing obstacles. They also showed that the flame slows down once it has propagated past the region containing obstacles, indicating that the enhanced propagation rate requires the turbulence and flow field distortions caused by the obstacles. By leaving a gap between the wall and the obstacles of only 4% of the obstacle height, Urtiew et al. [5] were able to increase the flame speed by a factor of 3.5 over that with obstacles mounted flush with the wall. Their schlieren photographs show that the combustion-generated velocity field pushes the flame through the gaps under the obstacles, causing it to jet into the unburned pockets of gas trapped behind the obstacles and producing an increased burning rate in these regions. In this case, the interaction of the flame and the flow field is enhanced by the small gap under the obstacles. This flame-flow field interaction can be reduced by providing venting along the length of the channel, producing lower overpressures and slower speeds past the obstacles because some of the gas is vented through the openings [6, 7]. Simulations of obstacle-generated flame acceleration have been limited. Moen et al. [3] de-

FLAME ACCELERATION veloped an analytic expression for the maximum overpressures produced when flames propagate in tubes containing obstacles, and their model shows good agreement with experimental results. The analytical model by Taylor [8] predicts the steadystate flame propagation speed that occurs when the extension of the flame surface due to the propagation of the leading edge equals the destruction of the flame area due to burning in the pockets behind the obstacles. Taylor's model can also identify continually accelerating flames. Among the more detailed transient simulations, Marx [9] has developed a two-dimensional unsteady model to simulate large-scale flame acceleration experiments. In this numerical model the flame is artificially thickened so that it can be resolved on the numerical grid. The increase in the local burning rate as the flame propagates into the eddy regions behind obstacles is specified by a turbulent burning submodel. Similar modeling work has been performed by Brandeis [10]. Although the effects of compressibility are included, these models do not predict the detailed interaction of the flame with the eddies, In this article results are presented that show the early stages of the propagation of a flame in a channel containing obstacles. The results were obtained from a numerical model that combines the vortex dynamics technique with a wrinkled laminar flame algorithm. The vorticity in the flow is assumed to originate from shedding of boundary layers at the edges of obstacles. Because these regions of vorticity dominate the rotational flow field in the unburned gas, the vorticity in attached boundary layers and the vorticity generated at the flame are ignored in these simulations. The flame is treated as an interface that can be distorted by the flow field and that propagates into the unburned gas, causing it to expand. This combination allows for an examination of the coupling between combustion-generated flow field and flame distortion. MODEL ASSUMPTIONS The numerical model simulates flame propagation in a planar channel that is open at one end and contains baffles. The presence of obstacles in the

113 path of a premixed flame causes two processes that act to increase the burning rate. One is the contraction of the streamlines as the flow goes around each obstacle. These spatial gradients of velocity increase the flame surface as the flow carries the flame through the contraction regions. The second effect is the formation of a turbulent eddy downstream of each obstacle. When the flame propagates into an eddy, further enhancement of the flame area occurs. The numerical model is used to illustrate the importance of the interaction of a flame with various confining geometries. Although the model is restricted to two dimensions, several physical mechanisms act to produce a three-dimensional flame front. Small-scale three-dimensional turbulence should develop in the eddy regions. Ashurst and Meiburg [11] used three-dimensional vortex dynamics to observe that these smaller structures are easily formed in both the large twodimensional coherent structures of shear layers and the braid regions between them. Other mechanisms that could generate a three-dimensional flame from two-dimensional initial conditions include thermodiffusive instabilities responsible for cellular flames and hydrodynamic instabilities caused by the density jump across the flame. Both current computer capabilities and the size of the physical domain of interest here restrict the simulations to two dimensions. The two-dimensional results presented in this article demonstrate the importance of the interaction of a flame with the obstacle-generated turbulence, which could produce fast flames in lean cornbustion mixtures. The two dimensional restriction may underpredict the growth rate of the flame because fine-scale turbulence and three-dimensional flame folding are ignored in the model. This article focuses on the interaction of the flame with the vorticity that has been created in the unburned gas by the combustion-driven flow past obstacles. This interaction causes the overall burning rate to increase because it increases the surface area of the flame. In the model, the flame is treated as a zero-thickness interface that propagates into the unburned gas at a constant burning velocity SL. The density ratio across the flame is fixed by the combustible mixture. It is assumed

114

P . K . BARR

that the time periods are too short for the vortical flow behind the obstacles to significantly dissipate, and thus no viscous damping is included. All vorticity in the computed flow field originates from the shedding of the boundary layers for obstacles, The model uses slip boundary conditions along the walls, thus neglecting the details of the wall boundary layers. The slip velocity at an obstacle edge determines the magnitude of the vorticity being released into the flow. Furthermore, in this model no vorticity is added at the flame front. Tsuruda and Hirano [12] used schlieren photography to demonstrate that turbulence is generated along an accelerating flame, This generation is greatest when the flame density gradient is normal to the pressure gradient. Using the discrete vortex method, Pindera and Talbot [13] included this baroclinic generation of vorticity in their simulation of a steady flame anchored by a rod. Tsuruda and Hirano [12, 14] and Ashurst and McMurtry [15] have studied how this vorticity source term changes the propagation of a small flame segment. Although the acceleration mechanism described in this article could be enhanced by the baroclinic generation of vorticity, the inclusion of this effect would dramatically increase the computational costs for the simulations presented here. It is left for future work. In spite of these simplifying assumptions, the model does reproduce the experimentally observed interaction of a flame with a twodimensional region of vorticity that was set up by the combustion-generated flow past the edge of a single obstacle, as shown below. Predictions of flame propagation past a series of obstacles in a channel display features that are similar to those observed experimentally. These results show that the interaction of the flame with the vorticity shed behind a series of obstacles is sufficient to set up a positive feedback mechanism that can cause a flame to accelerate, NUMERICAL METHOD In the numerical model the velocity field is decomposed into two components: U

=

U p q-

U r ,

(1)

where U p is the potential velocity induced by the volume expansion due to burning and U r is the rotational velocity induced by the shedding of a boundary layer off an obstacle. A wrinkled laminar flame algorithm is combined with the discrete vortex method. The flame interface algorithm uses a fixed numerical grid to determine both the flame position and the potential flow field. The growth of the vorticity regions behind the obstacles is simulated by introducing discrete vortices at the obstacle edges and tracking the vortices as they interact with each other and with the potential flow field. Ghoniem et al. [16] developed a model that combines both flame propagation and vortex dynamics algorithms to simulate flame stabilization behind a rearward facing step. Flame Propagation Algorithm and Potential Velocity Field In this two-dimensional planar model the flame is treated as a discontinuity that can be distorted by the combustion-generated flow field. A fixed orthogonal grid is used to record the location of the burned and unburned gases, and the flame interface is reconstructed from this information using a hybrid numerical scheme [17, 18]. The procedure is based on locating the flame such that the burned gas in neighboring cells forms a continuous zone, thereby minimizing numerical diffusion of the two gases. The method combines the Simple-Line-Interface-Calculation method (SLIC) [19, 20], for advancement of the flame position due to either advection or self-advancement, with an algorithm from the Volume-of-Fluid method (VOF) [21] for determining the local flame orientation. The initial flame shape is used to define the volume fraction of burned gas in each cell, Fij. The velocity of the flame front is decomposed into two components: one normal to itself at the laminar burning speed, and the other in a direction and of a magnitude governed by the combustiongenerated flow field. The amount and location ~Jf the newly burned gas is determined by using the hybrid interface tracking algorithm to simulate combustion along the flame interface. The flame front is identified by locating all values of Fij be-

FLAME ACCELERATION

115

tween 0 and 1 (or neighboring cells with fractions of 0 and 1), and, thus, the size of the mesh imposes a lower limit on the resolution of flame wrinkling, The flame front propagates normal to itself at the constant laminar burning speed S L for a time period A t . The amount of mass burned in this time step is then m on, j _ m no,j- I = ~

(FTj - Fnj-')Ou Vij,

(2)

dling these irregular boundaries is used [22]. The solution to Eq. 3 is the velocity potential ~, which is related to U p by U p = VO.

(5)

The potential flow velocity is combined with the rotational component to determine the velocity field in the channel.

ij where the superscripts n - 1 and n denote values for the previous and current time steps, pu is the density of the unburned gas, and Vij is the volume of cell i, j . The assumption of constant density in both the burned and unburned gases results in the restriction to low speed flows, that is, speeds below Mach 0.4. The velocity potential ~b resulting from the expansion of the newly burned gas is governed by the Poisson equation ~2 f~ = S i j ,

(3)

where the term Sij represents a source of volume to account for the expansion of the gas in cell i j . The value of this term is found from '

sij =

(Fnj _ AFijn t

-

I) (Pu Pb Pb)

'

(4)

which is located where F i j has increased. Because of the density difference across the flame, this term is always a positive source term. These source terms are actually located one cell length behind the flame to ensure that expansion occurs in the burned gas. (See the Appendix in Ref. 18 for a discussion of numerical accuracy and grid sensitivity.) A slip boundary condition is imposed along all solid walls. For complex geometries, uniform flow is assumed at the outlet. In simpler configurations, where all obstacles are far from the open boundaries, the velocity distribution along the outlet scales with the inverse of the distance to the flame front. The geometry of the obstacle configuration produces an irregular domain for the flow field. In order to solve the Poisson equation over this domain, a fast Poisson solver capable of han-

Rotational

Velocity

Field

When the combustion process pushes gas past an obstacle, a region of vorticity is created downstream of each obstacle as the boundary layer is shed from the edge. The formation of these eddy regions is simulated by introducing discrete vortices into the flow field at the ends of the obstacles (in planar coordinates these discrete vortices represent lines of vorticity or straight vortex tubes). The rotational velocity field is found from the superposition of the flow induced from all of the vortices as determined by the Biot-Savart integral. In the model the vorticity that is shed from the obstacle edge is simulated by discrete elements, or vortices. The rotational velocity field at a point (x, y) from a single vortex k is . F~

Uk r = ! ~

-

(Y - Y0~)

(X - Xok)2 ; ~ -_ - Y0k)2 + rcore~2 -

. Fk ( x - x0,) + J 2~ (x - x0,)2 ÷ (Y - Y0k)2 + rcorek" (6) where i andj are unit vectors in the x and y directions, and (x0k, Y0k) is the location of the vortex center. A finite core size rcore ~. is used to remove the singular point at the vortex center. The circulation strength of each discrete vortex Fk accounts for both the magnitude of the velocity field and the frequency of vortex shedding. Circulation is defined by a line integral about a closed path c of the tangential velocity component along the path. This integral is simplified by assuming a onedimensional velocity profile that varies from zero at the edge of the obstacle to U e at the edge of the boundary layer. The path of integration c is chosen to account for the time period between vortex

116

P.K. BARR

shedding A t. By selecting a rectangular path that has a height greater than the boundary layer and a length equal to the distance covered by a particle traveling at the mean velocity of the fluid enclosed by the rectangle, or A t U e / 2 , this integral is approximated by U • dc ~ ~1 2Ue A t.

rk = f

(7)

c

Once released, the strength of each vortex remains constant. The effect of the solid walls is incorporated into the rotational velocity field by superimposing an irrotational velocity field Uric on the flow induced by the vortices. In this flow field the velocity component normal to the walls is the negative of that from the vortices. When combined with the vortical flow this produces a slip condition along the walls: m Uk r, (8) k=l where m is the total number of vortices. Because U B c is irrotational, the only source of vorticity is from the boundary layer that was shed from the edges of the obstacles, The rotational velocity field U r from Eq. 8 must be combined with the potential velocity field U p from Eq. 5 to produce the velocity field that convects both the flame front and each discrete vortex. This convective velocity field changes both the locations of the vortices and the flame shape, Combustion along this distorted flame gives the volume sources for the next time step. The size of this time step is determined by the Courant condition: the flame advection algorithm will not allow the flame to propagate further than one cell in a single time step. Convection of the discrete vortices is performed in a Lagrangian, or grid-free, manner using first-order time integration, U r = UBc + E

F L A M E PROPAGATION PAST A SINGLE OBSTACLE The interaction of a flame with a single region of vorticity has been studied experimentally by Yip et al. [23] (also described in Strehlow et al.

ignition /

/

e°mbusti~i"

generatedflow

shed vorticity

~ 2 cm I

-""

- -

10 cm - -

Fig. 1. Schematicdiagram of the two-dimensionalgeometry used by Yip et al. [23].

[24]). In their two-dimensional geometry only minor confinement is imposed by two solid perpendicular walls. A shorter horizontal wall forms a small channel (see schematic diagram in Fig. 1). The combustible mixture was ignited at the closed end of this channel, and the gas that expands during combustion pushes flow down the channel past the edge of the channel, and out through the free boundaries. This flow past the edge of the channel exit causes a continuous sheet of vorticity to be released into the flow, which grows in size and strength as the gas flows out the channel exit. Because of both the quiescent initial conditions and the dimensions of the channel, a single vortex is present when the flame reaches the channel exit, as shown in the schlieren photographs in Fig. 2a from Strehlow et al. [24]. In the computer simulation, a uniform grid of 250 × 150 cells is used over the 25 cm × 15-cm region bounded on two adjacent sides by the solid walls, and open to the atmosphere on the other two sides. The length scales of the geometry are set to reproduce experimental conditions. Because the obstacle configuration must be aligned with the numerical grid, it is one cell thick. The properties of the combustible gas in the model represent the stoichiometric propane-air-oxygen mixture used in the experiment. This mixture has an oxygen mole fraction of o~ =O2/(O2+N2)=0.3, where ot -- 0.21 for pure air. The burning velocity SL for this oxygen-enriched mixture is 95 cm/s

FLAME ACCELERATION

117

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t ic it y flame

~

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(a) (b) Fig. 2. Comparisonof schlierenphotographs(Fig. 2a) and numerical results (Fig. 2b). For both sets of results the geometry is as shownin Fig. 1. The time intervalbetweenframesis 0.5 ms. Experimentalresults are from Strehlowet al. [24]. [25] and the density ratio across the flame Pu/pb is 9.46 [23]. When a discrete vortex is released into the flow it is assigned a core size rcore (see Eq. 6), which is randomly selected from a maximum of 0.25 cm to a minimum of 1% of this size. The vortex is shed from the computational cell

adjacent to the edge of the obstacle; the location within this cell is also selected at random. The calculated time development of the flame-vortex interaction is shown in Fig. 2b. As in the experimental results presented in Fig. 2a, the time increment between plots is 0.5 ms. The earliest plot from the numerical simulation was selected as the initial plot because it has a similar flame position as in the first photo from the experimental results. This figure shows both the development of the vorticity region (indicated by the vectors at the end of the obstacle representing the location and velocity of each vortex) and the propagation of the flame (indicated by the bolder line segments forming the flame). The experimental and the numerical results show similar development of the flame-vortex interaction. In both, the swirling motion of the vortex distorts the flame, increasing its surface area and thus the burning rate. The vortex pulls the flame in a spiral, creating separate spirals of burned and unburned fluid. As can be seen by comparing Figs. 2a and 2b, the numerical prediction shows excellent agreement with the experimental results. The vortex wraps the flame around at similar rates. Eventually, the flame consumes the unburned spirals within the vorticity region. At the latest time, the schlieren photograph indicates that the spirals of unburned fluid in the vortex are gone. Small-scale three-dimensional turbulence present in the experiment might increase the burning rate faster than that predicted by the numerical model. As can be seen by comparing the bottom flame shapes in Figs. 2a and 2b, the model predicts a faster horizontal spread of the flame than that observed in the experiment. This is consistent with the fact that in the model both the wall boundary layers and heat transfer are ignored. The excellent agreement with the experimental results of Strehlow et al. [24] indicates that the model captures the main features of flame propagation through regions of vorticity at early times. In this configuration, as the flame spirals around the vortex, much of the flame is propagating perpendicular to the pressure gradient. The baroclinic generation of vorticity along the flame is greatest for this orientation of the density gradient and the

118

P.K. BARR

pressure raient'seeRef15' Thefact at e II numerical results agree so well with the experi-

i

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mental results indicates that this mechanism for generating vorticity is not significant under these conditions,

r

L.

~,

~~

OF OBSTACLES IN A CHANNEL

~

L~~

r ..

L~),

The propagation of a flame past obstacles in a channel imposes a higher degree of confinement than that from the configuration previously discussed. In this case the planar channel has a length 5.1H and height H. The numerical grid is 204 ×40 cells in all of the multichamber computations presented here. In most of the results, the combustible mixture is assumed to be 15% H2 in air, which has a density ratio pu/pb of 4.5. Because results are presented in normalized units (distances are normalized bythechannelheightHandtimesare scaled by SL/H), the laminar burning speed SL is identically equal to one. The maximum core size for the discrete vortices is 0.01H. The acceleration of the flame as it interacts with flow past a series of staggered obstacles is demonstrated in Fig. 3, showing the progress of the flame at equal time intervals (O.03H/SL). Because the flame algorithm reconstructs the front in a cellby-cell manner, a continuous flame surface is not explicitly defined, giving the flame an unconnected appearance. However, the algorithm ensures that the burned and unburned fluids remain separated. In this case the flame propagates through five square chambers separated by the staggered obstacles that are each 0.025H (one cell) Wide and 0.6H high, creating a blockage ratio BR of 60%. The combustible mixture has been ignited in a plane, as indicated in the top frame. Successive frames show both the development of the turbulent recirculation regions and the propagation of the flame as it moves through the channel. Figure 3 shows that the acceleration of the flame starts when its passes the first obstacle. At this stage the flame has not encountered any vorticity, and so the speedup is entirely due to the higher gas velocity caused by the obstacle blockage, or sudden contraction in the channel. As the flame propagates through the second chamber it encoun-

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FLAME PROPAGATION

PAST A SERIES

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I ~ :~,-h I~.~,-~:, =#'e.~ . . . I ,...~.. ~.-',/ , f--~'~" ~ • , '=~:] " ~|%~..___i-~'~',:. , 1 | " "---~.~"/II ~2h~--' ~.~."*~" ' ~ " ~ ) l ...... : " ' ,, Fig. 3. Instantaneousprofilesof flame propagationpast a series of staggeredobstacles. Time intervalbetweenprofilesis 0.03 H/S£ after planar ignition (top frame). The height of the obstaclesis 60% of the channelheight. The densityratio ~ / ~ is 4.5. Flame shape is marked by the heavy line segmerits; locationsof the discrete vortices are indicatedby the scaled velocity vectors.

FLAME ACCELERATION ters the vortices that have been shed off the first obstacle. Because the initial flow velocities were low, these vortices are not strong enough relative to the expansion that occurs along the flame to cause significant distortion of the flame front as it propagates through this chamber. When the flame reaches the third chamber the combustiongenerated velocities have increased because of the larger flame surface area, which now extends back to the first chamber. The gas pushes through the channel to the open exit, shooting the flame along the diagonal to the next chamber and leaving pockets of trapped unburned gas that will burn at rates determined to a large extent by the turbulent eddies. The distortion of the flame front caused by the turbulence increases the burning rate in all of the chambers, as can be seen in the bottom frame, The strongest turbulent flow is found in the farthest chambers, where the highest velocities have occurred. It is in these chambers that the flame is most distorted by the turbulence, resulting in an enhanced burning rate. This enhanced burning will act to further accelerate the flame by producing larger flow velocities and, thus, stronger turbulence behind downstream obstacles, producing higher burning in subsequent trapped pockets, These effects act together to push the tip of the flame at an increasing speed, The flow features displayed in Fig. 3 are also discussed by Moen et al. [4]. They describe "a standing eddy region between the obstacles, which is separated from the outer flow region by a shear layer. As the flame encounters this flow field it will become 'stretched' due to the gradient in the flow velocity field, leading to the formation of a largescale 'flame-fold' consisting of a curved leading flame front in the outer flow region with a trailing flame in the trapped pockets or the standing eddy regions between the obstacles." The results presented in Fig. 3 dispute only the notion of "standing eddies" because in these results, as the eddies evolve, portions of them can be convected downstream. Therefore, the physical picture of steadystate eddies standing behind obstacles gives an inaccurate representation of accelerating flow past obstacles, Similar flow features are displayed when all obstacles are located on the bottom of the channel.

119 Results from this simulation are shown in Fig. 4; all conditions except obstacle placement are the same as in Fig. 3. Again, the flame does not accelerate until it encounters the first obstacle. After that, it jets along the top of the channel, leaving the trapped pockets to burn later. These trapped pockets were also recorded by Chan et al. [6] in their schlieren photographs taken in similar conditions. They reported rapid turbulent burning of these trapped pockets, which can be seen in the bottom frame of Fig. 4 in the results presented here. The propagation of a flame in another geometry shows the same initial behavior. Figure 5 presents the propagation in a channel containing symmetric obstacles with the same blockage ratio as the previous conditions. Now, the flame sweeps down the center of the channel before the gas in each chamber burns. In this case there are twice as many trapped pockets and roughly twice the flame surface area to burn them in comparison with the first two geometries (Figs. 3 and 4). This results in an enhanced acceleration of the flame front as it propagates beyond the computational domain. In all of these results, at early times the vortices move in a spiral pattern behind each of the obstacles, as shown in the top frames of Figs. 3-5. As the flow velocities increase, the spiral pattern disappears and the discrete vortices are convected down the channel. This can be seen in the bottom frames of these figures, where the velocity vectors of some of the discrete vortices indicate that they are propagating into the next chamber. The convection of vorticity can have a dramatic impact on the burning rate in a geometry where the flame has a chance to catch up to the displaced vorticity. Such is the case in Fig. 6, where the presence of the obstacles in just the left portion of the channel allows the convected vorticity to accumulate in the unrestricted right portion. The flame jets past the obstacles and "explodes" when it reaches the unobstructed section. This could be the cause of the significant overpressures recorded by Moen et al. [3] for geometries containing just a single obstacle. The propagation rate is very sensitive to the shape of ignition. Results obtained by using a single line ignitor instead of the planar ignitor (Fig. 7) show that initially the flame has a cir-

120

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Fig. 4. Instantaneous profiles of flame propagation past a series of obstacles on the bottom of the channel. All other parameters are the same as in Fig. 3.

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Fig. 5. Instantaneous profiles of flame propagation past a series of symmetric obstacles. All other parameters are the same as in Fig. 3.

FLAME ACCELERATION I

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121 Fig. 8. Thus, the acceleration of a flame ignited by four line sources is greater than that ignited by a planar source. Figure 9 compares the rates of burning for the one-line, four-line, and planar ignition cases. The dot at the top of each curve indicates when the flame has reached the open end of the channel. Because the burning rates have been normalized by the initial burn rate for the plane ignition case, these curves also measure the increases in the surface areas of the flames relative to the initial planar fame area. As shown in this

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II 1'~ -''~ " " ' ' - ' ~ "~ " ' < ' - " : : " ' - " -~" Fig. 6. Instantaneous profiles of flame propagation past a series of staggered obstacles located in the left half of the channel. All other parameters are the same as in Fig. 3.

cular cross section, which becomes distorted as it approaches the first obstacle. Because the flame has not yet reached the top and bottom walls as it passes the first obstacle, the flame surface area is much greater than in the case of planar ignition for a similar position of the leading edge. This additional surface area produces higher velocities and a faster propagating flame, as indicated by the relopening.ative amount of time required to reach the channel When a planar ignition source is approximated by four line sources, these individual flames do not merge into a planar flame before they are affected by the presence of the first obstacle, as shown in

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~

f./ ....

L/

~ ) . ~ . , . f ~~ - x,~. . , . ) i

"~~ ~ 1 ~ ~(~"f~i

(-"'-~'~"

; - r-'l

" " ~ '

I

Fig. 7. Instantaneous profiles of flame propagation past a series of staggered obstacles. Ignition is by a single line source. All otherparameters are the sameas in Fig. 3.

122

P . K . BARR

11-one-line ~>~ ignition ]111 frplana I r casebecausetheflamesel°ngatebef°rereaching~/lligniti°n plot, although initially the one- and four-line ignition cases have lower burn rates than the planar ignition cases, both cases surpass the planar ignition

the first obstacle (Figs. 3, 7, and 8). For the fourline ignition case, the fast rise and subsequent decrease in burn rate at early times < 0.7) is caused by the growth and then merging of the individual flame kernals, as shown in Fig. 8. The height of the obstacles affects the strength of the positive feedback mechanism. This effect

~ o

(tSL/H

~ ;~

~

four-line g

]

tgnttton ' "'

/

~

/ J

O I

I

~

[

[ [

I

i

!

0.0

I

i

~: ?

f.~~

[~:~)'

•---'1

L:_~:.;.;o

['~!~!'

[: iif ~'"

t

I

il

I

0.1 t SL/H°2 0.3

Fig. 9. Normalizedburn rate curves showingthe effect of ignition type (planar, one- and four-line) on flame propagation. All other parameters are the same as in Fig" 3' which sh°ws instantaneousprofiles for planar ignition. Profiles for one- and four-line ignition are shown in Figs. 7 and 8, respectively. Burn rates are normalizedby the initial value from the planar ignition case. is shown , in Fig. 10. Although the presence obstacles of any size distorts the flame and introduces turbulence in the flow, these effects are weaker for smaller obstacles. For the lowest blockage ratio, where the channel is obstructed by only

BR

b- • . . _~ -~ X l .,. -.

i,I' .,f

:7~..,,,

i.:..,'i

I' /~'' "L~'~~~ .) /

"----~___.,-' -I

\~t:'~e.I

Fig. 8. Instantaneousprofiles of flame propagation past a sedes of staggeredobstacles. Ignition is by four line sources. All other parameters are the same as in Fig. 3.

0.4

60%

..~=~~ 9-

o

4~

i

i

O.O o., tSL/H°'Z

i o.a

0.4

Fig. 10. Normalized burn rate curves showing the effect of obstacle blockage ratio (BR=20%, 40%, and 60%) on flame propagation.All other parameters are the same as in Fig. 3, whichshows instantaneousprofiles for BR=40%. Burn rates are normalizedby the initial value.

FLAME ACCELERATION tra.

123

T

pu/pb 5.5 =

/

¢~

laminar burning speed should be significantly affected by the turbulent mixing, resulting in either localized flame quench or possibly flame extinction. Their results show a sharp transition in flame acceleration at mixtures of 13% hydrogen in air. For the higher hydrogen concentrations, where the reactions are faster, quenching does not appear to limit the maximum obtainable flame speed. It is in this regime, where the combustion rate along the flame is not affected by the flow conditions, that the results presented here are valid.

~l 4. /

' 1 3 . 5 t ~

j

~

•~

t~

~,

o

0.0

,

0.1

,

0.2

,

0.3

0.4

t SL/H Fig. 11. Normalized burn rate curves showing the effect of density ratio (ou/p~ = 3.5, 4.5, and 5.5, corresponding to H2-10~:, 15cZ, and 20% in air) on flame propagation. All other parameters are the same as in Figure 3, which shows instantaneous profilesfor H2=15%. Burn rates are normalized by the initial value,

20%, the increase in the flame propagation rate is greater than that which can be explained by just the area contraction at the obstacles. Even in this configuration, the flame distortion due to both the higher velocities past the obstacles and the interaction with the turbulent eddies causes a significant increase in the surface area of the flame and enhances the propagation rate. In an experimental study of flame acceleration using very small obstacles (BR=0.16), Moen et al. [3] observed overpressures greater than 1 bar. Without the obstacles the maximum overpressure was 0.12 bar. The effect of the fuel concentration is depicted in Fig. 11 for three density ratios--pu ~Oh = 3.5, 4.5, and 5.5--which correspond to fuel mixtures of 10%, 15%, and 20% H2 in air, respectively, The case with the highest density ratio provides the flame with the largest push and results in the fastest flame, as indicated by the earlier onset of acceleration. These results are valid only under the conditions that the laminar burning speed SL remain constant over the entire flame surface and that the flow velocities are within the low Mach number restriction imposed by the assumptions in the velocity potential. The results of Lee et al. [26] indicate that for the leanest mixture, H2 = 10%, the

CONCLUSIONS The effect of obstacles on the propagation of a premixed flame in a channel has been simulated. The model combines the discrete vortex dynamics method to simulate the time development of the eddy regions downstream of each obstacle with a wrinkled laminar flame algorithm to track the flame as an interface between the burned and unburned fluid. This simulation, though limited to low speed flows, includes the phenomena that resuit in the acceleration of a flame as it propagates in a channel past obstacles. Results from this model show that the presence of obstacles strongly influences the propagation rate of the flame. This is because both the strong convective flow patterns and the introduction of regions of high turbulence act to increase the burning rate of the flame. Results presented in this article and in Moen et al. [3] show that even small obstacles can dramatically increase the flame propagation speed. The dynamic coupling of the flame front and the flow field described by Moen et al. [4] are displayed in the results presented here, showing both the strong convective flow field responsible for the acceleration of the leading edge of the flame and the intense burning in the trapped pockets of reactants. These results show that the vorticity that was shed from each obstacle can be convected away from it, counter to the description by Moen et al. [4] of "standing eddies." During the early stages, flame propagation rates are very sensitive to the shape of ignition. Results presented here show that flames initiated from multiline sources do not merge to form a planar flame before reaching the first obstacle. Rather,

124 the combustion-generated flow field causes them to elongate, increasing the total surface area and causing the flame to accelerate faster than in the case of planar ignition. Because this model is twodimensional and planar, point ignition could not be simulated. However, the combustion-generated flow field that causes the line-ignited flames to elongate will also distort the point-ignited flames and inhibit them from merging to form a planar flame. Thus, results from multipoint ignition that is used to simulate planar ignition in experiments can be misleading if the data of interest occur during the early stages before a steady propagation rate is realized, The sensitivity of the flame to the type of ignition that was demonstrated in the simulated resuits also indicates that flames of this type must be three dimensional. Any spanwise perturbation along the two-dimensional front will continue to grow, resulting in a three-dimensional flame. The eddy structures will also be three dimensional. However, the two-dimensional results presented here provide significant insight into the phenomena of obstacle-generated flame acceleration. Although in principle it is possible to extend this model to a third dimension, it would be expensive in terms of computer resources. The simulation presented in Fig. 3 required 2.3 hours of CPU time on a Cray IS computer. In the twodimensional model the amount of computer time increases linearly with the complexity of the flame front (as measured by the number of cells through which it passes), and it increases with the square of the number of vortices (because the velocity at each vortex is determined by the velocity induced from every other vortex). The addition of a third dimension would require smaller domains for a given computer system and budget. Improvemerits in either numerical techniques or computer hardware would make three-dimensional simulations more attractive. Improved vortex dynamics methods for three dimensions are currently being developed (see Winckelmans and Leonard [27]). Results presented here show a positive feedback mechanism of flame acceleration caused by a series of obstacles located in a channel. Although in these results only a short section of the channel has been simulated, the model would predict that in

P.K. BARR longer sections the coupling between the flame and the flow field continues to accelerate the flame. As the flame propagation rate increases, effects due to compressibility, flame quench, baroclinic generation of vorticity along the flame front, and viscous damping of the vortices could become important. The vortex dynamics model described in this paper provides detailed insight into the flame-vortex interactions that can lead to initial flame acceleration. I would like to thank Bill Ashurst f o r his continued assistance with both the vortex dynamics m e t h o d and the development o f the model. His comments on the manuscript are also appreciated. ! would also like to thank Ken Marx and Billy Sanders f o r technical discussions on f l a m e acceleration. This work was partially supported by both the U.S. Department o f Energy, Office o f Basic Energy Sciences, Division o f Chemical Sciences and the U.S. Nuclear Regulatory Commission.

REFERENCES 1. Baker,W. E., Cox, P. A., Westine, P. S., Kulesz, J. J., and Strehlow, R. A., Explosion Hazards and Evaluation, Elsevier, New York, 1983. 2. Lee, J. H. S., Knystautas, R,, and Chan, C. K., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, p. 1663. 3. Moen,I. O., Lee, J. H. S., Hjertager, B. H., Fuhre, K., and Eckhoff, R. K., Combust. Flame 47:31 (1982). 4. Moen, I. O., Donato, M., Knystautas, R., and Lee, J. H., Combust. Flame 39:21 (1980). 5. Urtiew, P. A., Brandeis, J., and Hogan, W. J., Combust. Sci. Technol. 30:105 (1983). 6. Chan, C., Moen, I. O., and Lee, J. H. S. Combust. Flame 49:27 (1983). 7. Sherman, M. P., Tiezen, S. R., Benedick, W. B., Fisk, J.w., and Carcaiss, M., Prog. Astronaut. Aeronaut. 106:66 (1986). 8. Taylor, P. H., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, p. 1601. 9. Marx, K. D., Sandia National Laboratories Report NUREG/CR-4855 SAND87-8203, 1987. 10. Brandeis, J., Combust. Sci. Technol. 44:61 (1985). 11. Ashurst, W. T., and Meiberg, E. J. Fluid Mech., 189:87 (1988). 12. Tsuruda, T., and Hirano, T., Combust. Sci. and Technot. 51:323 (1987).

FLAME 13.

14. 15. l6. 17. 18. 19.

20. 21.

ACCELERATION

Pindera, M.-Z., and Talbot, L., Twenty-First Symposium (International)on Combustion, The Combustion Institute, Pittsburgh, 1986, p. 1357. Tsuruda, T., and Hirano, T., Prog. Astronaut. Aeronaut. 105:110 (1986). Ashurst, W. T., and McMurtry, P. A., Combust. Sci. Technol. 66:17 (1989). Ghoniem, A. F., Chorin, A. J., and Oppenheim, A.K., Philos. Trans. R. Soc. Lond. A 304:303 (1982). Barr, P. K., and Ashurst, W. T., Sandia National Laboratories Report SAND82-8773, 1983. Barr, P. K., and Witze, P. O. SAE Paper 880129, 1988. Noh, W. F., and Woodward, P., in Lecture Notes in Physics (A. I. van de Vooren and P. J. Zandbergen, Eds.), Springer-Verlag, New York, 1976, Vol. 59, p. 330. Chorin, A. J., J. Comp. Phys. 35:1 (1980). Hirt, C. W., and Nichols, B. D., J. Comp. Phys. 39:201 (1981).

125 22. 23.

24.

25.

26. 27.

Grcar, J. F., private communication, 1983. Yip, T. W. G., Strehlow, R A., and Ormsbee, A. I., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, p. 1655. Strehlow, R. A., Ormsbee, A. I., Yip, T. W. G., and Adiasar, M., Annual Report to the Gas Research Institute, Report No. GR181/0035, 1983. Duggar, G. L., and Graab, D. D., Fourth Symposium (International) on Combustion, The Combustion Institute, 1952, p. 302. Lee, J. H. S., Knystautas, R., and Freiman, A., Combust. Flame 56:227 (1984). Winckelmans, G., and Leonard, A., in Mathematical Aspects o f Vortex Dynamics (R. E. Caflisch, Ed.), SIAM, Philadelphia, 1989, p. 25.

Received 7 March 1989; revised 14 August 1989