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Flame Speeds in Combustible Vortex Rings
Flame Speeds in Combustible Vortex Rings SATORU ISHIZUKA,* TAKAHIRO MURAKAMI, TAKASHI HAMASAKI, KIMINORI KOUMURA, and RYO HASEGAWA
Department of Mechanical Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739, Japan Flame speeds in vortex rings of a stoichiometric methane/air mixture have been investigated. The flame speeds have been measured by a high-speed video camera, and the maximum tangential velocities in the vortex have been measured by a hot-wire anemometer. Results show that the flame velocity is approximately proportional to the maximum tangential velocity and its proportionality factor is about unity. By considering conservation of mass, linear momentum, and angular momentum across the flame, a simple theory has been proposed. A good agreement has been obtained between the theory and the experiment. The vortex bursting mechanism and the baroclinic torque mechanism have been discussed. © 1998 by The Combustion Institute
INTRODUCTION The problem of flame propagation along a vortex axis has attracted our keen interest, and many studies have been made in these decades [1–15]. McCormack [1, 2] was the first researcher to note this problem and he found with his coworkers that the flame speed in a vortex ring increased approximately linearly with the vortex strength up to about 15 m/s. However, the mechanism for the enhanced flame speed was not clear. In modeling turbulent combustion, Chomiak [3] has proposed a mechanism that the hydrodynamic pressure jump across the flame causes the rapid flame propagation along a vortex axis. This mechanism is now called the vortex bursting mechanism. Chomiak et al. [3, 4] have made a detailed explanation on this mechanism. Their theoretical results are summarized as
F
V f 5 V umax 2k 1
ru rb
G
1/2
(1)
in which V f is the flame velocity, V umax is the maximum tangential velocity in the Rankine’s combined vortex, ru and rb are the densities of the unburned and burned mixtures, respectively. The value of k 1 is unity when the vortex of infinite diameter is burned, whereas it is about one-third when the vortex within twice the diameter of the forced vortex core is burned and the mean pressure over the cross area is *Corresponding author. 0010-2180/98/$19.00 SSDI 0010-2180(97)00266-6
considered to work to drive the burned gas. In Chomiak’s original theory [3], the value of k 1 is one-half. In short, the flame speed is proportional to the maximum tangential velocity and also to the square root of the density ratio. This theoretical result is quite convincing and explains the experimental results well [1– 4]. However, an experiment using a vortex flow within a tube has shown that although the flame speed increases almost linearly with V umax, its slope is one-half or one-third the square root of the density ratio [7]. To elucidate the validity of the vortex bursting mechanism, the pressure jump across the flame has been measured in the vortex flow [12]. The results show that the pressure jump does exist and its magnitude is the order of ruV u2max, as predicted in the vortex bursting theory. But, the flame speed is much lower than the predicted value. In the vortex bursting mechanism, the pressure jump (. ruV u2max) is assumed to be converted into the kinematic energy of the burned gas (. rbV 2f /2), resulting in the expression of Eq. 1. Thus, if any part is wrong, it is this conversion part. Recently, assuming conservation of mass, linear momentum, and angular momentum across the flame, a simple theory has been proposed to account for the flame speed in a rotating tube. A successful agreement has been obtained between the theory and the experiment [15]. However, because of complex phenomena in a tube, its application is limited to the flame near the open end. COMBUSTION AND FLAME 113:542–553 (1998) © 1998 by The Combustion Institute Published by Elsevier Science Inc.
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Fig. 1. The schematic of the experimental apparatus.
In this study, an experiment has been carried out on a combustible vortex ring to obtain the relation between the flame speed V f and the maximum tangential velocity V umax. To discuss their relation rigorously, both the flame speed and the maximum tangential velocity were measured by a high-speed video camera and by a hot-wire anemometer, respectively. The simple analysis, originally developed for the propagating flame in a forced vortex flow within a rotating tube [15], has been extended to the flame in a vortex ring with assuming the tangential velocity distribution of the Rankine’s combined forced/free vortex. EXPERIMENTAL Figure 1 shows the schematic of the experimental apparatus. The vortex rings were formed by use of a mechanically driven piston at one end of a cylindrical cylinder with a circular orifice at the other end. The cylinder was 100 mm in diameter and four interchangeable plates with an orifice, the diameter (D o) of which were 30, 40, 50, and 60 mm, were used to generate vortex rings of different strength. The cylinder size and orifice sizes are smaller than those used in the experiments by McCormack et al. [2] or by Asato et al. [8], in which the cylinder diameters are 220 mm and the orifice diameters are 70 ; 110 mm. The piston was interconnected with a
pneumatically driven piston and a compressed air was used to drive this piston. The pressure P of the compressed air was varied from 0.3 to 0.8 MPa with 0.1 MPa stepwise to vary the vortex strength. The piston movement was fixed at 15 mm, since this gave vortex rings of good shape for the four orifice diameters and a wide range of the driving pressure. The fuel used was methane. Methane and air were supplied from batteries of capillary flowmeters, calibrated by a wet gas meter and a soap bubble meter, and a homogeneous mixture of methane and air was fed to the cylinder. Only a stoichiometric mixture was used in this experiment. A cylindrical hood of 100-mm diameter and 50-mm long was attached to the orifice plate and the same mixture was fed to the hood, otherwise dilution of the combustible mixture by entrainment of the ambient air into the vortex was unavoidable. This hood had a shutter to seal the mixture from the ambient air, and its movement was synchronized with an electromagnetic valve that allowed the pressure to be applied suddenly to the piston. However, the hood device is not perfect for dilution, because the Reynolds number defined by UD/n, where U and D are the translational velocity and diameter of the vortex ring, respectively and n is the kinematic viscosity of the mixture, becomes the order of 104 as later shown, and hence, the vortex rings produced in
544 this experiment are turbulent; i.e., the vortex ring still entrains the ambient air in the core instability stage, which begins at 5 ; 10D o downstream from the orifice and also in the similarity phase, which is achieved around 15 ; 70D o downstream from the orifice [16 –18]. Thus, the hood prevents only the entrainment of the ambient air into the vortex core at its initial stage. To suppress the influence of dilution as little as possible, a pair of rods were placed at a position 200-mm downstream from the orifice and the vortex rings were ignited by the corona discharge between them. This distance is enough for the vortex ring to reach almost constant ring diameter and almost constant translational velocity (according to the detailed measurements [8], 2D o is enough) and also this distance is the lower limit in which the vortex core is formed and the core instability almost starts for D o 5 60 mm or has started for D o 5 30 mm. Although it may be better if the vortex rings were ignited in their similarity phase because the rings reach some moderate state, preliminary experiments have shown that the dilution effects are so severe that they do not ignite the vortex rings. A Schlieren system was set up to take the Schlieren pictures of the flame. Two mirrors, 250 mm in diameter and 5 m focal length, were used. As a light source, a mercury vapor lump was used. The propagating flame was recorded by a digital high-speed color video camera (nac, MEMRECAMci). The maximum camera speed was 2000 pps (pictures per second). The shutter speed was 1/2000 s21 (open) and the recording time was 1.41 s at 2000 pps. The diameter of vortex ring D was measured from Schlieren pictures of vortex rings of methane/air mixtures, while the translational velocity U and the maximum tangential velocity V umax of the vortex ring were measured by a constanttemperature hot-wire anemometer (Kanomax, model-7004). The hot-wire probe was 5 mm in diameter and 2 mm in the sensing length. The output of the hot wire was calibrated by a standard nozzle, which had a uniform velocity distribution at the exit. The details of measurement were described with data presentation.
S. ISHIZUKA ET AL.
Fig. 2. The variations of the vortex ring diameter D with (a) the driving pressure P(D o 5 60 mm) and with (b) the orifice diameter D o(P 5 0.8 MPa).
RESULTS Characteristics of Vortex Ring The diameter of vortex ring D (cm), which was defined as the distance between the centers of the almost circular core, was measured for various driving pressures and orifice diameters. Figure 2a shows the variation of the ring diameter D with the driving pressure P for D o 5 40 mm. This measurement was made at a position 200 mm from the orifice, i.e., the ignition point. Figure 2a shows that the ring diameter is almost constant (5 cm) regardless of the driving pressure P. Very similar results have been obtained by Asato et al. [8]. Using other orifice plates, additional measurements were made. Figure 2b shows the variation of the ring diameter D with the orifice diameter D o for the condition of P 5 0.8 MPa. It is seen again that the ring diameter is almost constant regardless of the orifice di-
FLAME SPEEDS IN VORTEX RINGS ameters, although it slightly decreases for D o 5 40 mm. These results are somewhat different from the results in the experiment by Asato et al. [8], in which the ring diameter decreases from 100 to 75 mm as the orifice diameter is increased from 70 to 100 mm. Next, the translational velocity U (m/s) and the maximum tangential velocity V umax (m/s) were measured. As mentioned in the Introduction, the value of V umax is very important to discuss the flame speed. However, a propagating flame damages the hot-wire probe, and hence, the V f -V umax relation cannot be obtained directly in a burning vortex ring. However, together with the flame speed V f , the translational velocity U can be measured from their high-speed Schlieren pictures. Thus, we shall first examine the relation between U and V umax in the cold-air vortex ring to obtain the V f -V umax relation in the burning vortex ring. It is notable that the stoichiometric methane/air mixture used in this experiment contains only 9.5% methane and the rest is air. To obtain the U-V umax relation directly, two hot-wire probes were placed along a way where the ring passes and both U and V umax are measured at the same time for a traveling vortex. The translational velocity U was obtained simply by dividing the distance between the two probes (2.5 cm) by the time required for the ring to pass. For this sake, the outputs of the hot-wire probes were recorded by the transient memories (Autonics, APC-204, 4 channels, 64 Kwords/channel, minimum sampling time: 1 ms). To minimize interference with the ring, the hot-wire probes were inserted from down and set at a position where the bottom part of the ring may pass, and the maximum tangential velocity was measured by traversing a set of the two probes vertically or horizontally to find their maximum outputs. As shown in Fig. 3, there are two ways to set the hot-wire probe. One is to set the hot wire probe as its wire becomes perpendicular to the moving direction of the vortex ring (case I), and the other is to set it as its wire becomes parallel to that direction (case II). In the perpendicular case (case I), if the upper part of the forced vortex core passes the probe (denoted as case Ia), the output of the hot wire first monotonically increases and after
545
Fig. 3. Methods for measuring the maximum tangential velocity by hot-wire anemometry.
taking its maximum, decreases. But, if the center of the forced vortex core passes (denoted as case Ib), its output has two peaks. If the lower part of the forced vortex core passes the probe (denoted as case Ic), its output becomes again one peak, although its maximum becomes smaller than the maximum in case Ia. The peak values in case Ia, case Ib, and case Ic correspond to U 1 V umax, (U 2 1 V u2max)1/2, and V umax 2 U, respectively. It may be notable that the peaks in case Ib are smaller than the expected values corresponding to (U 2 1 V u2max)1/2, because velocity is changing rapidly in magnitude and in direction when the center of the vortex core passes, resulting in an awfully poor response of the hot wire. In case II, the output of the hot wire has two peaks when the center of the forced vortex core passes (case IIb), but a very moderate peak when the upper (case IIa) or the lower (case IIc) part of the forced vortex passes. Although the wire diameter is very small (5 mm), the sensing length is about 2 mm. As shown later, the core radius is about 10 mm. Thus, due to a lack of spatial dissolution, the peak values become smaller in case II than in case I.
546
S. ISHIZUKA ET AL. in which G is the vortex strength (circulation) and d is the diameter of the vortex core. If we assume a tangential velocity distribution of Rankine form (i.e., free/forced vortex), the maximum tangential velocity V umax is given as V umax 5
Fig. 4. The relation between the translational velocity U and the maximum tangential velocity V umax of the vortex ring.
From the above preliminary results, we have adopted the perpendicular and upper way, case Ia. The position of the probes are traversed vertically or horizontally to get their maximum for each experimental condition of P and D o. To take this procedure and owing to the experimental scatters, over 20 runs were made for each experimental condition. The value of V umax is obtained by subtracting the translational velocity U from the maximum output value, which may correspond to V umax 1 U. All measurements were made by varying the driving pressure P for each orifice diameter D o. The results show that the translational velocity U and the maximum tangential velocity V umax are increased with increasing P. Their relations are summarized in Fig. 4. It is seen that V umax is approximately proportional to U for each orifice diameter, although there are large scatters. As the orifice diameter D o becomes smaller, the range of U and V umax shifts larger. According to Lamb’s theory [19], the translational velocity U is given as U5
F
G 8D 1 ln 2 2pD d 4
G
(2)
D/d G 5 2U . pd ln ~8D/d! 2 0.25
(3)
That is, V umax is proportional to U as long as the ratio d/D is constant. However, this relation is for laminar vortex rings, while the vortex rings produced in this experiment are turbulent since the Reynolds number given as UD/ n is the order of 104 for most of D o and P [16 –18]. Thus, it is not appropriate to apply Eq. 3 to the present turbulent vortex rings. However, a linear relation can be seen in Fig. 4 within the experimental scatters. If we assume the validity of Eq. 3 with noting that d in this linear relationship is not the core diameter in the laminar vortex sense but some measure of core diameter of the turbulent vortex ring, a least square fitting gives the d/D ratio as 36% for D o 5 60 mm, 39% for D o 5 40 mm, and 48% for D o 5 30 mm, although the experimental results for D o 5 60 mm do not seem to pass the origin comparing with the broken line for d/D 5 36% in Fig. 4. Using a 10-in. inside diameter Plexiglas tube and a 4-in. diameter orifice, Johnson [20] measured the core diameter to find that the core/ ring diameter ratio was about 8.65%. Comparing this, the core sizes in the present rings are very large. However, in the experiment by Sullivan et al. [21], in which a vortex ring was generated by pulsating air through a sharpedged orifice using a loudspeaker, Laser Doppler Velocimeter (LDV) measurements show that there are two types of vortex rings, a relatively thick core ring, in which the core/ring diameter ratio is 27%, and a thin core ring, in which the core/ring diameter ratio is 7.5%. Therefore, it seems that turbulent vortex rings with a relatively thick core are formed in this experiment. Flame Velocities in a Vortex Ring Figure 5 shows a Schlieren sequence of the vortex ring combustion. Owing to the bulk of
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547
Fig. 5. Schlieren sequence of the vortex ring combustion (P 5 0.8 MPa, D o 5 40 mm, 2000 pps).
the vortex generator, the light beam from the first mirror passed through the vortex at an angle of about 30° to the normal to the plane of the ring. Hence, the vortex ring appears elliptical. Figure 5a is a Schlieren picture with only the ignition electrode and a rising hot gas, formed by the corona discharge, in the field of view. In this picture, the vortex ring comes from the right and goes to the left. Beside the electrode, two small rods were seen at the bottom, which were placed 2.5 cm apart on the straight centerline normal to the orifice plate to measure the translational ring velocity U. The combustible ring was ignited at the bottom (Fig. 5b), and after a short expansion period, two flame fronts start out from the region of ignition to propagate, in the opposite directions, one another along the vortex axis (Fig. 5c). After passing
about a quarter the ring diameter (Fig. 5d), the two fronts were approaching one another (Fig. 5e) and, finally, both flames coalesce on the far side of the ring (Fig. 5f). It is important to note that the ring diameter D is almost constant during combustion. From these figures, two kinds of flame speed were calculated. One is the mean flame speed from the starting of propagation (Fig. 5c) to the meeting of the two flame fronts (Fig. 5f), and the other is the mean flame speed during when the flames travel a latter quarter part of the vortex ring, i.e., from Fig. 5d to Fig. 5f. They are termed as V f,mean and V f,1/4 , respectively. Since the corona discharge may enhance the flame speed and, in addition, since it is not guaranteed that the flame speed is constant during propagation, the measurements of Vf,mean and
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S. ISHIZUKA ET AL.
Fig. 7. The relation between the flame speed V f and the maximum tangential velocity V umax.
Fig. 6. The variation of the flame speed V f with (a) the driving pressure P(D o 5 40 mm) and with (b) the orifice diameter D o(P 5 0.8 MPa).
Vf,1/4 are useful to know the flame behavior accurately. Figures 6a and b, respectively, show the variations of the flame speeds with the driving pressure P for D o 5 40 mm and those with the orifice diameter D o at P 5 0.8 MPa. With an increase in the pressure or with a decrease in the orifice diameter, the flame speeds increase. This indicates that V f increases with an increase in V umax, since V umax increases with increasing P or decreasing D o. Although the value of V f,mean are slightly larger than those of V f,1/4., they are almost the same within the experimental scatters. Thus, it seems that the influence of the corona discharge is negligible and the flame speed is almost constant during the propagation. Finally, the relation between the flame speed
V f and the maximum tangential velocity V umax was obtained. Measurements were made at various vortex strengths obtained by varying the driving pressure P and the orifice diameter D o. The maximum tangential velocity V umax was calculated by putting the measured value of U into Eq. 3, with assuming the d/D ratio as 36, 39, and 48% for D o 5 60, 40, and 30 mm, respectively. The results were shown in Fig. 7. In this figure, the relations predicted by Eq. 1 also were shown by the solid lines. As in the previous experiments [7, 8, 11, 15], the flame speeds are much less than the predicted values. In Fig. 7, there appears a broken line and a broken curve, which are well fitted with the experimental results. This broken line and curve are obtained by considering conservation of mass, linear momentum, and angular momentum across the flame, discussed in the following section. DISCUSSION Chomiak [3] tried to explain the phenomenon of rapid flame propagation along the vortex axis
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549
on the basis of the vortex breakdown concept. Recently, the phenomenon of vortex breakdown itself has been reproduced faithfully by numerical simulations and its essential feature is understood as the generation of negative azimuthal vorticity [22, 23]. Therefore, a detailed numerical simulation would reproduce the bursting flame and its mechanisms could be understood well. Here, however, following the previous model developed for the flame in a rigid-body rotation [15], a very simple model of integral nature has been developed in the sense that the local changes of neither density nor pressure within the flame zone are cared for. Figure 8 schematically shows the present model. We take the axis of rotation as the z-axis. As shown in Fig. 5, the diameter D of the vortex ring remains unchanged before and after combustion, and the flame propagation is symmetric. Thus, the ends of the combustible tube, one of which corresponds to the ignition point and the other of which corresponds to the meeting point of the two flame fronts, can be thought to be fixed by the solid walls, the distance between which is pD/2. In the model by Daneshyar and Hill [4], the axial velocity is assumed to be zero. However, we admit the existence of axial flow, which may be induced around the flame as in the spherically expanding flame in a quiescent mixture. We assume that the unburned gas and the burned gas flow at the velocities V u and V b from left to right, respectively, and to avoid confusion, we also assume that the flame moves from left to right at the velocity V f . Also, we denote the pressures in the unburned and burned gases as P u(r) and P b(r), respectively, in which the pressures are given as a function of the radial distance r. In the model by Daneshyar and Hill [4], the unburned mixture is assumed to expand only in the radial direction and there is no axial expansion. We admit, however, the axial expansion as well as the radial expansion. As for the axial expansion, it is taken into consideration in the relative velocity change from V u 2 V f to V b 2 V f. As for the radial expansion, we assume that the hu-sized unburned gas is burned to be the hb-sized burned gas. By considering the mass continuity across the flame, the following relation is obtained:
r u~V u 2 V f! ph 2u 5 r b~V b 2 V f! ph 2b
(4)
Next, we consider the momentum flux in the z-direction. Since the flame runs faster on the axis of rotation, we concentrate our attention on the axis of rotation. By considering the momentum flux before and after combustion, it can be assumed that the momentum flux before combustion should be equal to that after combustion, i.e., the unburned gas and the burned gas have the same momentum flux on the axis of rotation. Then, the following relation is obtained: P u~0! 1 r uV 2u 5 P b~0! 1 r bV 2b
(5)
As for rotation, we assume also that the angular momentum remains unchanged before and after combustion. Furthermore, we take the tangential velocity distribution of Rankine form and denote the rotational speed and the radius of the forced vortex core as Vu and hu/2 for the unburned, respectively, and Vb and hb/2 for the burned gases, respectively. That is, in this model, we consider the burning of the vortex tube whose diameter is twice the diameter the forced vortex core. This size may be reasonable, because the core/ring diameter ratio is 36 – 48% in the present vortex ring. Thus the following relation is obtained: 7p 7p r ~V 2 V f!V uh 4u 5 r ~V 2 V f!V bh 4b 32 u u 32 b b (6) In the vortex bursting mechanism, the pressure ahead of the flame P u(0) and that behind the flame P b(0) are given. In the Rankine’s combined vortex, they are given as follows; P u~0! 5 P ` 2 14 r uV 2uh 2u
(7a)
P b~0! 5 P ` 2 14 r bV 2bh 2b
(7b)
in which P ` is the pressure at infinity and the unburned gas for r . hb in the burned gas side is treated as it has the same density rb as the burned gas. Using the relations Vu 2 Vf 5 Su
(8)
and V umax 5 12 V uhu,
(9)
in which S u is the burning velocity of the combustible mixture, a quadratic equation is obtained for V f , i.e.,
550
S. ISHIZUKA ET AL.
Fig. 8. A model for the propagating flame in a combustible vortex ring.
~ r u 2 r b!V 2f 1 2 r uS u
S
D S
S D
h 2u r uh 4u z 1 2 2 V f 1 r uS 2u 1 2 hb r bh 4b 2 r uV u2max 1 2
uV fu 5
D
r bh 2u 50 r uh2b
(10)
S
r uS u 1 2 Vf 5 2 6
r u2 r b
F
uV fu 5 S u 1 V umax 1 1
D S
1 r uh2u r ur bS 2u 1 2 r u2 r b r bh2b
S
1 r u(ru 2 r b)V u2max 1 2
D
D
2
r bh2u r uh2b
DG
1/2
(11)
in which the solution corresponding to the sign (1) in the (6) sign should be abandoned and the flame is considered to propagate from the right to the left in Fig. 8 at velocity uV f u from the physical consideration. In an extreme case of hu 5 hb, i.e., the diameter of the combustible mixture remains unchanged and it expands only in the axial direction, the flame velocity becomes
1/2
.
(12)
In another extreme case of ruh2u 5 rbh2b, i.e., the mixture expands only in the radial direction, the flame velocity becomes
S
Solutions can be obtained easily as
h 2u h 2b
S
ru 2 S 1 V u2max rb u
rb ru
D
1/2
.
(13)
In Fig. 7, Eq. 12 is shown by a broken curve and Eq. 13 is shown by a broken line. A good agreement has been obtained between the theory and the experiment. However, what is curious in Fig. 7 is that although the expansion ways are completely different, the results, Eq. 12 and Eq. 13, are almost the same; i.e., the flame speed V f is almost proportional to the maximum tangential velocity V umax, and its proportionality factor is about unity. This curiosity may be understood by knowing the fact that the pressure difference is given as Pb 2 Pu
5
5
S D S D rb ru r2b 12 2 ru
ruVu2max 1 2
for hu 5 hb,
(14a)
ruVu2max
for ruh2u 5 rbh2b,
(14b)
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551
and hence, they coincide with each other within the order of rb/ru, i.e., 15% for the stoichiometric methane/air mixture. However, it seems still curious in Fig. 7 that not only Eq. 13 but also Eq. 12 coincides with the experimental results. In Eq. 12, the unburned gas must expand in the axial direction. This expansion ratio reaches about seven for the stoichiometric mixture. However, the experimental observation reveals that the ring diameter (the total length of the vortex tube) remains unchanged during combustion. This contradiction can be solved by considering a two-step process that in the first step the hu-sized unburned gas is burned and expands axially to be the hu-sized burned gas, and in the second step, this hu-sized burned gas is compressed axially and expands radially at the same time to be the hb-sized burned gas. This process can be seen in the Schlieren sequence in Fig. 5. Anyway, what is most surprising in Fig. 7 is a good agreement between the theory and the experiment. This agreement becomes much better if we remember to use a rough fitting for the U 2 V umax relation. For example, in Fig. 4, the values of V umax are underestimated in the range 2 , V umax , 4 for D o 5 60 mm. If we shift the corresponding data point to the right to correct the underestimation in Fig. 7, they enter in the area between Eqs. 12 and 13. The present model should be criticized as very crude because the model is one-dimensional and the axisymmetric flame shape and the momentum flux aside the axis of rotation are not taken into consideration. But, this model includes most essential parts of the propagation flame, hence, resulting in a good agreement between the theory and the experiment. Finally, we briefly discuss the relevant study to this problem. Recently, Hasegawa and Nishikado [14] have studied numerically the effect of density ratio on the flame propagation along a vortex tube and concluded that (1) the baroclinic torque expressed as dV 1 5 2 ¹ r 3 ¹p dt r
(15)
in which V is the radial speed, t is a time, ¹r is a density gradient, and ¹p is a pressure gradient, produces vorticity in the flame zone and
this vorticity provokes the flame to accelerate along the vortex with a velocity similar to the observed propagation velocity, and (2) the theory on the basis of the pressure difference across the flame does not explain the inverse dependency of the propagation velocity on the density ratio, the dependency on the 1.6 power of the circumferential velocity or the dependency on the diameter of the vortex tube. The conclusion (2) is derived on the basis of Eq. 1. If the linear momentum conservation is taken into consideration, the pressure difference theory gives the results of Eqs. 11–13. These results may not contradict their numerical results. The reasons are as follows. First, as shown in Eq. 13, the flame speed is decreased as the density ratio ru/rb is increased. Comparing Eq. 1, this is reversed completely. Second, the flame speed at small tangential velocities is given from Eq. 12 as, uV fu . S u
SD S ru rb
1/2
11
rb V2 1 2 r uS 2u umax
D
(16)
while the value of uV f u asymptotes to V umax as V umax goes to infinity. That is, the power dependency varies from 2 to 1. The solution, which is realized, may be situated between Eqs. 12 and 13, and hence, it has somewhat the nature of the inverse dependency of the density ratio and also the 1 ; 2 power dependency on the maximum tangential velocity. Third, in the previous theory on the flame propagation in a forced vortex flow within a rotating tube (hence, hu 5 hb is assumed) [15], the flame speed is given as uV fu 5
S
k2 2 ru ~YS u! 2 1 V rb 2 umax
D
1/2
.
(17)
in which Y is the flame area increase factor and k 2 is the flame diameter factor, both of which may be considered as unity for simplicity in the present case. It should be noted that in the previous theory, the burning is limited within the forced vortex core, i.e., 0 , r # hu/2, resulting in a denominator of 2 in the V umax term of Eq. 17. On the other hand, if burning is expanded out to twice the diameter of the forced vortex core, the denominator becomes unity as in Eq. 12 for hu 5 hb. That is, the flame speed increases as the diameter of the burning vortex is increased. Therefore, the present the-
552 oretical results do not seem to contradict the numerical results obtained by Hasegawa and Nishikado [14]. However, an issue is left. In the numerical simulation by Hasegawa and Nishikado [14], a vortex tube of very small diameter, the order of which is the laminar flame thickness (0.17 mm), is considered, whereas the vortex diameter in this experiment is very large (the core diameter is approximately 25 mm). Thus, the Reynolds number, which is defined by V umaxd/ n in Ref. 14, is below 100 in their simulation, whereas it becomes the order of 105 in this experiment, i.e., there is an essential difference between them. This issue is related with Chomiak’s theory [3], because a vortex tube of Kolmogorov scale is considered. Since the present theory assumes the establishment of laminar flame front as indicated by Eqs. 4 and 8, there is a conceptual difference between the combustion in laminar small-scale vortices and the combustion in turbulent vortex rings that have a large diameter core. Therefore, the high propagation velocity (Eq. 1) proposed on the basis of the vortex breakdown concept may not be denied by the results obtained in turbulent vortex rings. If the flame size in the turbulent vortex ring becomes small, their speeds seem to decrease. As seen in Fig. 7, the flame speed is fallen down and below the predicted curve or line for V umax . 10 m/s. Such a saturation in flame speed can be seen in the previous vortex flow experiment [7] and also in the experiment by Asato et al. [8]. In fact, a careful observation has indicated that the shape of the leading edge of the propagating flame becomes slender when the flame speed is fallen down. In the experiment by Asato et al. [8], the flame speeds are below the predicted values. For example, the flame speed is 7 m/s for V umax 5 20 m/s, although it should be approximately 20 m/s according to Eqs. 12 or 13. In the paper by Asato et al. [8], the much lower flame speed compared with Chomiak’s prediction [3] (Eq. 1) is attributed to the smallness of the ratio of the flame diameter to the core diameter, because the pressure difference across the flame is decreased as the ratio is decreased. However, the reason why their flame speeds become still lower than those predicted by the present theory (Eq. 11–13) may be explained also on the basis of flame size. In the experiment by Asato
S. ISHIZUKA ET AL. et al. [8], the Reynolds number, which is defined by UD/ n, is large. For example, the typical ring diameter and translational velocity are 10 cm and 7.5 m/s in Ref. 8, respectively, and hence, the Reynolds number becomes 4.7 3 104. This value is larger than that for the present vortex rings, which is about 1.25 3 104 for D 5 5 cm and U 5 4 m/s. Thus, it seems that a turbulent vortex with a thin core is established in Ref. 8 [16 –18, 21]. In fact, the core diameter is assumed around 10% the ring diameter [2, 8] and in Johnson’s experiment the core/ring diameter ratio is about 8.65% [20]. This may, together with the effect of large tangential velocity, result in a small flame diameter and therefore in lower flame speed. In this experiment, the core is large and the propagating flame is blunt at its nose. For small flames in turbulent vortex rings the present theory (Eqs. 11–13) should be modified. However, for this modification, detailed observations on the flame shape are indispensable. CONCLUSIONS Flame speeds in combustible vortex rings of a stoichiometric methane/air mixture have been investigated. Flame velocities have been measured by a high-speed video camera, and the maximum tangential velocities in the vortex have been measured by a hot-wire anemometer. Results show that over the range of the maximum tangential velocity available the flame velocity is approximately proportional to the maximum tangential velocity and its proportionality factor is about unity. By considering conservation of mass, linear momentum, and angular momentum across the flame, a simple theory has been proposed. The result is given by Eq. 11, which is simplified as Eq. 12 for the axial expansion case and as Eq. 13 for the lateral expansion case, although their difference is small. A good agreement has been obtained between the theory and the experiment. We would like to express our sincere thanks to Professor Hiroyuki Hiroyasu, the head of the Center for Technology Research and Development (CTRD) of Hiroshima University, for stimulating discussion and for use of the high-speed color video camera at CTRD. We would like to express
FLAME SPEEDS IN VORTEX RINGS our sincere thanks also to one of the reviewers who drew our attention to the work by Asato et al. [8] and also pointed out the difference between combustion in laminar small-scale vortices and that in turbulent vortex rings. This research is supported in part by the Yazaki Memorial Foundation for Science and Technology. We are grateful to Professor Shiro Taki for helpful discussions.
553 10.
11. 12. 13. 14.
REFERENCES 1. 2. 3.
4. 5. 6. 7. 8.
9.
McCormack, P. D., Proceedings of the Royal Irish Academy 71, Section A, 6:73– 83 (1971). McCormack, P. D., Scheller, K., Muller, G., and Tisher, R., Combust. Flame 19:297 (1972). Chomiak, J., Sixteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1977, p. 1665. Daneshyar, R. J., and Hill, P. G., Prog. Energy Combust. Sci. 13:47 (1987). Sivashinsky, G. I., Rakib, Z., Matalon, M., and Sohrab, S. H., Combust. Sci. Technol. 57:37 (1988). Sheu, W. J., Sohrab, S. H., and Sivashinsky, G. I., Combust. Flame 79:190 (1990). Ishizuka, S., Combust. Flame 82:176 (1990). Asato, K., Takeuchi, Y., and Kawamura, T., Proceedings of the Eleventh Australasian Fluid Mechanics Conference, University of Tasmania, Hobart, Australia, 14 –18 December, 1992, p. 167. Ishizuka, S., and Hirano, T., Prog. Astro. Aero. 151:284 (1993).
15.
16. 17. 18. 19. 20.
21. 22. 23.
Asato, K., Takeuchi, Y., and Wada, H., Proceedings of the Russian-Japanese Seminar on Combustion, The Russian Section of The Combustion Institute, Moscow, 1993, p. 73. Atobiloye, R. J., and Britler, R. E., Combust. Flame 98:220 (1994). Ishizuka, S., and Hirano, T., Nensho-no-Kagaku-toGijutsu 2:15 (1994). Hasegawa, K., Nishikado, K., and Chomiak, J., Combust. Sci. Tech., 108:67 (1995). Hasegawa, K., and Nishikado, K., Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsuburgh, 1997, p. 291. Sakai, Y. and Ishizuka, S., Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1997, p. 847. Maxworthy, T., J. Fluid Mech. 51:15 (1972). Maxworthy, T., J. Fluid Mech. 64:227 (1974). Maxworthy, T., J. Fluid Mech. 81:465 (1977). Lamb, H., Hydrodynamics, 6th ed., Dover, London, 1945, p. 241. Johnson, G. M., Research on the Propagation and Decay of Vortex Rings, ARL TR 70-0093, Aerospace Research Lab., Wright-Patterson Air Force Base, Ohio, 1970. Sullivan, J. P., Windnall, S. E., and Ezekiel, S., AIAA Journal 11:1384 (1973). Lopez, J. M., J. Fluid Mech. 221:533 (1990). Brown, G. L., and Lopez, J. M., J. Fluid Mech. 221:553 (1990).
Received 23 April 1997; accepted 15 June 1997
Department of Mechanical Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739, Japan Flame speeds in vortex rings of a stoichiometric methane/air mixture have been investigated. The flame speeds have been measured by a high-speed video camera, and the maximum tangential velocities in the vortex have been measured by a hot-wire anemometer. Results show that the flame velocity is approximately proportional to the maximum tangential velocity and its proportionality factor is about unity. By considering conservation of mass, linear momentum, and angular momentum across the flame, a simple theory has been proposed. A good agreement has been obtained between the theory and the experiment. The vortex bursting mechanism and the baroclinic torque mechanism have been discussed. © 1998 by The Combustion Institute
INTRODUCTION The problem of flame propagation along a vortex axis has attracted our keen interest, and many studies have been made in these decades [1–15]. McCormack [1, 2] was the first researcher to note this problem and he found with his coworkers that the flame speed in a vortex ring increased approximately linearly with the vortex strength up to about 15 m/s. However, the mechanism for the enhanced flame speed was not clear. In modeling turbulent combustion, Chomiak [3] has proposed a mechanism that the hydrodynamic pressure jump across the flame causes the rapid flame propagation along a vortex axis. This mechanism is now called the vortex bursting mechanism. Chomiak et al. [3, 4] have made a detailed explanation on this mechanism. Their theoretical results are summarized as
F
V f 5 V umax 2k 1
ru rb
G
1/2
(1)
in which V f is the flame velocity, V umax is the maximum tangential velocity in the Rankine’s combined vortex, ru and rb are the densities of the unburned and burned mixtures, respectively. The value of k 1 is unity when the vortex of infinite diameter is burned, whereas it is about one-third when the vortex within twice the diameter of the forced vortex core is burned and the mean pressure over the cross area is *Corresponding author. 0010-2180/98/$19.00 SSDI 0010-2180(97)00266-6
considered to work to drive the burned gas. In Chomiak’s original theory [3], the value of k 1 is one-half. In short, the flame speed is proportional to the maximum tangential velocity and also to the square root of the density ratio. This theoretical result is quite convincing and explains the experimental results well [1– 4]. However, an experiment using a vortex flow within a tube has shown that although the flame speed increases almost linearly with V umax, its slope is one-half or one-third the square root of the density ratio [7]. To elucidate the validity of the vortex bursting mechanism, the pressure jump across the flame has been measured in the vortex flow [12]. The results show that the pressure jump does exist and its magnitude is the order of ruV u2max, as predicted in the vortex bursting theory. But, the flame speed is much lower than the predicted value. In the vortex bursting mechanism, the pressure jump (. ruV u2max) is assumed to be converted into the kinematic energy of the burned gas (. rbV 2f /2), resulting in the expression of Eq. 1. Thus, if any part is wrong, it is this conversion part. Recently, assuming conservation of mass, linear momentum, and angular momentum across the flame, a simple theory has been proposed to account for the flame speed in a rotating tube. A successful agreement has been obtained between the theory and the experiment [15]. However, because of complex phenomena in a tube, its application is limited to the flame near the open end. COMBUSTION AND FLAME 113:542–553 (1998) © 1998 by The Combustion Institute Published by Elsevier Science Inc.
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Fig. 1. The schematic of the experimental apparatus.
In this study, an experiment has been carried out on a combustible vortex ring to obtain the relation between the flame speed V f and the maximum tangential velocity V umax. To discuss their relation rigorously, both the flame speed and the maximum tangential velocity were measured by a high-speed video camera and by a hot-wire anemometer, respectively. The simple analysis, originally developed for the propagating flame in a forced vortex flow within a rotating tube [15], has been extended to the flame in a vortex ring with assuming the tangential velocity distribution of the Rankine’s combined forced/free vortex. EXPERIMENTAL Figure 1 shows the schematic of the experimental apparatus. The vortex rings were formed by use of a mechanically driven piston at one end of a cylindrical cylinder with a circular orifice at the other end. The cylinder was 100 mm in diameter and four interchangeable plates with an orifice, the diameter (D o) of which were 30, 40, 50, and 60 mm, were used to generate vortex rings of different strength. The cylinder size and orifice sizes are smaller than those used in the experiments by McCormack et al. [2] or by Asato et al. [8], in which the cylinder diameters are 220 mm and the orifice diameters are 70 ; 110 mm. The piston was interconnected with a
pneumatically driven piston and a compressed air was used to drive this piston. The pressure P of the compressed air was varied from 0.3 to 0.8 MPa with 0.1 MPa stepwise to vary the vortex strength. The piston movement was fixed at 15 mm, since this gave vortex rings of good shape for the four orifice diameters and a wide range of the driving pressure. The fuel used was methane. Methane and air were supplied from batteries of capillary flowmeters, calibrated by a wet gas meter and a soap bubble meter, and a homogeneous mixture of methane and air was fed to the cylinder. Only a stoichiometric mixture was used in this experiment. A cylindrical hood of 100-mm diameter and 50-mm long was attached to the orifice plate and the same mixture was fed to the hood, otherwise dilution of the combustible mixture by entrainment of the ambient air into the vortex was unavoidable. This hood had a shutter to seal the mixture from the ambient air, and its movement was synchronized with an electromagnetic valve that allowed the pressure to be applied suddenly to the piston. However, the hood device is not perfect for dilution, because the Reynolds number defined by UD/n, where U and D are the translational velocity and diameter of the vortex ring, respectively and n is the kinematic viscosity of the mixture, becomes the order of 104 as later shown, and hence, the vortex rings produced in
544 this experiment are turbulent; i.e., the vortex ring still entrains the ambient air in the core instability stage, which begins at 5 ; 10D o downstream from the orifice and also in the similarity phase, which is achieved around 15 ; 70D o downstream from the orifice [16 –18]. Thus, the hood prevents only the entrainment of the ambient air into the vortex core at its initial stage. To suppress the influence of dilution as little as possible, a pair of rods were placed at a position 200-mm downstream from the orifice and the vortex rings were ignited by the corona discharge between them. This distance is enough for the vortex ring to reach almost constant ring diameter and almost constant translational velocity (according to the detailed measurements [8], 2D o is enough) and also this distance is the lower limit in which the vortex core is formed and the core instability almost starts for D o 5 60 mm or has started for D o 5 30 mm. Although it may be better if the vortex rings were ignited in their similarity phase because the rings reach some moderate state, preliminary experiments have shown that the dilution effects are so severe that they do not ignite the vortex rings. A Schlieren system was set up to take the Schlieren pictures of the flame. Two mirrors, 250 mm in diameter and 5 m focal length, were used. As a light source, a mercury vapor lump was used. The propagating flame was recorded by a digital high-speed color video camera (nac, MEMRECAMci). The maximum camera speed was 2000 pps (pictures per second). The shutter speed was 1/2000 s21 (open) and the recording time was 1.41 s at 2000 pps. The diameter of vortex ring D was measured from Schlieren pictures of vortex rings of methane/air mixtures, while the translational velocity U and the maximum tangential velocity V umax of the vortex ring were measured by a constanttemperature hot-wire anemometer (Kanomax, model-7004). The hot-wire probe was 5 mm in diameter and 2 mm in the sensing length. The output of the hot wire was calibrated by a standard nozzle, which had a uniform velocity distribution at the exit. The details of measurement were described with data presentation.
S. ISHIZUKA ET AL.
Fig. 2. The variations of the vortex ring diameter D with (a) the driving pressure P(D o 5 60 mm) and with (b) the orifice diameter D o(P 5 0.8 MPa).
RESULTS Characteristics of Vortex Ring The diameter of vortex ring D (cm), which was defined as the distance between the centers of the almost circular core, was measured for various driving pressures and orifice diameters. Figure 2a shows the variation of the ring diameter D with the driving pressure P for D o 5 40 mm. This measurement was made at a position 200 mm from the orifice, i.e., the ignition point. Figure 2a shows that the ring diameter is almost constant (5 cm) regardless of the driving pressure P. Very similar results have been obtained by Asato et al. [8]. Using other orifice plates, additional measurements were made. Figure 2b shows the variation of the ring diameter D with the orifice diameter D o for the condition of P 5 0.8 MPa. It is seen again that the ring diameter is almost constant regardless of the orifice di-
FLAME SPEEDS IN VORTEX RINGS ameters, although it slightly decreases for D o 5 40 mm. These results are somewhat different from the results in the experiment by Asato et al. [8], in which the ring diameter decreases from 100 to 75 mm as the orifice diameter is increased from 70 to 100 mm. Next, the translational velocity U (m/s) and the maximum tangential velocity V umax (m/s) were measured. As mentioned in the Introduction, the value of V umax is very important to discuss the flame speed. However, a propagating flame damages the hot-wire probe, and hence, the V f -V umax relation cannot be obtained directly in a burning vortex ring. However, together with the flame speed V f , the translational velocity U can be measured from their high-speed Schlieren pictures. Thus, we shall first examine the relation between U and V umax in the cold-air vortex ring to obtain the V f -V umax relation in the burning vortex ring. It is notable that the stoichiometric methane/air mixture used in this experiment contains only 9.5% methane and the rest is air. To obtain the U-V umax relation directly, two hot-wire probes were placed along a way where the ring passes and both U and V umax are measured at the same time for a traveling vortex. The translational velocity U was obtained simply by dividing the distance between the two probes (2.5 cm) by the time required for the ring to pass. For this sake, the outputs of the hot-wire probes were recorded by the transient memories (Autonics, APC-204, 4 channels, 64 Kwords/channel, minimum sampling time: 1 ms). To minimize interference with the ring, the hot-wire probes were inserted from down and set at a position where the bottom part of the ring may pass, and the maximum tangential velocity was measured by traversing a set of the two probes vertically or horizontally to find their maximum outputs. As shown in Fig. 3, there are two ways to set the hot-wire probe. One is to set the hot wire probe as its wire becomes perpendicular to the moving direction of the vortex ring (case I), and the other is to set it as its wire becomes parallel to that direction (case II). In the perpendicular case (case I), if the upper part of the forced vortex core passes the probe (denoted as case Ia), the output of the hot wire first monotonically increases and after
545
Fig. 3. Methods for measuring the maximum tangential velocity by hot-wire anemometry.
taking its maximum, decreases. But, if the center of the forced vortex core passes (denoted as case Ib), its output has two peaks. If the lower part of the forced vortex core passes the probe (denoted as case Ic), its output becomes again one peak, although its maximum becomes smaller than the maximum in case Ia. The peak values in case Ia, case Ib, and case Ic correspond to U 1 V umax, (U 2 1 V u2max)1/2, and V umax 2 U, respectively. It may be notable that the peaks in case Ib are smaller than the expected values corresponding to (U 2 1 V u2max)1/2, because velocity is changing rapidly in magnitude and in direction when the center of the vortex core passes, resulting in an awfully poor response of the hot wire. In case II, the output of the hot wire has two peaks when the center of the forced vortex core passes (case IIb), but a very moderate peak when the upper (case IIa) or the lower (case IIc) part of the forced vortex passes. Although the wire diameter is very small (5 mm), the sensing length is about 2 mm. As shown later, the core radius is about 10 mm. Thus, due to a lack of spatial dissolution, the peak values become smaller in case II than in case I.
546
S. ISHIZUKA ET AL. in which G is the vortex strength (circulation) and d is the diameter of the vortex core. If we assume a tangential velocity distribution of Rankine form (i.e., free/forced vortex), the maximum tangential velocity V umax is given as V umax 5
Fig. 4. The relation between the translational velocity U and the maximum tangential velocity V umax of the vortex ring.
From the above preliminary results, we have adopted the perpendicular and upper way, case Ia. The position of the probes are traversed vertically or horizontally to get their maximum for each experimental condition of P and D o. To take this procedure and owing to the experimental scatters, over 20 runs were made for each experimental condition. The value of V umax is obtained by subtracting the translational velocity U from the maximum output value, which may correspond to V umax 1 U. All measurements were made by varying the driving pressure P for each orifice diameter D o. The results show that the translational velocity U and the maximum tangential velocity V umax are increased with increasing P. Their relations are summarized in Fig. 4. It is seen that V umax is approximately proportional to U for each orifice diameter, although there are large scatters. As the orifice diameter D o becomes smaller, the range of U and V umax shifts larger. According to Lamb’s theory [19], the translational velocity U is given as U5
F
G 8D 1 ln 2 2pD d 4
G
(2)
D/d G 5 2U . pd ln ~8D/d! 2 0.25
(3)
That is, V umax is proportional to U as long as the ratio d/D is constant. However, this relation is for laminar vortex rings, while the vortex rings produced in this experiment are turbulent since the Reynolds number given as UD/ n is the order of 104 for most of D o and P [16 –18]. Thus, it is not appropriate to apply Eq. 3 to the present turbulent vortex rings. However, a linear relation can be seen in Fig. 4 within the experimental scatters. If we assume the validity of Eq. 3 with noting that d in this linear relationship is not the core diameter in the laminar vortex sense but some measure of core diameter of the turbulent vortex ring, a least square fitting gives the d/D ratio as 36% for D o 5 60 mm, 39% for D o 5 40 mm, and 48% for D o 5 30 mm, although the experimental results for D o 5 60 mm do not seem to pass the origin comparing with the broken line for d/D 5 36% in Fig. 4. Using a 10-in. inside diameter Plexiglas tube and a 4-in. diameter orifice, Johnson [20] measured the core diameter to find that the core/ ring diameter ratio was about 8.65%. Comparing this, the core sizes in the present rings are very large. However, in the experiment by Sullivan et al. [21], in which a vortex ring was generated by pulsating air through a sharpedged orifice using a loudspeaker, Laser Doppler Velocimeter (LDV) measurements show that there are two types of vortex rings, a relatively thick core ring, in which the core/ring diameter ratio is 27%, and a thin core ring, in which the core/ring diameter ratio is 7.5%. Therefore, it seems that turbulent vortex rings with a relatively thick core are formed in this experiment. Flame Velocities in a Vortex Ring Figure 5 shows a Schlieren sequence of the vortex ring combustion. Owing to the bulk of
FLAME SPEEDS IN VORTEX RINGS
547
Fig. 5. Schlieren sequence of the vortex ring combustion (P 5 0.8 MPa, D o 5 40 mm, 2000 pps).
the vortex generator, the light beam from the first mirror passed through the vortex at an angle of about 30° to the normal to the plane of the ring. Hence, the vortex ring appears elliptical. Figure 5a is a Schlieren picture with only the ignition electrode and a rising hot gas, formed by the corona discharge, in the field of view. In this picture, the vortex ring comes from the right and goes to the left. Beside the electrode, two small rods were seen at the bottom, which were placed 2.5 cm apart on the straight centerline normal to the orifice plate to measure the translational ring velocity U. The combustible ring was ignited at the bottom (Fig. 5b), and after a short expansion period, two flame fronts start out from the region of ignition to propagate, in the opposite directions, one another along the vortex axis (Fig. 5c). After passing
about a quarter the ring diameter (Fig. 5d), the two fronts were approaching one another (Fig. 5e) and, finally, both flames coalesce on the far side of the ring (Fig. 5f). It is important to note that the ring diameter D is almost constant during combustion. From these figures, two kinds of flame speed were calculated. One is the mean flame speed from the starting of propagation (Fig. 5c) to the meeting of the two flame fronts (Fig. 5f), and the other is the mean flame speed during when the flames travel a latter quarter part of the vortex ring, i.e., from Fig. 5d to Fig. 5f. They are termed as V f,mean and V f,1/4 , respectively. Since the corona discharge may enhance the flame speed and, in addition, since it is not guaranteed that the flame speed is constant during propagation, the measurements of Vf,mean and
548
S. ISHIZUKA ET AL.
Fig. 7. The relation between the flame speed V f and the maximum tangential velocity V umax.
Fig. 6. The variation of the flame speed V f with (a) the driving pressure P(D o 5 40 mm) and with (b) the orifice diameter D o(P 5 0.8 MPa).
Vf,1/4 are useful to know the flame behavior accurately. Figures 6a and b, respectively, show the variations of the flame speeds with the driving pressure P for D o 5 40 mm and those with the orifice diameter D o at P 5 0.8 MPa. With an increase in the pressure or with a decrease in the orifice diameter, the flame speeds increase. This indicates that V f increases with an increase in V umax, since V umax increases with increasing P or decreasing D o. Although the value of V f,mean are slightly larger than those of V f,1/4., they are almost the same within the experimental scatters. Thus, it seems that the influence of the corona discharge is negligible and the flame speed is almost constant during the propagation. Finally, the relation between the flame speed
V f and the maximum tangential velocity V umax was obtained. Measurements were made at various vortex strengths obtained by varying the driving pressure P and the orifice diameter D o. The maximum tangential velocity V umax was calculated by putting the measured value of U into Eq. 3, with assuming the d/D ratio as 36, 39, and 48% for D o 5 60, 40, and 30 mm, respectively. The results were shown in Fig. 7. In this figure, the relations predicted by Eq. 1 also were shown by the solid lines. As in the previous experiments [7, 8, 11, 15], the flame speeds are much less than the predicted values. In Fig. 7, there appears a broken line and a broken curve, which are well fitted with the experimental results. This broken line and curve are obtained by considering conservation of mass, linear momentum, and angular momentum across the flame, discussed in the following section. DISCUSSION Chomiak [3] tried to explain the phenomenon of rapid flame propagation along the vortex axis
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549
on the basis of the vortex breakdown concept. Recently, the phenomenon of vortex breakdown itself has been reproduced faithfully by numerical simulations and its essential feature is understood as the generation of negative azimuthal vorticity [22, 23]. Therefore, a detailed numerical simulation would reproduce the bursting flame and its mechanisms could be understood well. Here, however, following the previous model developed for the flame in a rigid-body rotation [15], a very simple model of integral nature has been developed in the sense that the local changes of neither density nor pressure within the flame zone are cared for. Figure 8 schematically shows the present model. We take the axis of rotation as the z-axis. As shown in Fig. 5, the diameter D of the vortex ring remains unchanged before and after combustion, and the flame propagation is symmetric. Thus, the ends of the combustible tube, one of which corresponds to the ignition point and the other of which corresponds to the meeting point of the two flame fronts, can be thought to be fixed by the solid walls, the distance between which is pD/2. In the model by Daneshyar and Hill [4], the axial velocity is assumed to be zero. However, we admit the existence of axial flow, which may be induced around the flame as in the spherically expanding flame in a quiescent mixture. We assume that the unburned gas and the burned gas flow at the velocities V u and V b from left to right, respectively, and to avoid confusion, we also assume that the flame moves from left to right at the velocity V f . Also, we denote the pressures in the unburned and burned gases as P u(r) and P b(r), respectively, in which the pressures are given as a function of the radial distance r. In the model by Daneshyar and Hill [4], the unburned mixture is assumed to expand only in the radial direction and there is no axial expansion. We admit, however, the axial expansion as well as the radial expansion. As for the axial expansion, it is taken into consideration in the relative velocity change from V u 2 V f to V b 2 V f. As for the radial expansion, we assume that the hu-sized unburned gas is burned to be the hb-sized burned gas. By considering the mass continuity across the flame, the following relation is obtained:
r u~V u 2 V f! ph 2u 5 r b~V b 2 V f! ph 2b
(4)
Next, we consider the momentum flux in the z-direction. Since the flame runs faster on the axis of rotation, we concentrate our attention on the axis of rotation. By considering the momentum flux before and after combustion, it can be assumed that the momentum flux before combustion should be equal to that after combustion, i.e., the unburned gas and the burned gas have the same momentum flux on the axis of rotation. Then, the following relation is obtained: P u~0! 1 r uV 2u 5 P b~0! 1 r bV 2b
(5)
As for rotation, we assume also that the angular momentum remains unchanged before and after combustion. Furthermore, we take the tangential velocity distribution of Rankine form and denote the rotational speed and the radius of the forced vortex core as Vu and hu/2 for the unburned, respectively, and Vb and hb/2 for the burned gases, respectively. That is, in this model, we consider the burning of the vortex tube whose diameter is twice the diameter the forced vortex core. This size may be reasonable, because the core/ring diameter ratio is 36 – 48% in the present vortex ring. Thus the following relation is obtained: 7p 7p r ~V 2 V f!V uh 4u 5 r ~V 2 V f!V bh 4b 32 u u 32 b b (6) In the vortex bursting mechanism, the pressure ahead of the flame P u(0) and that behind the flame P b(0) are given. In the Rankine’s combined vortex, they are given as follows; P u~0! 5 P ` 2 14 r uV 2uh 2u
(7a)
P b~0! 5 P ` 2 14 r bV 2bh 2b
(7b)
in which P ` is the pressure at infinity and the unburned gas for r . hb in the burned gas side is treated as it has the same density rb as the burned gas. Using the relations Vu 2 Vf 5 Su
(8)
and V umax 5 12 V uhu,
(9)
in which S u is the burning velocity of the combustible mixture, a quadratic equation is obtained for V f , i.e.,
550
S. ISHIZUKA ET AL.
Fig. 8. A model for the propagating flame in a combustible vortex ring.
~ r u 2 r b!V 2f 1 2 r uS u
S
D S
S D
h 2u r uh 4u z 1 2 2 V f 1 r uS 2u 1 2 hb r bh 4b 2 r uV u2max 1 2
uV fu 5
D
r bh 2u 50 r uh2b
(10)
S
r uS u 1 2 Vf 5 2 6
r u2 r b
F
uV fu 5 S u 1 V umax 1 1
D S
1 r uh2u r ur bS 2u 1 2 r u2 r b r bh2b
S
1 r u(ru 2 r b)V u2max 1 2
D
D
2
r bh2u r uh2b
DG
1/2
(11)
in which the solution corresponding to the sign (1) in the (6) sign should be abandoned and the flame is considered to propagate from the right to the left in Fig. 8 at velocity uV f u from the physical consideration. In an extreme case of hu 5 hb, i.e., the diameter of the combustible mixture remains unchanged and it expands only in the axial direction, the flame velocity becomes
1/2
.
(12)
In another extreme case of ruh2u 5 rbh2b, i.e., the mixture expands only in the radial direction, the flame velocity becomes
S
Solutions can be obtained easily as
h 2u h 2b
S
ru 2 S 1 V u2max rb u
rb ru
D
1/2
.
(13)
In Fig. 7, Eq. 12 is shown by a broken curve and Eq. 13 is shown by a broken line. A good agreement has been obtained between the theory and the experiment. However, what is curious in Fig. 7 is that although the expansion ways are completely different, the results, Eq. 12 and Eq. 13, are almost the same; i.e., the flame speed V f is almost proportional to the maximum tangential velocity V umax, and its proportionality factor is about unity. This curiosity may be understood by knowing the fact that the pressure difference is given as Pb 2 Pu
5
5
S D S D rb ru r2b 12 2 ru
ruVu2max 1 2
for hu 5 hb,
(14a)
ruVu2max
for ruh2u 5 rbh2b,
(14b)
FLAME SPEEDS IN VORTEX RINGS
551
and hence, they coincide with each other within the order of rb/ru, i.e., 15% for the stoichiometric methane/air mixture. However, it seems still curious in Fig. 7 that not only Eq. 13 but also Eq. 12 coincides with the experimental results. In Eq. 12, the unburned gas must expand in the axial direction. This expansion ratio reaches about seven for the stoichiometric mixture. However, the experimental observation reveals that the ring diameter (the total length of the vortex tube) remains unchanged during combustion. This contradiction can be solved by considering a two-step process that in the first step the hu-sized unburned gas is burned and expands axially to be the hu-sized burned gas, and in the second step, this hu-sized burned gas is compressed axially and expands radially at the same time to be the hb-sized burned gas. This process can be seen in the Schlieren sequence in Fig. 5. Anyway, what is most surprising in Fig. 7 is a good agreement between the theory and the experiment. This agreement becomes much better if we remember to use a rough fitting for the U 2 V umax relation. For example, in Fig. 4, the values of V umax are underestimated in the range 2 , V umax , 4 for D o 5 60 mm. If we shift the corresponding data point to the right to correct the underestimation in Fig. 7, they enter in the area between Eqs. 12 and 13. The present model should be criticized as very crude because the model is one-dimensional and the axisymmetric flame shape and the momentum flux aside the axis of rotation are not taken into consideration. But, this model includes most essential parts of the propagation flame, hence, resulting in a good agreement between the theory and the experiment. Finally, we briefly discuss the relevant study to this problem. Recently, Hasegawa and Nishikado [14] have studied numerically the effect of density ratio on the flame propagation along a vortex tube and concluded that (1) the baroclinic torque expressed as dV 1 5 2 ¹ r 3 ¹p dt r
(15)
in which V is the radial speed, t is a time, ¹r is a density gradient, and ¹p is a pressure gradient, produces vorticity in the flame zone and
this vorticity provokes the flame to accelerate along the vortex with a velocity similar to the observed propagation velocity, and (2) the theory on the basis of the pressure difference across the flame does not explain the inverse dependency of the propagation velocity on the density ratio, the dependency on the 1.6 power of the circumferential velocity or the dependency on the diameter of the vortex tube. The conclusion (2) is derived on the basis of Eq. 1. If the linear momentum conservation is taken into consideration, the pressure difference theory gives the results of Eqs. 11–13. These results may not contradict their numerical results. The reasons are as follows. First, as shown in Eq. 13, the flame speed is decreased as the density ratio ru/rb is increased. Comparing Eq. 1, this is reversed completely. Second, the flame speed at small tangential velocities is given from Eq. 12 as, uV fu . S u
SD S ru rb
1/2
11
rb V2 1 2 r uS 2u umax
D
(16)
while the value of uV f u asymptotes to V umax as V umax goes to infinity. That is, the power dependency varies from 2 to 1. The solution, which is realized, may be situated between Eqs. 12 and 13, and hence, it has somewhat the nature of the inverse dependency of the density ratio and also the 1 ; 2 power dependency on the maximum tangential velocity. Third, in the previous theory on the flame propagation in a forced vortex flow within a rotating tube (hence, hu 5 hb is assumed) [15], the flame speed is given as uV fu 5
S
k2 2 ru ~YS u! 2 1 V rb 2 umax
D
1/2
.
(17)
in which Y is the flame area increase factor and k 2 is the flame diameter factor, both of which may be considered as unity for simplicity in the present case. It should be noted that in the previous theory, the burning is limited within the forced vortex core, i.e., 0 , r # hu/2, resulting in a denominator of 2 in the V umax term of Eq. 17. On the other hand, if burning is expanded out to twice the diameter of the forced vortex core, the denominator becomes unity as in Eq. 12 for hu 5 hb. That is, the flame speed increases as the diameter of the burning vortex is increased. Therefore, the present the-
552 oretical results do not seem to contradict the numerical results obtained by Hasegawa and Nishikado [14]. However, an issue is left. In the numerical simulation by Hasegawa and Nishikado [14], a vortex tube of very small diameter, the order of which is the laminar flame thickness (0.17 mm), is considered, whereas the vortex diameter in this experiment is very large (the core diameter is approximately 25 mm). Thus, the Reynolds number, which is defined by V umaxd/ n in Ref. 14, is below 100 in their simulation, whereas it becomes the order of 105 in this experiment, i.e., there is an essential difference between them. This issue is related with Chomiak’s theory [3], because a vortex tube of Kolmogorov scale is considered. Since the present theory assumes the establishment of laminar flame front as indicated by Eqs. 4 and 8, there is a conceptual difference between the combustion in laminar small-scale vortices and the combustion in turbulent vortex rings that have a large diameter core. Therefore, the high propagation velocity (Eq. 1) proposed on the basis of the vortex breakdown concept may not be denied by the results obtained in turbulent vortex rings. If the flame size in the turbulent vortex ring becomes small, their speeds seem to decrease. As seen in Fig. 7, the flame speed is fallen down and below the predicted curve or line for V umax . 10 m/s. Such a saturation in flame speed can be seen in the previous vortex flow experiment [7] and also in the experiment by Asato et al. [8]. In fact, a careful observation has indicated that the shape of the leading edge of the propagating flame becomes slender when the flame speed is fallen down. In the experiment by Asato et al. [8], the flame speeds are below the predicted values. For example, the flame speed is 7 m/s for V umax 5 20 m/s, although it should be approximately 20 m/s according to Eqs. 12 or 13. In the paper by Asato et al. [8], the much lower flame speed compared with Chomiak’s prediction [3] (Eq. 1) is attributed to the smallness of the ratio of the flame diameter to the core diameter, because the pressure difference across the flame is decreased as the ratio is decreased. However, the reason why their flame speeds become still lower than those predicted by the present theory (Eq. 11–13) may be explained also on the basis of flame size. In the experiment by Asato
S. ISHIZUKA ET AL. et al. [8], the Reynolds number, which is defined by UD/ n, is large. For example, the typical ring diameter and translational velocity are 10 cm and 7.5 m/s in Ref. 8, respectively, and hence, the Reynolds number becomes 4.7 3 104. This value is larger than that for the present vortex rings, which is about 1.25 3 104 for D 5 5 cm and U 5 4 m/s. Thus, it seems that a turbulent vortex with a thin core is established in Ref. 8 [16 –18, 21]. In fact, the core diameter is assumed around 10% the ring diameter [2, 8] and in Johnson’s experiment the core/ring diameter ratio is about 8.65% [20]. This may, together with the effect of large tangential velocity, result in a small flame diameter and therefore in lower flame speed. In this experiment, the core is large and the propagating flame is blunt at its nose. For small flames in turbulent vortex rings the present theory (Eqs. 11–13) should be modified. However, for this modification, detailed observations on the flame shape are indispensable. CONCLUSIONS Flame speeds in combustible vortex rings of a stoichiometric methane/air mixture have been investigated. Flame velocities have been measured by a high-speed video camera, and the maximum tangential velocities in the vortex have been measured by a hot-wire anemometer. Results show that over the range of the maximum tangential velocity available the flame velocity is approximately proportional to the maximum tangential velocity and its proportionality factor is about unity. By considering conservation of mass, linear momentum, and angular momentum across the flame, a simple theory has been proposed. The result is given by Eq. 11, which is simplified as Eq. 12 for the axial expansion case and as Eq. 13 for the lateral expansion case, although their difference is small. A good agreement has been obtained between the theory and the experiment. We would like to express our sincere thanks to Professor Hiroyuki Hiroyasu, the head of the Center for Technology Research and Development (CTRD) of Hiroshima University, for stimulating discussion and for use of the high-speed color video camera at CTRD. We would like to express
FLAME SPEEDS IN VORTEX RINGS our sincere thanks also to one of the reviewers who drew our attention to the work by Asato et al. [8] and also pointed out the difference between combustion in laminar small-scale vortices and that in turbulent vortex rings. This research is supported in part by the Yazaki Memorial Foundation for Science and Technology. We are grateful to Professor Shiro Taki for helpful discussions.
553 10.
11. 12. 13. 14.
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Received 23 April 1997; accepted 15 June 1997