In the first system studied, the spherical vessel has a radius of 1.8 m, the burning velocity and the expansion ratio have values of 17.5 m sec1 and 10, respectively, the unburned gas density has an initial value of 1.13 (kg m3) the local sound speed in the unburned gas is 354 m sec-1, the specific heat ratios for the unburned and burned gases are 1.4 and 1.2, respectively, and the initial radius of the burned gas volume is 0.028 m. For flame speed of 175.0 m sec1, or about 50 pct the sound speed, the flame travels the initial 0.028 m in 0.16 msec; this, therefore, constitutes the starting (initial) time for the numerical calculations. In this time (0.16 msec), the leading edge of the sonic disturbance ahead of the expanding flame created at the instant of ignition has traveled 0.57 m, or 31.6 pct of the radius of the sphere. As already mentioned, the flame acts like a porous spherical piston, leaking gas through the surface at a mass flux of pu Su and expanding initially at a constant velocity equal to the flame speed. Taylor has shown that weak shock waves first appear ahead of an expanding spherical piston when the surface radial velocity exceeds 0.5 Mach. Furthermore, to avoid the nonphysical acoustical oscillations at solution start referred to earlier, the expansion ratio was increased adiabatically from 1 to 10 according to the expression

Gas velocity histories calculated at distances of 0.0, 0.3, 0.6, 0.9, 1.2, 1.5, and 1.8 m (wall) from the center of the vessel are shown in figure 2. In the same figure the flame trajectory is indicated by a solid line starting at 0.028 m at 0.16 msec and arriving at the wall (1.8 m) at 15 msec. The burned gases are to the right of this flame line and the unburned gases to the left. Also shown in this figure are Mach lines for the leading edges of some of the more interesting features of the aerodynamic history. cases, the density dependence of the sound speeds is shown in parentheses on the Mach lines.

In all

The nomenclature used in labeling the various Mach lines is as follows: the letter C or R refers to compression or rarefaction, respectively. (or t) subscript indicates that this wave originated after one reflection (or transmission) from the flame of the parent wave, C, at the wall or center. Subsequent r's (or t's) indicate the ancestral history of the wave. example, to arrive at the wave Ctr, the parent wave C was first reflected at the wall to create C,, and a transmission through the flame then created Ct, which in reflection from the center created Cptr. In general, the more subscripts on a wave label the weaker the wave. The italic letters A, B, C, etc., identify coincident events from figure to figure, which arise from FSS (finite sound speed) theory only since the ISS (infinite sound speed) theory cannot generate wave phenomena. The event labeled A in figure 4 and figure 2 corresponds to the intersection of the flame front and the C, wave returning from the sphere wall.

The parent Mach line C in figure 2, obtained by connecting the leading edges of the initial disturbance ahead of the expanding flame, has exactly the sound speed (354 m sec1) of the unburned gas, as it should if the solution At 5.23 msec, C reflects from the vessel wall creating C.. Cr travels back toward the flame, intersecting it at 6.58 msec (point A) at which

were correct.

FIGURE 2. Calculated FSS velocity histories at 0, 0.3, 0.6, 0.9, 1.2, 1.5, and 1.8 m

from the center where (S,/C.) = 0.5.

.9

1.5

FLAME RADIUS, m

1.8

time a reflected rarefaction wave R., and a transmitted compression wave Crt are created. Up to this time (6.58 msec) the flame has been expanding at a speed (175 m sec) equal to that associated with an unconfined system. That is, the propagating flame has not yet realized that a sphere wall exists until the return of wave C..

The R travels at an average speed of 372 m sec-1 (owing to the increased unburned gas density associated with the compression wave behind C and C), reflects off the wall at 9.26 msec, and creates Rrrr, which travels back toward the flame at 380 m sec-1, intersects the flame at 9.65 msec, and creates the reflected wave C and the transmitted wave Rrrt (not labeled in the figures). The compression wave Crt moves through the burned gas back toward the sphere center at a velocity of 1,042 m sec1; it reflects at the center at 7.69 msec. Beyond 10 msec, the wave diagram becomes too complex to follow accurately.

Also seen in figure 2 is the initial exponential growth of the flame radius as a result of the artificial functional dependence of the expansion ratio on the flame radius according to equation 7. The figure also shows, particularly at the 0.3-m location, the growth of an oscillatory disturbance within the burned cavity, corresponding to a frequency near 1,000 Hz which is close to the full-wavelength resonance for the burned cavity. The solid line flame trajectory line in figure 2 is associated with the FSS model, whereas the dashed flame trajectory line is associated with the ISS model, as discussed later in the report.

Figure 3 shows a different perspective of the particle velocity history, which is plotted against the radial distance. Whereas figure 2 gave a temporal distribution of various radii, figure 3 shows the spatial distribution at various times. Such a plot better shows, particularly in the topmost curve, the air mass sloshing within the spherical vessel; otherwise this plot shows the same events as figure 2.

Figure 4 is a plot of flame speed as a function of time. In this plot we see the rapid initial flame acceleration, due to the use of equation 7, up to about 2 msec. The dashed flame velocity trace shows the unreal startup oscillations that result from mismatch in initial conditions across the flame front when S, does not increase gradually, as when using equation 7. Although these oscillations eventually dampen, they nevertheless propagate away from the origin and produce unreal wave phenomena (not shown in figure 4) at later times if equation 6 is used. The flame speed reaches a maximum (ES, = 175 m sec-1) where it levels off until event A. C, (figs. 2-3) being a compression wave heading into the flame front on the unburned side causes the particle velocity reversal and therefore the deceleration of the flame front velocity shown following event A. Subsequent to event A, the flame speed drops rapidly as C, passes over the flame until event B, at which time the flame speed attains a minimum value of about 35 m sec-1 The intersection of C, with the flame front generates two waves traveling away from the flame front, namely Crt and Rr. These waves then return to the flame at events labeled C and D, respectively, giving rise to four waves, and so it goes on ad infinitum.

FIGURE 3. Velocity profiles at different times for (S,/C.) = 0.5.

Events B, D, and H, at which times the flame suddenly changes the direction of acceleration, seemed puzzling at first. These events were clearly not associated with the primary wave C nor with any of its offspring, although they can be reasonably connected by a Mach line (not shown).

The disturbance that causes the velocity reversal at B could not have traveled through the burned gas mixture because figure 2 shows the burned gas between the flame front and Crt is quiescent. By erecting a Mach line, subsequently meeting the flame at J (3.6 msec), we find that neither figure 2 nor

1.8

1.6

1.4

1.2

1.0

6

50

FSS theory

FIGURE 4. Time histories of flame trajectory and flame speed for (S/C.) 0.5 using ISS and FSS theories,

FLAME RADIUS, m

3 shows anything significant

occurring at the flame at 3.6 msec. After some deliberation, it was decided that the reversal of flame velocity at B had its origin in the interaction of the disturbances following C, C., and Rr. Figure 2 shows at 8.4 msec, when wave M passes the 1.5-m station, a slight depression in the particle E velocity record caused by the superposition of the disturbances behind C and C, and amplified by Rrt since Rr moves the gas in the same direction as C, but opposite to that of C. When this depression or slower moving gas zone reaches the flame front, the flame moves forward faster because the unburned gas ahead of the flame is moving away slower, resulting in the flame acceleration from B to C shown in figure 4. A similar process we believe is repeated at events E and H. However, if this is the correct explanation for these three events, there is no reason why they should be connectable by a single Mach line; possibly it is coincidental that they can be so connected.

The se

Figure 4 shows that after 8 msec the flame front undergoes rapid oscillatory motion, frequently reversing its direction. phenomena are also apparent from the flame trajectory history shown in the same figure. These phenomena have frequently been observed in experiments with flames propagating in tubes. We will examine this oscillatory problem more closely when we consider the case of a lower flame speed, where the oscillatory motion is more readily examined.

Figure 5 shows the corresponding wave diagram plot for the static overpressure history; static overpressure is defined as the difference between the static pressure at any location and time and the initial pressure. This plot shows the oscillatory behavior of the pressure at the wall (1.8-m stations) in