Propagation of a premixed flame in a divided-chamber combustor

C O M B U S T I O N A N D F L A M E 7 7 : 1 0 1 - 1 2 1 (1989)

101

Propagation of a Premixed Flame in a Divided-Chamber Combustor R. J. CATTOLICA, P. K. BARR, and N. N. MANSOUR* Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94550

A study of the propagation of lean premixed ethylene-air flames in a divided-chamber combustion vessel was conducted to compare experimental observations with a numerical simulation based on a flame sheet-vortex dynamics model in axisymn~tric coordinates. The temporal and spatial development of flames exiting a small cylindrical prechamber were observed in the experiments using laser-schlieren videography. The equivalence ratio of the ethylene-air mixture was varied from 0.5 to 0.65. The corresponding flame speeds S~ for this range of equivalence ratios were measured in a complementary study and were shown to increase from 10 to 24 cm/s, respectively, while the ratio of unburned to burned gas densities varied by less than 20%. The associated increase in gas velocity with equivalence ratio was used to change the estimated Reynolds number in the experiment from 1870 to 8090. Good agreement between the predicted and measured prechamber flame propagation rates was obtained. For the lowest Reynolds number experiment the calculated spatial and temporal development of the flame in the main combustion chamber agreed with the experimental observations. For all Reynolds numbers the numerical results scale with the characteristic time scale obtained by normalizing with the burned gas flame speed Sb. In the experimental results this normalized time scale is not an accurate measure of similar flame development. As the Reynolds number was increased in the experiment the flame development was more rapid. At the highest Reynolds number conditions, a combination of the simplifying assumptions in the flame sheet-vortex dynamics model results in a slower development of the flame-vortex interaction without the experimentally observed shear layer instabilities.

IN T R O D U C T I O N The influence o f fluid mechanics on flame propagation has been the subject o f numerous studies in combustion research. As the degree o f complexity o f the fluid mechanics in a combustion p r o b l e m is increased the level o f sophistication in modeling the combustion chemistry is typically reduced because o f computational necessity. In modeling the combustion o f premixed f u e l - a i r mixtures the fiat-flame burner p r o b l e m with one-dimensional fluid mechanics includes the most detailed chemical kinetics and transport phenomena [1]. Although this modeling approach can be extended to steady, two-dimensional problems [2], the extension to three-dimensional and/or unsteady prob-

* Present address (N.N.M.): Computational Fluid Dynamics Branch, NASA Ames Research Center, Moffett Field, CA 94035. Copyright © 1989 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010

lems [3] is difficult and requires the introduction o f a number o f simplifications that do not model in sufficient detail the fluid physics effects on flame structure. F o r turbulent combustion problems, stochastic methods [4, 5] can be used to model the fluid mechanics with the reactive flame front viewed as a flame sheet with infinitely fast chemistry. This latter method is not applicable to unsteady combustion problems that require a deterministic approach. A deterministic approach based on the concept of a flame sheet [6, 7] has been used in modeling premixed flame propagation using the discretevortex method [8, 9]. Applications o f this method to turbulent flame development in a combustion tunnel [10], turbulent j e t ignition [11, 12], and flame acceleration past obstacles [13, 14] has reproduced qualitative features o f these combustion systems. Most o f these applications were twodimensional simulations that combined distributed

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102 vortex elements with distributed volume sources to describe the fluid mechanics. Flame propagation was introduced through the flame speed Su, which was assumed constant and normal to the flame. For turbulent combustion a two-dimensional planar solution and a constant flame speed are severely restrictive in obtaining quantitative results. Turbulent flow is inherently three dimensional and the flame speed in such an environment will depend strongly on flame stretch (flow divergence and/or flame curvature). To fully understand the applicability of the vortex-dynamics method and its associated assumptions in predicting flame propagation, a systematic study of a combustion system is required in which the character of the flow can be changed in a controlled manner from laminar to turbulent conditions. Although general laws governing flame propagation in tubes and vessels are well known [15], specific details of the influence of fluid mechanics on flame propagation remain to be discerned. In studying the coupling of fluid mechanics and combustion, observation of the development of unsteady flame propagation in a combustion vessel has proved very useful in elucidating the fundamental processes that are involved. In our initial experiments, the propagation of a spherical, laminar, adiabatic flame into a quiescent gas was observed with advanced spectroscopic imaging techniques [16]. The distribution of the OH concentration through the flame front was measured and used to validate the concentration predict from an unsteady one-dimensional flame model with detailed chemistry [17]. To introduce more complex fluid mechanics into the experiment a prechamber was added to the combustion vessel, and the interaction of the flame propagating through a starting vortex-ring structure was investigated. The effect of flow field elements, such as a stagnation point, a vortex, and a shear layer, on the chemical structure of the flame was examined [18] with spectroscopic imaging of the hydroxyl molecule. Studies of the effect of Reynolds number on flame propagation and flame structure in this system were conducted [19, 20] to examine the coupling of fluid mechanics and combustion. With a constant fuel-air mixture ratio, the flow conditions in the divided-chamber combustion

R . J . CATTOLICA ET AL. experiment were changed from laminar to turbulent by reducing the diameter of the orifice connecting the prechamber with the main combustion vessel. Dramatic changes in the structure of the flame and the rate of flame propagation in this experiment were observed with laser-schlieren and fluorescence imaging techniques. The propagating flame produced in the dividedchamber combustor is ideally suited for simulation with the discrete-vortex method. Under the conditions studied in this article the ethylene-air flame is thin relative to the size of the combustor, so it can be treated as a discontinuity in the flow field. For laminar flame conditions at low Reynolds number, the constant flame speed assumption used in the flame sheet-discrete-vortex method should be valid. As the Reynolds number of the system is increased, both the constant flame speed assumption and the assumption of axisymmetry will eventually break down, and the two-dimensional, flame sheet-discrete-vortex method will fail. Comparison of the observed spatial and temporal develoFment of flames propagating in this combustion system at different Reynolds numbers with the numerical simulations based on the flame sheet-vortex dynamics method will test the limits of applicability of the underlying assumptions. EXPERIMENT

The divided-chamber flame propagation experiment was conducted in the combustion vessel illustrated in Fig. 1. The 2-liter vessel was divided by a small cylindrical prechamber of 39 mm diameter and 38 mm length, with a spark electrode pair located 1.75 mm from the closed end and a circular orifice of 2.54 cm diameter leading to the main combustion chamber at the other end. The vessel has optical access with four orthogonal 100mm-diameter fused silica windows and was instrumented with a pressure transducer. Previous experiments in the divided-chamber combustor increased the Reynolds number by reducing the prechamber orifice diameter, with the fuel-air mixture held constant. This resulted in large changes in the Reynolds number. To obtain fine control of the Reynolds number in the current experiment, the preehamber orifice diameter was

FLAME PROPAGATION IN DIVIDED-CHAMBER COMBUSTOR

103

GASOUTLET 4-7 GASINLET

t

ESSURETRANSDUCER

PRECHAMBER - -

REACTANTS

PRODUCTS SPARK ELECTRODE Fig. 1. Divided-chamber combustion vessel with cylindrical prechamber.

held fixed and the fuel-air equivalence ratio was varied to increase the flame speed and the resulting gas velocity. In the experiment, five ethylene-air mixture ratios were used. Ethylene-air was chosen for the experiments because of the low flame speeds attained under fuel lean conditions. Listed in Table 1 are the fuel-air equivalence ratios ~, burned-gas flame speed Sb, measured flame exit time, estimated gas exit velocity through the prechamber orifice, and Reynolds numbers.

TABLE 1 Ethylene-Air Equivalence Ratio, Burned Gas Flame Speed, Flame Exit Time, Gas Velocity, and Reynolds number

¢a 0.50 0.525 0.55 0.60 0.65

Burned gas flame speed Flameexit Sb (era/s) time (ms) 55. 67. 81. 123. 157.

40. ° 33. 26.5 16.5 12.

Exit gas velocity Reynolds (cm/s) number 110. 135. 165. 254. 475.

a Corrected for induction time (see Appendix).

1870. 2300. 2830. 4330. 8090.

The gas velocity ahead of the flame exiting the prechamber was calculated from the measured ethylene-air flame speed and density ratio across the flame front (see Appendix). The gas velocity through the orifice was estimated from the area ratio, assuming steady, one-dimensional, flat flame propagation. The estimated Reynolds number of the starting jet from the prechamber was based on the unburned reactant density and viscosity, with the orifice diameter used as the characteristic length. Because in the experiment the flame was ignited by a point source in the prechamber and initially propagates as a hemispherical front, and thus at a faster rate because of increased surface area compared to a flat flame, the velocities and Reynolds numbers listed in Table 1 should be taken only as relative indicators of the actual parameters. Because this is an unsteady problem no single value of velocity or Reynolds number characterizes the starting jet flow through the prechamber orifice. The use of a characteristic velocity based on the burned gas flame speed or the resulting gas exit velocity derived from a simple one-dimensional approach is a convenient choice. It is important to note, however, that substantial differences exist between this "charac-

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R . J . CATTOLICA ET AL.

teristic" Reynolds number and the actual unsteady process, which is certainly not one dimensional. The propagation of the flame in the main combustion chamber was recorded with a laser schlieren videography system illustrated in the schematic diagram in Fig. 2. A 2-W, cw argon-ion laser operated at a wavelength of 514.9 nm was used to bacldight the main combustion vessel. The laser light was formed into a series of 0.5-#s pulses synchronized with the framing rate (2000 Hz) of a high-speed video camera (Kodak-Spin Physics model SP2000) using an acoustic-optic modulator. The schlieren visualization was recorded on 32-track magnetic tape. With playback and digitization, each frame from the laser schlieren record was stored on a minicomputer. The flame exit time for each equivalence ratio in Table 1 was determined from the time between spark ignition of the flame at the bottom of the prechamber and the first appearance of the flame at the orifice leading to the main combustion chamber. The accuracy of the flame exit time measurement was one video frame time, i.e., 500/zs. For the leanest equivalence ratio studied (4, = 0.5), the measured flame exit time was corrected for a 15ms induction time observed in a separate experiment to determine the flame speed (see Appendix). In Fig. 3 individual frames taken from the video

recording of the spatial development of the propagating flame are presented for each equivalence ratio given in Table 1. The frames were selected to maintain comparable penetration distance into the main combustion vessel. The time given with each frame was referenced to the spark-discharge time. The number in parentheses below each frame is the time normalized by a characteristic time based on the length of the prechamber (38 mm) and the burned gas flame speed Sb listed in Table 1. The flame development for the 4, = 0.50, ethylene-air mixture is shown in Fig. 3a over the time period from 44 to 72.5 ms after ignition. This time was corrected for the induction time effect discussed previously. On the normalized time base the flame visualization in this figure is from ~ = 0.634 to 1.044. The laminar development of this flame is similar to that observed previously in a study of methane-air flames [19]. The flame initially propagates through the prechamber orifice along the axis of the starting jet of gas ejected from the prechamber (T = 0.634-0.696) located at the bottom of the field of view. As the vortex-ring structure [18] of the starting jet is encountered the flame develops a symmetric "mushroom"-like shape (~ = 0.785) and proceeds to propagate around the core of the vortex (~" = 0.871 to 0.877). Later in time (T = 1.044), the interaction

ARGON.ION LASER 514.5 nm

, J

SPIN PHYSICS

01-

......

Fig. 2. Schematic diagram of laser-schlieren videography setup.

FLAME PROPAGATION IN DIVIDED-CHAMBER COMBUSTOR of the flame with the starting jet is completed as the flame that is drawn around the vortex-ring core and that in the stem of the structure burn together. For an ethylene-air equivalence ratio of cI, = 0.525 the spatial development of the flame in Fig. 3b is identical to that for the leanest case at 4, = 0.50. Although the temporal development of the flame in the main combustion vessel is 30% faster (referenced to the exit time on a real-time basis), on the normalized time scale the development is nearly identical, with only a 7% increase in development speed based on the normalized times from the last frames. In Fig. 3c the flame development for 4, = 0.55 is similar to that for the two leaner mixtures, differing significantly only in the last frame (r = 0.910) as wrinkling appears in the flame front. Although the propagation of the flame through the starting jet structure is a factor of 2 faster on a real-time basis than the 4, = 0.50 case, on a normalized time basis the agreement is within 13 % at r = 0.910, again with the increased equivalence ratio condition producing the faster flame development in the normalized time scale. For these three leanest mixtures the general development rate of the flame in the main combustion vessel was similar when placed on the normalized time scale based on the burned gas flame speed, although this rate increased slightly as the equivalence ratio was raised. This trend of increasing the rate of development continues for an equivalence ratio of • = 0.60, as is indicated by the normalized times shown in Fig. 3d. Also, at this equivalence ratio the higher velocity of the gas ejected from the prechamber orifice results in a vortex-ring structure with higher velocities in the vortex core, producing a noticeable effect on the flame shape. This is exhibited in the spatial development of the flame at r = 0.676 and 0.740, where the flame is rapidly convected into the vortex core. At r = 0.805 and 0.837 the higher gas velocity results in the appearance of KelvinHelmholtz instabilities appearing as waves on the shear layer [21, 22] along the stem of the structure. In Fig. 3e the visualization of the flame development for this highest velocity case (4, = 0.65) exhibits the same general behavior with the appearance of instabilities in the shear layer,

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although in spite of these instabilities the flame still maintains an axisymmetric character. Also, the time scale for flame development continues to be more rapid. In comparing with modeling results it is important to understand the pressure change during the interaction of the flame with the starting jet from the prechamber. In Fig. 4 the pressure histories in the divided-chamber combustor for the five ethylene-Mr mixtures listed in Table 1 are plotted. As the equivalence ratio was increased the peak pressure in the system was higher because of the increase in energy released by combustion. The time to reach the peak pressure reflects the increase in both the flame speed and the density ratio for the richer mixtures. In all cases the pressure in the chamber has risen less than 10% during the interaction of the flame with the starting jet. NUMERICAL SIMULATION AND THEORY

The numerical model of flame propagation combines a wrinlded laminar flame model with the discrete-vortex method. The flame interface algorithm requires a fixed numerical grid to determine both the flame position and the resulting combustion-generated flow field. The growth of the vorticity region produced in the main chamber by the gas flowing through the prechamber orifice is simulated by introducing individual vortex rings at the orifice edge and tracking them as they interact with each other and with the combustiongenerated flow field. Previous combinations of these models have been used in planar geometry. In this article both the flame interface model and the vortex dynamics method are developed for the axisymmetric case. Flame Model and Prechamber Simulations

The simulation of flame propagation in the divided-chamber combustor is performed in two stages: first we model the laminar flame propagation in the prechamber, which determines the flow rate into the main chamber, and second we use this flow rate as a boundary condition for the main

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chamber simulation. By separating the prechamber from the main chamber, the numerical grid in each section is rectangular, which simplifies the problem, and unless backflow from the main chamber to the prechamber takes place, the flow in the main chamber will not significantly affect the prechamber results. Although the experiments were performed in a closed chamber, in these simulations we assume that the gas does not compress because over the time period of interest the experimentally measured pressure rise is small (less than 10%) as shown in the pressure curves of Fig. 4 for the corresponding time periods of Fig. 3. We model the premixed flame as an axisymmetric discontinuity separating the burned from the unburned gases. These gases have different (but spatially uniform) densities, limiting the calculations to low Math number flows. We specify the initial flame kernel position by specifying the location of burned gas within the prechamber. The domain of interest is discretized into computational cells located at (r i, zj). In each cell the burned gas fraction is known. The flame interface that separates the unburned from the burned gases is reconstructed from the burned gas volume fraction within each computational cell. The basic philosophy of the reconstruction method is to place the burned gas in neighboring cells together,

thereby minimizing diffusion of the two gases [23]. This scheme is based on a combination of the Simple Line Interface Calculation method (SLIC) [24, 25] for advancement of the flame position due to either advecfion or burning, and an algorithm from the Volume of Fluid method (VOF) [26] for determining the local flame orientation. To simulate the flow in the prechamber we assume that during the combustion process the flame propagates with two velocities, one normal to itself at the flame speed Su (from Table A1 in the Appendix), and the other in a direction (and of a magnitude) governed by the combustion-generated flow field. We determine both the amount and the location of the newly burned gas by using both the SLIC and VOF algorithms to simulate combustion. The local expansion of the freshly burned gas is treated as a volume source located in the burned gas behind the flame. The strength of the source, sij, in a cell depends on both the volume fraction of unburned gas within this cell that is consumed by the flame, AF,-.b, and the ratio of the unburned to burned gas densities (also listed in Table A1) as Sij = AFu~b(PU/Pb --

1)/dt,

(1)

where the total amount of freshly burned gas depends on both the flame surface area, the flame speed, and the time step size. The flow field in the prechamber is assumed to

FLAME PROPAGATION IN DIVIDED-CHAMBER COMBUSTOR be irrotational; therefore we use UP = V~,

(2)

where the potential ~b satisfies V26 = s~i.

(3)

Because the thermodynamic pressure is assumed to remain constant, the gas does not compress and the amount of volume created during burning must exit the prechamber through the orifice, where we assume a uniform velocity profile. The exit velocity is determined by integrating the volume expansion over the entire prechamber volume Uz~i, = ~

1 I

~rro 2

v

sij d r ,

(4)

where ro is the radius of the orifice. The boundary layer in the prechamber is expected to remain attached and all the vorticity generated during combustion in this region will be concentrated at the walls. Thus, the assumption of slip boundary conditions in the prechamber should be valid. We obtain the velocity potential generated by the source distribution by solving the resulting Poisson equation using second-order approximation. The velocity field at the flame location is found by interpolating from the velocities on each cell face. The advection causes the flame to distort, changing both the location and area of the flame. Combustion along this distorted flame gives us the volume sources for the next time step. The model of a flame as an interface separating two different (but uniform) density gases is unstable to perturbations of all wavelengths, as has been shown by landau and Lifshitz [27]. This hydrodynamic instability of the flame front causes any perturbations introduced along the flame to grow. Nonphysical perturbations can be formed with this numerical model due to both the finite computational mesh and the discrete-volume sources. Because these perturbations are caused by the numerics and not the physics they are of the same scale as the numerical grid. Thus, we smooth the flame front by making the local flame speed a linear function of the local curvature normalized by the grid size, similar to the approach of

109

Markstein [28]. The amount of smoothing is determined from calculations of freely expanding, spherical flames. This type of flame model was used to simulate the formation of a cusped, or "tulip," flame during combustion in a vessel [29]. These simulations showed that the cusp forms when the flame burns towards the side walls, producing a distortion in the flame front that continues to grow. In this case the source of the perturbation is physical, the confinement of the wall produces an increase in the velocity field that distorts the flame. In the numerical simulations the prechamber geometry has the same dimensions as in the experiment (see Fig. 1), and the combustible gas is an ethylene-air mixture at an equivalence ratio ~I, of 0.55 (see Table 1, as well as the Appendix). We place the initial flame kernel at the location of the spark electrodes in the experiment, and select the kernel size to be resolvable on the computational grid (20 by 40 cells in the radial and axial directions). This initial flame is shown in Fig. 5a, along with the predicted flame shape at 2-ms intervals. Note that in the model we track the location of the burned fluid, storing the volume fraction of burned fluid in each cell. From these fractions the flame front is reconstructed with line segments. This process causes the front to appear discontinuous, as can be seen from the figure. Because the flow field is governed by a velocity potential, the velocity through the orifice is directly proportional to the flame surface area. Figure 5b shows the variation in orifice velocity for the flame propagation depicted in Fig. 5a (solid line). The velocity increases dramatically as the flame area grows from the initial spherical kernel. When the flame reaches the orifice and enters the main chamber (indicated by the arrow on the curve) the velocity is still large because combustion is still occurring in the prechamber (see Fig. 5a). When the flame reaches the side walls the velocity drops quickly. Because the walls are assumed to be adiabatic they do not quench the combustion and the flame continues to propagate at the same flame speed until all of the unburned gas is consumed. This sharp drop is further enhanced by the flame model because, due to the limited resolution, the model tries to align the

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Fig. 5. Prechamber flame development and gas velocity through exit orifice predicted by the numerical simulation: (a) Point ignition for i = 0.55 with S~ = 14 cm/s at 2-ms intervals. £o) Velocity profiles for (a): solid line--14 cm/s; dashed line--28 cm/s; arrow indicates flame exit time. (c) Point ignition for i = 0.55 with S. = 14 cm/s and density ratio increased by 50% at 2-ms intervals. (d) Velocity profile for (c): arrow indicates flame exit time. (e) Plane ignition for i = 0.55 with S. = 14 cm/s at 2-ms intervals. (f) Velocity profile for (e): arrow indicates flame exit time.

80

FLAME PROPAGATION IN DIVIDED-CHAMBER COMBUSTOR flame with grid lines, causing much of the flame surface to reach the wall at the same time. Replacing the combustible gas mixture with one that has the same density ratio and a different flame speed causes the time scale (and velocity magnitudes) to change, but the flame shape remains the same. This can be shown by using a characteristic length and the laminar burning velocity to normalize Eq. 3. For a gas having twice the flame speed as the ethylene-air mixture at cI, = 0.55 the flow through the orifice is the dashed curve in Fig. 5b. Note that by doubling the burning velocity, the peak velocity through the orifice increases by a factor of 2, and the time period is cut in half relative to the results for the • = 0.55 case. Although the flame shape is the same as those shown in Fig. 5a, the time interval is 1 ms. However, if the combustible mixture is replaced with one that has the same flame speed, but 50% higher density ratio, both the time scale and the flame shape change, as shown in Figs. 5c and 5d. The time interval separating the flame profiles in Fig. 5c is the same as that in Fig. 5a, 2 ms, but it takes less time for the flame to reach the orifice. Note that although the flame speed is the same as in Fig. 5a, the integrated volumetric flow through the orifice is not necessarily the same because differing amounts of burned and unburned gases can be passed through the orifice. The shape of the initial flame kernel has a

111

significant impact on flame propagation in the prechamber. Figure 5e shows the flame profiles when combustion is initiated as a flat flame, for the same ethylene-air mixture used in Fig. 5a (~ = 0.55). In this case, the surface area of the flame remains fairly constant, producing a constant flow through the orifice, as shown in Fig. 5f. As can be seen from both Figs. 5e and 5f, it takes much longer for the flame to reach the orifice when it is initiated as a flat flame than as a point, even though the surface area of the initial flame is much greater for the flat flame.' In this case the distortion of the flame as it approaches the orifice does not produce much of an increase in orifice velocity, as can be seen in Fig. 5f, because this distortion occurs near the centefline. Because we used a flat flame profile to estimate the Reynolds numbers in Table 1, these values are too low. The relative values, however, are qualitatively correct. The effect of the equivalence ratio on the simulation of the prechamber combustion and the gas velocity exiting the prechamber orifice is shown in Fig. 6 for the mixtures listed in Table 1. In each of these cases the flame is initiated at the spark location, as was described previously. As is expected, the velocity through the orifice is the greatest for the mixture with the fastest flame speed. The times for flame arrival at the orifice, also indicated on Fig. 6, can be compared with the experimentally measured values. This compari-

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son, presented in Fig. 7, shows good agreement, indicating that the flame propagation model gives a reasonable simulation of the prechamber combustion.

tal data near the nozzle where the jet is axisymmetric. The velocity field in the main chamber is tal~en as the sum of two velocities:

Vortex Dynamics Simulation of Main Chamber Flame Propagation

O= Or+ OP,

Initially, as the combustion in the prechamber pushes the unburned gas through the orifice into the main chamber, the resulting flow field is that of an axisymmetric starting jet. Although the combustion chamber is closed, at this stage the volume of this starting jet is much smaller than the chamber volume. As the velocity increases, the strength of the shear layer changes, forming a sizable region of vorticity in the main chamber before the flame passes through the orifice. It is this interaction of the flame with the vorticity that distorts the flame surface as it propagates through the main chamber. In our numerical simulations, we model the developing shear layer in the main chamber by superimposing individual vortex rings. The use of ring-vortex elements to simulate startup jets was done by Acton [30] to investigate the effects of harmonic excitation in the coalescence of eddies. Her results show good agreement with experimen-

where 0 f is the rotational velocity induced by the vortex rings and OP is the potential velocity induced by the volume sources due to burning. The velocity field satisfies the following relations: V " 0 = V " 0 p = Sij

(5a)

and V x O - - V x Or= oo,

(5b)

where co is the vorticity field in the chamber. In this work, a set of vortex rings is used to model the vorticity region caused by the gas flowing from the prechamber into the main chamber. As given by the Biot-Savart law, the velocity induced by a vortex ring at a point P in the flow field is found from the integral

l i-.o/x

(6)

FLAME PROPAGATION IN DIVIDED-CHAMBER COMBUSTOR where Q/~is the vector to the point p from the set of points Q along ~, the circumference of the vortex ring. To avoid large velocities near a ring, we give the ring a finite core size o [9]. The velocity field at (r, Z) produced by a single vortex ring is then given by radial:

rg f"

Urr = 2--~- o

( z - Z o ) cos 0 dO x [(z - Zo)2 + R 2 + r 2 + oto2 _ 2Rr cos 0] 3/2' (7a) axial: FR f"0 Vzr=G; ( R - r cos 0) dO [(Z - Zo)2+ R 2 -t- r 2 +

O[O "2 - -

2Rr cos 0] 3 / 2 ' (Tb)

where R and Zo are the radius and position of the ring, respectively, F is the circulation strength, and o is the core radius, ot is a distribution factor chosen so that the self-induced velocity of the ring is the same as that of a ring with Gaussian distribution ofvorticity, o~ = 0.413. The core size is assumed to satisfy the relation

o2R = constant.

(8)

Using the vorticity equation at the orifice and the boundary layer assumptions, it can be shown that the rate of change of circulation at the orifice is given by

dF = ~1 Uz
(9)

where Uz(in) is the mean velocity through the orifice that was computed during the prechamber simulation (Eq. 4) and tit is the time interval. We assume that a vortex element of strength dP enters the chamber after every dt time interval. Thus, as

113

the velocity through the orifice increases (Fig. 6), the circulation strength of the individual vortex rings entering the chamber also increase. We continue to release these vortices into the main chamber throughout the simulation, modeling the unsteady evolution of the vorticity region. Using the change of variables method, the integrals in Eqs. 7a and 7b are recast in terms of elliptic integrals. The elliptic integrals are in turn evaluated using polynomial approximations [31]. The combined effect of the injected gas motion (modeled as potential flow) and all the vortex rings produces the velocity field that convects each vortex element. We assume the cylindrical side boundary to this main chamber is open, allowing gas to flow out of the domain. In our simulations the flame does not get close to these side walls, although the velocity normal to the wall is small because the total open area is large relative to the flame surface area. The results from the prechamber simulations showed that initially the flow through the orifice consists of only unburned gas. Thus, the vorticity region develops before the flame reaches the main chamber. In addition to tracking the orifice velocity from the prechamber calculation, we also record the time of arrival and the radius of the flame at the orifice, and use these results as boundary conditions in the main chamber simulations. Again, we use the zero-thickness flame model described for the prechamber model, adding the combustion-generated volume expansion to the model of the flow field in the main chamber. Although in the experiment the geometry of the main chamber is not axisymmetric (Fig. 1), the flame remains symmetric, as can be seen in Fig. 3. In the simulations of the main chamber we use a cylindrical geometry, matching the height from the orifice to the top wall (6.7 cm), the orifice size (2.54 cm), and the total volume (1,400 cm3). Because the volume of the main chamber is larger than that of the prechamber we use a finer grid, in this case 40 by 40 cells in the radial and axial directions, respectively. This grid size provides reasonable resolution of the flame shape in a moderate amount of time, taking typically 40 rain of CPU time on a Cray 1 for the computations presented in the next section.

114 RESULTS AND DISCUSSION

A series of frames in Fig. 8 shows both the experimentally recorded (right) and the numerically predicted (left and center) flame development for an ethylene-air equivalence ratio of • = 0.5 using the prechamber-driven orifice flow in Fig. 6. The time interval between frames is 4 ms, starting 42 ms after "ignition" when the flame has exited the orifice. In this figure we present the numerical results in two frames, separating the flame shape (center) and the vorticity field (left) at each time interval. The time from ignition is also indicated, both in real time, t, measured in milliseconds, and in the normalized time, r, used for the experimental results. Because in the numerical simulation we use a small time step the individual vortex-ring markers overlap, producing the solid wide curve shown in the first few frames. In addition to the position of the vortices, we also mark the boundary of the unburned gas that has been pushed into the main combustion chamber from the prechamber. In the simulation this set of marker points, originally positioned along the orifice opening, is advected by the total flow field. As can be seen from the series of plots, during the time period covered by the simulation the flame reaches, but does not pass, this boundary. Because this boundary marks the edge of the gas displaced from the prechamber, and not a change in gas properties, the flame will propagate past it as it continues to burn. In the simulation of the main chamber fluid mechanics shown in Fig. 8, by the time the flame reaches the orifice, the flow field in the main chamber caused by the prechamber flame propagation has developed a significant rotational motion, indicated by the roll-up of the vortex rings and motion of the ejected prechamber fluid. Both the experimental and numerical results show that the flame propagates through the center of the toroidal distribution of vorticity and is pulled around the core and spirals into its center. The combustion process continues to reduce the amount of unburned gas within the toroid, until it is consumed. In Fig. 8 the numerically predicted time development agrees well with the experimental results, indicating that the combination of the

R . J . CATTOLICA ET AL. discrete-vortex dynamics method and the flame interface model with a constant flame speed provides a good simulation of this flame-vortex interaction. The experimental results show that the flame remains smooth throughout the development; the resolution of the numerical solution causes some flame wrinkling when the flame spiral is less than two cells apart, as can be seen in the top frames. Other secondary differences between the flame shapes are obvious from the comparison. Several factors may be responsible for the differences in the details of the flame shape between the experimentally observed and the computationally predicted interaction of the flame with the starting jet from the prechamber. The main chamber geometry used in the numerical simulation is a right circular cylinder, and thus is simpler than that of the experiment shown in Fig. 1. Also, the flame is stretched as it interacts with flow regions of the starting jet, such as the stagnation point at the apex of the starting jet and the high velocity core of the concentrated vorticity. Although in the simulation we assume a constant flame speed, this stretching can locally alter the flame speed, and, thus, it can change the combustion-generated flow field. We obtained good agreement between the predicted results and the observed flame propagation for this lowest Reynolds number laminar flame condition, despite neglecting the effects of the local fluid flow on the flame speed in the numerical simulation. As the fuel-air equivalence ratio was increased in the experiment, the increased velocities and Reynolds numbers characterizing the flame propagation began to bring into question some of the idealized assumptions that we have used in the numerical simulation. In Fig. 9 we present the flame development at progressively higher Reynolds number conditions from both the experiment and the numerical simulation. Each frame in the figure was chosen to represent a comparable stage of flame propagation at each of the five equivalence ratios. The time from ignition is also presented, both in milliseconds and in normalized time. The numerical simulations presented in Fig. 9 scale well with the normalized time scale based on the burned gas flame speed and the characteristic

FLAME PROPAGATION IN DIVIDED-CHAMBER COMBUSTOR

115

70.0 ms (1.005)

-J l' '~'\2J" I

66.0 ms (0.947) / J

ff

.

k i

62.0 ms (0.890)

~ - \

(; "~l

58.0 ms (0.833)

if ' ~

54.0 ms (0.775)

50.0 ms (0.718)

46.0 ms (0.660)

42.0 ms (0.603)

a

b

c

Fig. 8. Flame development in main combustion chamber for • = 0.50. (a) vorticity distribution, (b) predicted flame shape, and (c) laser-schlieren visualization (not same scale).

116

R . J . CATTOLICA ET AI.

II

i ~¸ ~

:

o II

I

o II

s

. ~ii

"

A

o

Q

II

\ ~:)"

FLAME PROPAGATION IN DIVIDED-CHAMBER COMBUSTOR length of the prechamber. This similar time scaling is shown by the solid horizontal curve in Fig. 10, a plot of normalized times from Fig. 9 versus equivalence ratio, Fig. 10. The temporal similarity in the flame development remains valid over the entire range of equivalence ratios, with the flames reaching a similar stage of development at the same normalized time (r - 0.94). In the experimental results presented in Fig. 9, the flames reach comparable flame shapes at different normalized times, and these times decrease linearly with increasing equivalence ratio, as shown by the slanted solid curve in Fig. 10. The error bars correspond to the time spacing between individual video frames (0.5 ms). Note that the numerical and experimental results agree at the lowest equivalence ratio, indicating that the assumptions in the model are valid for this lowest Reynolds number condition. The linear change in the experimental times also appears in the other sets of normalized times from the experimental results in Fig. 3. These results are indicated by the dashed curves in Fig. 10. The discrepancy between the numerical and the experimental results at higher equivalence ratios may be due to the increased influence of flame stretch on the burned

0

I-

kx

.{.

r---r, xI.. . . .

Jr

~ | "'"

0 0

0.45 0 . 5 0

0 . 5 5 0 . 6 0 0 . 6 5 0.70

Fig. 10. Normalized time for similar ethylene-air flame development as a function of equivalence ratio. Solid line is data from Fig. 9 for both the numerical (horizontal) and experimental results. Dashed lines are the normalized time from the other experimental results shown in Fig. 8.

117

gas flame speed. This effect should increase in importance at higher equivalence ratios due to the resulting larger flow velocities. It is possible to include the effects of flame stretch in the flame sheet model empirically by varying the local flame speed with the local flow field. This effect has been parameterized recently by accurate measurements of flame speeds in some stretched hydrocarbon flames [32]. In addition to the discrepancy in the use of the normalized time scale, at higher Reynolds numbers Kelvin-Helmholtz instabilities development in the experimental results (Fig. 9). The instability and breakup of the shear layer that appear along the flame "stem" in the experimental results at the higher equivalence ratios (~ = 0.60 and 0.65) are not observed in the simulations, although precursors of the flame wrinkling and instabilities are observed in the clustering of the vortex-ring elements that are shed from the prechamber orifice (see Fig. 8 at r = 0.890 and 0.947). There are several aspects of the numerical simulation that can affect both the breakdown of the normalized scaling and the appearance of instabilities in the shear layer. First, in the simulation we assume a uniform velocity profile of prechamber gas flowing through the exit orifice. As seen in Fig. 5a of the prechamber simulation, the flame shape is parabolic as it propagates through the exit orifice. This implies that the profile of the gas velocity exiting the orifice may also be parabolic. If this is the case it would have the effect of increasing the gas velocity on the centerline of the starting jet, altering both the time scale for the subsequent interaction with the flame and the resulting flame shape. This effect is expected to become more significant at the higher velocities associated with the richer mixtures, explaining the relative acceleration of the flamevortex interaction observed experimentally. However, because the numerical predictions of times that the flame exited the precharnber were in excellent agreement with the experimental measurements, the assumption of a flat velocity profile through the orifice is not unreasonable. The second aspect of the model that may lead to the discrepancies is the method used to define the strength and core size of the vortex elements. The

118 vortex rings are shed from the prechamber exit orifice with a strength defined by the velocity through the orifice (see Eq. 9). The assumption of a uniform velocity profile directly affects the strength of the vortex elements. In the simulation the core size of the vortex tings shed from the exit orifice was held constant at o = 0.04 cm, which provided good agreement with the experimental results presented in Fig. 8. Because this core size represents a viscous length scale, it should depend on both the viscosity and the velocity through the orifice at the time each vortex is released, and it may increase with time. The results from other simulations obtained with this numerical model show that if the core size of the vortex elements is reduced, the clustering of these elements is enhanced and the instability observed in the experiment can be promoted in the simulation. This indicates that a more sophisticated model of the vortex elements is needed, both in terms of the definition of core size and strength. Velocity measurements of the gas exiting the orifice in the experiment would provide details on the spatial and temporal development of the starting jet flow that could be incorporated into more sophisticated definition of the vortex elements used to simulate the fluid mechanics. Also, vorticity was introduced in the simulation only in the shear layer emanating from the prechamber orifice. For higher velocity flame propagation, particularly with strong velocity gradients transverse to the density gradient across the flame front, combustion-generated vorticity has been shown to be important [33]. This would most strongly affect the higher Reynolds number cases. New modeling approaches that build on the vortex-dynamics approach are being developed with the inclusion of more sophisticated descriptions of the flame. A hybrid approach [34] using vortex-dynamics to model the fluid mechanics and a simple one-step kinetic mechanism to describe both the flame structure and the flame speed has been used to study the effect of nonunity Lewis number on flame propagation. Although this initial approach is two-dimensional and assumes constant density, this type of modeling has the potential for coupling the fluid mechanics and combustion at higher Reynolds number.

R . J . CATTOLICA ET AL. CONCLUSIONS In this article we have discussed the experimental and numerical study of ethylene-air flame propagation in a divided-chamber combustor. Using an axisymmetric computational model that combines the discrete-vortex dynamics technique with a flame sheet algorithm based on a constant flame speed, we predicted both the prechamber flame propagation rates and the temporal and spatial development of the unsteady flame in the main chamber. Agreement with the experimentally measured flame propagation rates at the lowest equivalence ratio was excellent, as indicated by comparison of high-speed laser-schlieren videography with the numerically predicted results. At higher equivalence ratios, which produced higher flow velocities, the temporal agreement between the experimental and computational results diverged. We believe that this is due to the variation of the flame speed caused by flame stretch, an effect not included in the model. KelvinHelmholtz instabilities were observed in the experimental results at the highest equivalence ratios. The fact that the numerical results did not predict these indicates that a more sophisticated vortexdynamics model is needed to capture the physical phenomena that occur over a range of conditions.

We would like to thank Jon Meeks for his assistance in conducting these experiments. His careful operation of the schlieren videography system and subsequent photography of the results is most appreciated. This research was sponsored by the U.S. Department of Energy Office of Basic Energy Sciences, Division of Chemical Sciences. REFERENCES 1. Grcar, J. F., Kee, R. J., Smooke, M., and Miller, J. A., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, in press. 2. Kee, R. J., and Miller, J. A., Sandia National Laboratories Report, SANDS1-8241, August 1981. 3. Amsden, A. A., Butler, T. D., O'Rourke, P. J., and Ramshaw, J. D., S A E Transactions, Paper #850554, p. 4.1, 1985. 4. Bray, K. N. C., and Moss, J. B., Acta Astronaut. 4:227 (1977).

FLAME PROPAGATION IN DIVIDED-CHAMBER COMBUSTOR 5. 6.

7. 8. 9. 10. 11. 12. 13. 14.

15.

16.

17. 18. 19. 20.

21. 22. 23. 24.

25. 26. 27. 28.

29.

30. 31.

Libby, P. A., and Bray, K. N. C., Combust. Flame 30:33 (1980). Ghoniem, A. F., Chorin, A. J., and Oppenhiem, A. K., Eighteenth Symposium (International) on Combustion, Combustion Institute, Pittsburgh, 1981, p. 1375. Ashurst, W. T., and Barr, P. K., Combust. Sci. Technol. 34:227 (1983). Chorin, A. J., J. Fluid Mech. 57:785 (1973). Leonard, A., Annu. Rev. Fluid Mech. 17:523 (1985). Ghoniem, A. F., Chin, A. J., and Oppenhiem, A. K., Philos. Trans. R. Soc. Lond. A304:303 (1982). Chert, D. Y., Ghoniem, A. F., and Oppenhiem, A. K., NASA CR 168139, 1983. Ghoniem, A. F., Chen, D. Y., and Oppenhiem, A. K., A I A A paper No. 84-0572, 1984. Ashurst, W. T., and Barr P. K., Sandia National Laboratories Report, SAND82-8224, September 1982. ~ , J. H. S., Knystautas, R., Chan, C., Barr, P. K., Grcar, J. F., and Ashurst, W. T., Sandia National Laboratories Report SAND83-8655, September 1983. Guenoche, H., in Nonsteady Flame Propagation (G. H. Markstein, Ed.), Pergamon, London, 1964, pp. 167176. Cattolica, R. J., and Vosen, S. R., Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1985, p. 1273. Westbrook, C. K., Dryer, F. L., and Schug, K. D., Combust. Flame 51:299 (1983). Cattolica, R. J., and Vosen, S. R., Combust. Sci. Technol. 48:77 (1986). Cattolica, R. J., and Vosen, S. R., Combust. Flame 68:267 (1987). Cattolica, R. J., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1987. Patnik, P. C., Sherman, F. S., and Coreos, G. M., J. Fluid Mech. 73:215 (1976). Corcos, G. M., and Sherman, F. S., J. Fluid Mech. 73:241 (1976). Barr, P. K., and Ashurst, W. T., Sandia National Laboratories Report SAND82-8773, 1984. Noh, W. F., and Woodward, P., in Lecture Notes in Physics (A. I. van de Vooren and P. J. Zandbergen, Eds.), Springer-Verlag, Berlin, 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). Landau, L. D., and Lifshitz, E. M., Fluid Mechanics, Pergamon, London, 1959, pp. 474-478. Markstein, G. H., in Nonsteady Flame Propagation (G. H. Markstein, Ed.), Pergamon, London, 1964, pp. 22-24. Dunn-Rankin, D., Barr, P. K., and Sawyer, R. F., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1987. Acton, E., J. FluidMech. 98:1 (1980), Abramowitz, M., and Stegun, I. A., Handbook o f Mathematical Functions, 9th printing, National Bureau

32.

33.

34.

119

of Standards, Applied Mathematics Series, 55, Washington D.C., 1970. Law, C. K., Zhu, D. L., and Yu, G., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1987. Pindera, M-Z., and Talbot, L., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1987. Ashurst, W. T., Peters, N., and Smooke, M. D., Numerical Simulation of Turbulent Flame Structure with Non-Unity Lewis Number, Combust. ScL TechnoL (in press).

Received 22 June 1987; revised 10 August 1988

APPENDIX: ETHYLENE-AIR FLAME SPEED MEASUREMENTS The ethylene-air flame speeds for the range of lean mixtures used in this study are not available in the combustion literature [1A]. Because the flame speed is a required parameter in the vortexdynamics model of the divided chamber flame propagation experiment, it was necessary to measure it directly in an ancillary experiment. The approach used to determine flame speed is based on schlieren observations of flame kernel growth [2A] in a combustion vessel. The combustion vessel used in the primary experiment was reconfigured without the prechamber. A pair of electrodes located in the lower third of the chamber provided a spark source to initiate a laminar flame that propagates through the vessel. The growth of the radius of the flame kernel was measured from sequential frames from a schlieren videography recording of the flame propagation taken at 2000 frames per second with 500 ns temporal resolution. A straight-line fit to a plot of the measured flame radius versus time yields the flame propagation or space velocity Vs in the laboratory frame of reference. To minimize energy deposition effects from the spark and curvature effects on the flame speed, the space velocity was determined from the growth of the flame kernel from 32 to 48 ram. For the 100-mmdiameter (2-liter volume) combustion vessel the pressure rise was less than 2 % above the original ambient pressure (1 atm) during this period in the development of the flame. If the burned gas is considered to be confined and flame propagation is considered to be at constant pressure, then the

120

R. J. CATTOLICA ET AL. TABLE A I

Ethylene-Air Space Velocity, Density Ratio, and Flame Speed from Flame Kernel Growth Measurements

¢

V, (cm/s)

p./pb

0.50 0.525 0.55 0.60 0.65 0.70

55. 67. 81. 123. 157. 228.

5.47 5.58 5.80 6.12 6.53 6.69

S. (cm/s) 10. 12. 14. 20.1 24. 34.

+ ± ± ± + ±

0.4 0.5 0.6 0.8 1.2 2.2

measured space velocity of the flame is related [3A] directly to the flame speed Su by the density ratio of the gas across the flame front as (la)

S,, = V s l ( p u l / ~ ) ,

where Pb is the hot, burned-gas density and p. is the cold, unburned-reactant density. Note that due to the symmetry of the spherical flame shape, the space velocity Vs that was experimentally measured during the initial development of the flame is identical to the burned gas flame speed Sb used in the article. In Table A1 the measured space velocity, density ratio, and calculated flames speeds are given for a range of ethylene-air equivalence ratios from 0.5 to 0.7. The density ratios were

calculated using the density from computed equilibrium flame conditions [4A] and ambient density at 293 K for each equivalence ratio. The precision of the flame-speed measurements in Table A1 was limited by the temporal resolution of the videography recording of the observed flame propagation for equivalence ratios above 0.55. For the leaner equivalence ratios the precision was determined by the repeatability of the experiment. For all the results listed in Table A1, welldeveloped laminar, hemispheric flames were initiated by the spark electrodes. The only anomalous behavior was a pronounced induction time associated with the leanest ethylene-air mixture (~ = 0.5). From the schlieren observations this induction time appears to be associated with the effect of the initial turbulence generated by the spark ~ discharge. Flame initiation and propagation does not begin until the turbulence level is sufficiently reduced. The induction time measured from repeated experiments was 15 ms and varied as much as 1 ms. The flame speed measurements from Table A1 are plotted along with available literature values and the results of a theoretical model [5A] in Fig. A1. At an equivalence ratio of 0.7 the results of the current work agrees within 10% of measurements from a double-flame kernel experiment [3A] and a burner study [6A]. At an equivalence ratio of (~ = 0.6) the flame speed from the ,

55-

45

O this work A Raezer and Olsen (1951)

40

n

Linnett and Hoare (1951)

35

$

Westbrook, et al (1983) ~:~

60 o(D

E

"o o 30 (D Q. ¢n 25 (D E 20 m '*- 15

$

1II III

10 5

=

0.45

0155

O165

O175

0185

equivalence ratio Fig. AI. Ethylene-air flame speed at 1 arm pressure and ambient temperature (293 K) as a function of equivalence ratio.

F L A M E P R O P A G A T I O N IN D I V I D E D - C H A M B E R COMBUSTOR theoretical flame structure model with a full kinetic mechanism is 20% higher than the measurexl flame speed.

REFERENCES 1A. Miller,S. A., Ethylene, Ernst Benn Limited, London, 1969.

121

2A. Fristrum, R. M., and Westenberg, A. A., Flame Structure, McGraw-Hill,New York, 1965. 3A. Raezer, S. D., and Olsen, H. L., Combust. Flame 6:227 (1962). 4A. Miller,J. A., Kee, R. J., and Jefferson, T. A., Sandia Report SAND80-8003, 1980. 5A. Westbrook, C. K., Dryer, F. L., and Schug, K. D., Combust. Flame 51:299 (1983). 6A. Linnett,J. W., and Hoare, M. F., Trans. Faraday Soc. 179 (1951).