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Limiting acoustic oscillations in T-burners
Journal of Sound and Vibration (1974) 33(3), 305-317
LIMITING ACOUSTIC OSCILLATIONS IN T-BURNERS S. TEMKIN
Department of ~lechanical, Industrial and Aerospace Enghteerhlg, Rtttgers University, New Brunswick, New Jersey 08903, U.S.A. (Received 25 June 1973, and in revisedform 23 October 1973) A simple model is used to analyze finite-amplitude standing modes in T-burners. Mean flow effects are neglected in the model and it is assumed that a limiting condition results owing to non-linear energy transfer from the more unstable fundamental mode of the burner cavity to its harmonic. Results for the limiting pressure of the fundamental mode and its harmonic are found in terms of the growth constant, and of the damping constant of the second mode. This quantity is calculated on the assumption that the second mode losses are mainly caused by wall effects, and by radiation from the vent. It is shown that, in general, the vent does not result in large amounts of energy losses, except when a resonant field is excited inside the vent. In that condition, the amplitude of the first mode can be quite large. The theoretical prediction for the limiting amplitudes is compared with available experimental data, and it is found that while the experimental data display a resonant-like behavior about the theoretically predicted condition, the width of the resonant peak is much broader than that predicted theoretically. 1. INTRODUCTION T-burners are devices which are used to test the stability of solid propellants for rocket propulsion [1]. Basically, a T-burner is a cylindrical cavity closed at both ends by rigid caps and vented at its middle section as shown schematically in Figure I. The vent is connected to a pressurized tank of large dimensions. Samples of propellants to be tested are placed at one or both ends of the cavity, and are ignited by external means. I f the propellant is unstable, acoustic waves will be excited in the burner with frequencies corresponding to the acoustic modes 'of the cavity. For some geometries, radial and tangential modes are excited, but in most studies attention is focused on the fundamental longitudinal mode of the cavity. In order to determine how unstable a given propellant is at a frequency to, the time history ofthe acoustic pressure of the fundamental mode in a burner of length L = nao/to is recorded. (Here ao is the ambient speed of sound in the burner.) The initial growth and eventual decay of this pressure record are exponential, and yield data that can be used to compute the propellant's response function at that frequency. Although the method is based on the assumption that losses in the burner during the growth and decay stages are the same, its simplicity has proved very useful in determining the Ptopellcmt
Figure 1. Schematic diagram of a T-burner. 305
306
s. TEMKIN
intrinsic instability of solid propellants. However, as presently used, the procedure ignores one of the main characteristics of the pressure oscillations, namely, the limiting state they reach during burning. This limiting state is clearly important because it affords another measure of the propellant's instability, and because it shows the importance of non-linear mechanisms. Furthermore, since elimination of acoustic instability has not yet been accomplished, the most important questions that arise in practice relate precisely to the existence of this limiting state and to the values of the oscillatory pressure amplitude in this condition. Recently, several studies have appeared in the literature which consider the limiting amplitude state in T-burners. These studies include works by Weiner [2], Temkin [3], Culick [4, 5] and Perry [6]. Culiek's works are the most general from a theoretical point of view, since they include coupling effects between the burning propellant and the gas in the cavity. However, his results are difficult to interpret as they depend on several parameters which must be evaluated separately. Also, his results are based on the assumption that the limiting factor is the non-linearity of the damping mechanisms. This assumption may, of course, be correct, but there exist other non-linear effects which ought to be considered. Weiner [2] and Temkin [3] have studied the limiting condition using simple models for the acoustic field in the burner cavity. Thus, Temkin assumed that, in the limiting condition, the amplitudes of the waves would be so large that shock waves would be present in the cavity. However, in most tests in T-burners no shock waves are observed. Weiner assumed that in the limiting condition the wave profile is a sawtooth with superpose d small-amplitude even modes. That is, the profile he assumed had relative amplitudes P~, 1"2, Pd3, P4, P~/5, etc., for the first few modes in the cavity. The amplitudes of the first and second modes were found by computing the work done b y t h e harmonics on the fundamental. The results were given in terms of easily measured quantities, but the procedure used in the derivation and the severe assumption made about the profile have limited their usefulness. In this paper the limiting-amplitude condition in a T-burner is described by a model based on the assumption that this condition results from non-linear transfer of energy from the fundamental mode in the burner cavity to its harmonic. The amplitude of this harmonic mode is first calculated from an approximation to the equations of motion for the gas in the cavity. The limiting amplitude of the first mode is then found by means of a simple energy balance in terms of the growth constant of the fundamental mode and of the damping coefficient of the second longitudinal mode of the cavity. This damping coefficient is then calculated for a centrally vented burner, and the theoretical prediction for the limiting amplitude is compared with the experimental results of Perry. 2. ENERGY CONSIDERATIONS Consider the fundamental longitudinal mode in a cylindrical cavity of length L filled with a perfect gas. If purely longitudinal oscillations are assumed, the acoustic pressure, p~, and velocity, ul, at time t are given by
p~(x, t) = P~(t) sin (cot) cos (kx), u~(x, t) =
Pl(t) po ao
cos (cot) sin (kx),
(1) (2)
where Po is the mean gas density, co is the circular frequericy of the fundamental mode, and k = 7t/L is the wave number of that mode. The amplitude P~ is typically a slowly varying function of time. Initially, P~ grows exponentially with time. Its growth then slows gradually, and eventually a limiting state is reached. It is of interest to find the limiting value, P~r., of Pt. In some cases this can be done by simple energy-balance arguments. That is, the acoustic
OSCILLATIONS IN T-BURNERS
307
energy of the fundamental mode increases at a rate which is determined by the energy addition and removal from the oscillatory motion of the gas in the cavity. The limiting state, if it exists, is clearly obtained when these two energy rates are equal. The problem, therefore, reduces to calculating these quantities separately. Now, the instantaneous energy of the fundamental mode, per unit cross-sectional area of the cavity, is [7, see p. 250] L
L
Ex(t)=~po(u2dx+21~fp~dx. a
Substitution ofp~ and
(3)
~'poJ
0
0
ux from equations (1) and (2) yields L
Ex(t)=--~----poP~(t).
(4)
Now, since Px initially grows in an exponential manner, the acoustic energy of the first mode also grows exponentially. Therefore, the rate at which energy is transferred from the burning propellant to the fundamental mode is ~?,.,o = 2~(L/4Wo),%
(5)
where ~ is, approximately, the fractional energy added per cycle. Equation (5) is, in principle, valid only for small values of Px. However, since the energy transferred from the propellant to the acoustic waves is a small fraction of the total energy released by the combustion, the rate of energy addition to the waves would not be expected to change significantly owing to their growth. One may therefore provisionally assume that equation (5) is applicable even when the amplitude of the waves is not infinitesimal. That is, of course, an assumption which may be invalid during the growth o f the oscillations. The energy losses associated with the first mode are due to linear and non-linear mechanisms. The linear type consists mainly of viscous and thermal losses at the walls of the burner, and leads to an energy-loss rate which is also proportional to the instantaneous energy: i.e., El,out =
2fl~(L[4rpo)P?,
(6)
where fl~ is the linear damping coefficient for the first mode. If there are no other dissipative mechanisms, the energy of the first mode would change at a rate L',.,,, = 2~,(L]4wo)/2,
(7)
where ctg = e - fix. Clearly, ifcto > 0, the amplitude of the first mode will increase continuously until the assumed linear model ceases to be valid. What may happen, physically, is that if the fundamental mode reaches a sufficiently large amplitude, it will start to transfer energy to its harmonic. Some of this non-linearly transferred energy will then be dissipated by linear mechanisms associated with the second mode. If this energy loss is not sufficient to limit the growth of the fundamental mode, more modes will appear, and ifthe process is continued indefinitely, a weak shock wave will appear. This weak shock wave will result in a rate of energy dissipation proportional to Pax (as opposed to p2 produced by linear mechanisms), and will therefore result in a limiting condition [3]. However, in most T-burner tests, shock waves are not observed. This indicates that in those instances the limiting effects arise because of energy transfer to the first few harmonics. Furthermore, since a T-burner has one or more vents at its middle section, and this is a pressure maximum for the second mode, this mode may lose a great deal of energy by linear mechanisms such as radiation through the vent's opening. In that case, the first mode may not scatter energy to higher modes. Therefore, it is reasonable to consider here the ease where only one harmonic mode is excited non-linearly, and proceed to calculate its amplitude in the limiting state.
308
S. TEMKIN
3. AMPLITUDE OF SECOND MODE 3.1. BASICEQUATIONS The transfer of energy from the first longitudinal mode to its harmonic occurs because the amplitude of the fundamental mode is not infinitesimal, so that the non-linear convective terms in the gas equations of motion cannot be neglected. It is therefore necessary to consider the complete set of equations. However, in the acoustic approximation, the dissipative ,terms in these equations simply slow the energy-transfer process and limit the-growth of the highestorder modes. Therefore, consider first the gas equations of motion for an inviscid, nonconducting medium. Dissipative effects will be included at a later stage. In one spatial dimension, the governing equations are Op' 0
+
a
(p' .) = o,
(8)
0 (p' u) + ~ x (p' + p' u2) = 0,
(9)
p'(p')-' = constant.
(10)
Herep' is the pressure, p' .is the density, ~, is the ratio of specific heats of the gas, and u is the gas velocity. Equations (8) and (9) can be combined to yield a2p '
at 2
t92p '
a2
02
a~ ax---T = a~-~xZ( p' z,z) + - ~ (p'
.
alp').
(11)
This is an exact equation, and can be simplified to some degree since it contains terms which in the acoustic approximation are of third order. One first writes P'=Po+P,
P'=Po+p,
(12)
so that p and p represent the deviations of pressure and density from their mean values. To simplify the second term on the right-hand side ofequation (11), one expandsp' in terms ofp' about p' = Po, and uses equation (10) to evaluate the expansion coefficients, thus obtaining, to second order, ( ? - l)aZoP 2 p' = po + a2op q (13) 2po Similarly, to second order, p'U 2 ~ poll 2, SO that equation (l l) can be written as 02P 0I 2
2 azp
2 02 u2
(y-- 1) 2 02p2
ao 0 x 2 = poao-~x 2 + " " ~ p o a0 . -0-x-2 .
(14)
The significance ofthe two terms on the right-hand side ofthis equation is clear. They represent non-linear acoustic distortion arising from different sources. The first is simply due to nonlinear convection, while the second is caused by "medium non-linearity": i.e., the pressure and the density are not linearly related. The order of magnitude of these two terms is also different; thus, ifuo is a typical acoustical velocity and ao/co is a typical length, the magnitude of the first term is of the order ofpou2~ 2, while that of the second is (~ - l)pou2to2/2, so that the magnitude of the second term divided by that of the first is (V - 1)/2. For propellant gases this ratio is about 0.1. One can therefore neglect the medium non-linearity compared to the convective, and equation (14) reduces to a2 P Ol 2
2 a2 p a2 u2 a 00x---"S = poa2o Ox---T .
(15)
OSCILLATIONSIN T-BURNERS
309
3.2. APPROXIMATESOLUTION
Equation (15) can be solved, approximately, by the usual perturbation procedure. Let P = P l +P2 + "", u = ul + u2 + "", wherep2 and u2 are assumed to be much smaller thanpl and ul respectively. Substituting in equation (15) and collecting terms of equal order of magnitude yields a2Pt
2a2pt -----0, ao Ox'---~
~t2
~2P2 + 2fl2 ap2 2 azp2 2 02 hq2 Ot 2 - ~ - - a 0 0x----T=poao ax 2 .
(16)
(17)
The quantity//2 appearing in equation (17) is the temporal damping coefficient associated withp,, and the term 2f12(Op2/Ot) has been added, as usual, to include the effects of dissipation. Tile term cannot be derived from first principles as it accounts for dissipation in a global manner while it retains a one-dimensional geometry. The solution to equation (16) is given by equation (1) and represents the fundamental mode of the cavity. Equation (17) clearly shows how the fundamental mode drives its harmonic, for its solution represents acoustic waves due to a source of strength poa2(O 2 ll2[OX2). Now, what is of interest here is the solution forp2 when a limiting condition has been established. Thus one substitutes ul from equation (2) with PI = Ptr. (the limiting value of P~), and sets P2 = q~(t)cos(2kx) in equation (17) to obtain o;P~L[l + cos (2~ot)] dr(t) + 2fl2 dp'(t) + (209)2 ~b(t) = (i 8) )'Po The solution to this equation can be written as ~b= ~b~+ ~b2,where 41 ~b2 =
=
??,.
, 4~,po
.o,??L 4~'po b'2
sin (2~ot).
(19) (20)
The quantity q~l is associated with a small increase ofthe mean pressure in the cavity, and has no bearing on the computation ofthe acoustic energy dissipation rate associated withp2. Thus
toP?L
P2 = 4ypo fl, sin (2cot) cos (2kx).
(2 I)
This represents the second longitudinal mode of the cavity. The corresponding velocity and energy per unit cross-sectional area of the cavity are, respectively, u, ---
o~P?L
cos (2tot) sin (2kx),
(22)
~ 4)'po f12
(23)
4~po Po ao flz E2 = ~ 3.3. LIMmr~GCONDmON Equations (21)-(23) are applicable provided that P2L = [P21 '~ P~L, and can he used to measure the distortion o f the sound waves in the limiting condition. However, the main interest here is in the value of P~L, and this is obtained by equating the net energy input rate of the first mode to the energy dissipation rate of the second mode. This last quantity is given by /~2.,ut =
2fl2 E2.
(24)
310
s. TEMKIN
Thus, making use of equations (7), (23) and (24), one finds that the limiting amplitude of the fundamental mode of the cavity is Pit-
-
47 (~g --//2)1t29
/~0
(25)
O9
Weiner's result [2] is equivalent to equation (25) multiplied by 27/(7 + 1). For propellant gases the value of this ratio is about 1-1. The amplitude of the second mode, relative to that of the first, can be obtained from equations (21) and (25), the result being P2~. (agy/2 e l l = \fl---2] "
(26)
Also, from equations (25) and (26),
P2L 47%
(27)
Po
It can be seen from equation (26) that since in the perturbation procedure it was assumed that
P2, ~ PaL,the results are applicable only iffl2 >>%. That is, the second mode must be heavily damped if the results are to be valid. Equation (25) is one of the main findings of this paper; it gives PIL in terms of quantities which can be easily measured, or, in simple cases, obtained analytically. Thus, the growth constant % can be obtained from the early stages of the growth curve, the frequency is determined by the length of the burner, and the temporal damping Coefficient of the second mode can be determined from simple experiments, or from approximate analytical considerations. 4. ENERGY LOSSES OF THE SECOND MODE Now consider the energy dissipation rates of the second longitudinal mode in a T-burner with the purpose of evaluating the second-mode temporal damping coefficient, f12. Now, when several dissipation mechanisms are present, the total rate of energy dissipation of the second mode is the sum of the rates of energies dissipated separately by each mechanism. Since each of these is ofthe form 2fl2~//2, where the sub-index i represents the type of loss, it is clear that f12 is simply the sum of the separate damping coefficients: i.e., = 1
Also, since the losses occur both in the burner and in the vent, it is convenient to compute these contributions separately. 4.1. BURNER CAVITY LOSSES The losses in the burner cavity are due to dissipation in the gaseous phase, and dissipation at the walls of the burner. When there are no solid or liquid particles in the gaseous phase, and when the frequencies are not too large, one can assume that the dominant losses will be due to viscous friction and heat transfer at the walls. The dampin.g coefficient associated with this loss is the "wide-tube" damping coefficient given by
(28) where R is the cavity radius, v is the kinematic coefficient of viscosity of the propellant gases, and Pr is their Prandtl number. The notation fl2wbmeans that this is the contribution to the temporal damping coefficient of the second mode due to the wails of the burner. Equation
OSCILLATIONS IN T=BLrRNERS
311
(28) has received much attention in the past, and has been shown to give relatively good agreement with experimental results for low frequencies and large tube radii in cold cavities (see, for example, reference [8]), provided also that absorption at the closed ends is negligible. However, its application to T-burners is questionable mainly because the walls of the burner 9 are not at the same mean temperature as the combustion gases in the cavity. Also, losses at the closed ends of the cavity are not negligible. The temperature problem has been discussed in some detail by Perry [6], who heuristically modified equation (28) to account for the temperature difference. The modification changes the magnitude of the first bracket, but leaves the dependence on frequency and on cavity radius unchanged. It is therefore assumed here that the wall damping coefficient, fl2wb, is given by fl2wb = Aco 1/2,
(29)
where the quantity ,4 depends on gas properties, has the dimensions of (time) -1/2, and is inversely proportional to the cavity radius. 4.2. VENTLOSSES Vent losses are sometimes calculated on the assumption that the vent length is negligible. This reduces the problem to finding the energy radiated at the mouth of the vent. However, in some cases the vent's length is not negligible, so that viscous effects at the walls of the vent must be included. Further, at some frequencies the (fixed) length of the vent is comparable to the length of the burner. When this is the case, the pressure oscillations at the mouth of the vent are amplified by the standing waves that are set up in the vent, and this may result in a considerable increase of the energy loss, especially in the case of the second mode. Now, in order to find these losses, one must first obtain the acoustic field inside the vent. This is not easily done, for there are effects which are difficult to handle. However, considerable simplifications can be made when the vent is located at the middle section of the burner, has a small diameter, and the radiation losses are not too large. 4.2.1. Field inside vent The vent may be treated as a pipe of length I driven at one end, say z = 0, by the second mode in the burner cavity, and open at the other at z = I. The acoustic pressure field inside the vent, P2v will satisfy the linear wave equation together with some boundary conditions. These conditions are of the impedance type and involve quantities which are not known. However, if the vent diameter is small compared to the burner diameter, and the vent is centered along the burner, the boundary condition at z = 0 will be, approximately, P2v(0, t) = P2t. sin (2090,
(30)
where P2L = ogp2d4~'poflz [see equation (21)]. The boundary condition at z = 1 is more complicated since some acoustic energy is radiated from the open end of the pipe. That is, the acoustic pressure there is not identically zero as it would be in the absence of radiation. Furthermore, the vent is connected to a surge tank of finite size, so that reflected waves with unknown amplitude and phase will reach the vent. However, ifthe dimensions of the tank are very large, one can neglect the reflected waves. Also, if the energy radiated per cycle is a small fraction o f the total energy in the vent, one can neglect it in the first approximation, and write the boundary condition at z = I as p2v(l,t)~O. As stated above, P2~ is assumed to satisfy a linear wave equation, so that if wall damping in the vent is taken into account, P2v = ,4 exp (ko 2 t - iK2 z) + Bexp (i~o2t + iK2 z),
312
s. TEMKIN
where K2 = k2 + i/J2~. ao Here k2 = 2k, co2 = 2co, and f12~is the wall damping coefficient for the field in the vent (not to be confused with fl2w~+,the damping coefficient of the second mode in the burner due to wall effects in the vent). The quantity f12, is, like fl2wb,inversely proportional to the.radius, so that
f12~ = [32~b(R/a),
(31)
\vhere a is the radius of the vent. Applying the boundary conditions, one obtains, approximately, sin k2(l - z) sin (co2t - 0) P2v -----P2/., " 2 [sin k21 + (f12~l/ao) 2 cos2 k21] 1/2' P2L
u2~ = - -
COSk2(l - z) cos (092 t - 0)
Po ao [sin 2 k2 ! -t- (fl2v l/ao) 2 cos2 k2 l] i/2,
(32)
(33)
where tan0 = (ao/fl2vl)tank2l. The average acoustic energy in the vent associated with P2~ and u2~ is s ~
l-'+~fl~~"
(34)
4.2.2. Radiation losses t
The average power lost from the vent's termination by radiation to an infinite medium is given by [7, see p. 293]
~2, =
Po na +
4ao
(z~L(l, t)>,
(35)
where the brackets represent time average. Substituting u2~ from equation (33) and dividing the result by the cross-sectional area of the burner cavity yields
(coL/arpo) E2"=zrr/L'---R ] ~47pofl------~2}sin22kl+([32vllao)2COS22kl" [ a 2 ~2[ (.OP1L ~2
(36)
Now, E2, = 2f12,E2, and E2 is given by equation (23). Comparing these expressions one finds that the radiation damping coefficient of the second mode due to radiation to the outside is co(a2[LR) 2 f12, = n sin2 2kl + (f12~l/ao) 2 cos 2 2kl" (37) Now, this value of f12, was obtained on the assumption that radiation losses were relatively small. This is in fact the case, for the ratio of energy lost in one period to the energy in the vent is, from equations (34) and (36), of the order ofa2/Ll, and this is small. Similar remarks apply to the following computation of wall losses in the vent. 4.2.3. Wall losses in the vent Here one needs to compute the damping coefficient, f12,~, of the second mode in the burner cavity that results because of dissipation in the vent. Now, the average of the rate at which energy is dissipated in the vent by wall effects is d2wo = 2f12v e2v ,
(38)
O S C I L L A T I O N S IN T - B U R N E R S
,313
where fl~ is given by equation (31). Since the losses of the second mode in the burner cavity due to wall effects in the vent can be written as
F.2wv= 2fl2wvE2 = ~2w~./r~Rz,
(39)
one has, from equations (31), (34), (38) and (39),
(a/R) (ilL) [1 - sin ( 4kl)/4kl] fl, w -- fl2w, sin2 2kl + (f12J/ao) 2 c~ 2kl
(40)
Thus, the wails in the vent result in a damping coefficient for the second mode in the cavity which differs from that due to the walls in the burner. Two factors account for this difference: first, the expected geometrical factor appearing in the numerator ofequation (40), and second, the amplification factor in the denominator. 5. RESULTS 5.1. TOTALDAMPINGCOEFFICIENT If the predominant dissipation mechanisms of the second mode in the burner are those considered above, the total temporal damping coefficient of the second mode, fl,, can be obtained by adding the results given in equations (29), (37) and (40). However, in order to display the effect of the vent, one can compute from that result the ratio fl2/fl2wb and obtain Sin (4hilL)
1 + n(a3/LRl)(o)/fl2~,.b)
4rcl/L (40
s i n 2 ( 2 n l ) + ( nfl'wbl/LI2c~ r
(2 n-~) ' "
It is evident from this equation that one effect of the vent is, as expected, to increase the magnitude of the damping coefficient of the second mode in the burner cavity. The increase, as given by the second term on the right-hand side of the equation is, in general, very small. However, when ilL = N/2, where Nis a positive integer, the acoustic field in the vent is largely amplified, and this may result in a significant increase for f12. For example, when the first E)O
i
i
I
/A,0.25
IO
I0
Or2
J
04
06
----708
I
I0
l/L
Figure 2. Variations of second mode with vent's length for two values of wall damping.
314
s. TEMKIN
resonant condition is met in the vent (N-- 1), the leading term of equation (41) gives
B2
~ 1+ 8~
fl2wb
(aiRy
(fl2wb/f)2'
(42)
wherefis the frequency of the first longitudinal mode in the burner. For typical values of a, R and f, the ratio (a]R)3]([32wb/f)2 is, at least, of order one, so that the damping coefficient of the second mode is increased in this condition by at least one order of magnitude. Furthermore, in view of equation (25), it is to be expected that when this resonant condition in the vent is satisfied, the limiting pressure amplitude may reach large values (for constant %). The dependence of fl,/fl2,~b on IlL and on fl2wbis displayed graphically in Figure 2, where fl2/fl2,~b is plotted versus 1/L for constant air but for various values of A in equation (29). 5.2. FIRST-MODEDECAY
An important quantity in T-burner experimental work is the temporal damping coefficient of the first mode--fit in this work. This quantity is usually obtained from the decay records by approximating the envelope of the decaying oscillations by several exponential portions, each one y!elding a different value of fl~. There are at least two effects which may produce departures from a single exponential decay. First, as the propellant is extinguished, the temperature changes. This results in different values of other thermodynamic properties. In particular, the speed of sound decreases with temperature, and this results in a lower frequency ofoscillation. Second, non-linear effects, which may be present at larger oscillatory amplitudes, decrease in importance as the amplitudes diminish. Consider, for example, the non-linear effect described in this paper: i.e., the build-up of the second mode in the cavity by non-linear energy transfer. This mode has an amplitude proportional to P~ (where P1 is the amplitude of the first mode), and dissipates energy at a rate proportional to P~(dP,/dt). Immediately after extinguishment, this may be an important fraction of the total energy dissipation rate, but as P, decreases, it becomes less important. The effects of the second mode on the decay of the first can be studied by determining PI as a function of time after burn-out. Assume that, at t = to, the propellant is suddenly extinguished. Further, assume that the relationship between PI and P2 obtained for the limiting condition is applicable at all instants during the decay. Now, since e in equation (5) is identically zero after extinguishment, an energy balance yields d E , = ~ , out - E2 out.
dt ' ' Substitution from equations (4), (6), (23) and (24) yields
d---7-= - 2 fl, P? - f12 k @Po ] P•"
(43)
(44)
This differential equation can be integrated easily by writing it in terms of lIPS. The solution, in terms of P,, is Pi
P'< :[
,
e-Bj i
-e-:"Tl
(45)
where P,L, the limiting pressure amplitude, is the value of P, at t = to when extinguishment takes place. Note that ifP,L/Po ~ 1, there is an exponential decay with damping coefficient fix. However, for any finite value of P~L the initial decay is not exponential. Further, the asymptotic decay as t --~ oo is exponential and the damping coefficient is independent of P,L.
OSCILLATIONS IN T-BURNERS
315
IC
go (c
00~ OO
OI
02
03
04
05
06
t (secmds) Figure 3. D e c a y o f pressure oscillations for three values o f B =
(toPIJ4ypo)'[flafl2.
Equation (45) is displayed graphically, in Figure 3, in the usual semi-logarithmic plot, for several values of the parameter B = (o~Plt./4~po)Z/fllfl2, and for fll = 5 s -1. The value B = 0 corresponds to linear decay, and the slope of that line is the damping coefficient fit. For nonzero values of B, non-linear effects are present. These produce a faster decay, and delay the occurrence of the exponential-decay stage. It is clear that if non-linear effects are present, the decay constant should be obtained only from the slope of the eventual exponential portion of the decay curve.
6. COMPARISON WITH EXPERIMENTAL RESULTS It is of interest to compare the results for f12 (equation (41)) and for P,L (equation (25)) with experimental data. However, the author is not aware o f a n y systematic measurements of damping of even modes in centrally vented tubes, so that a comparison of f12 alone is not possible. On the other hand, Perry [6] has recently reported extensive measurements of limiting acoustic oscillations in T-burners, so that an overall comparison with experiments is possible. Perry's measurements were made at different mean pressures and with different propellants in T-burners with different diameters. The most complete set of measurements was made with A-13 propellant at a mean pressure of 21.4 atm (300 psig) in a 3.81 cm-diameter T-burner. The length, L, of the burner could be adjusted so that longitudinal waves with frequencies in the range 300-6400 Hz could be excited in the burner. The length of the vent, l, was about 12.7 cm for the low-frequency runs (up to 2600 Hz) and about 15-4 cm for the higher frequencies, and its diameter was 1.017 cm. From Perry's data, one can obtain the values of
3 !6
s. TEMKIN
P1L and % for various frequencies. Now, according to equation (25),
(%lf)"=
n\f ]
where f12 is predicted by equation (41). In order to provide a comparison of the analytical results with Perry's data, the experimentally determined values of (P~L/Po)[(%lf) 1t2 versus IlL are plotted in Figure 4. The quantity IlL is equivalent to a non-dimensional frequency.
I
i
i
I
I
I
I
f 06
i 07
i
I
//-
2C
~
L s
t y
-
o
I 00I
I 02
1
O-:3
I
04
\ I
0.5
i 0 8
iff.. 09 I0
t/L
Figure 4. Comparison of equations (25) and (41) with Perry's experimental data.
The open circles shown in the figure represent the average of several tests, and the length of the bars represents the spread of the data. The solid curve shown in Figure 4 represents the quantity 27(f12]f)t12/rc from equation (41) with values of a, R and I corresponding to those of Perry's burner. Also used in the reduction of equation (41) was equation (29), with A 2 = 0.60 s-1. This value of A nearly corresponds to Perry's decay data, and is about two times larger than that predicted by equation (28). The quantity A controls the height and width of the solid curve. Thus, a higher value of A would yield a broader curve of lower maximum. However, this maximum would still be centered at ill = 1/2. The reason for this maximum is that for that value of IlL, the field in the vent is a resonant one, so that in such a condition dissipation is increased as explained above. The experimental data shown in the figure also display a typical resonant behavior with a maximum of the same order as that predicted by equations (25) and (41), and also centered at about IlL = 1/2. This indicates that dissipation in the vent is in fact responsible for the limiting condition when ilL ~.. 112. However, it can also be seen that while the theoretical curve is very sharp, the experimental one is not. Several factors may account for this behavior, such as kinetic energy losses due to turbulence, mean flow losses, etc., but it appears that the dominant one is that outside a narrow frequency band near 1/L = 1/2, the vent does not dissipate much energy. That is, outside that narrow band f12 ~ fl2,,.b.Therel'ore, the assumption (fl2/~g)u2 < 1 used in deriving equation (25) is not satisfied, because % is of the same order of magnitude as fl2wb.This would imply, for example, that higher-order modes are present which may account for significant energy dissipation outside the resonant condition. In other words, the assumption that only one harmonic is excited is not valid outside that condition.
OSCILLATIONSIN T-BURNERS
317
It is of interest to point out that there are data published in the literature which seem to indicate that the vent's role is, in general, a sinall one. For example, Temkin [9] in a study of non-linear gas oscillations in a cylindrical tube observed that a small orifice ( i l l = 0"02), located near the middle section of the tube, had little effect on the total pressure amp!Rude. More relevant to the present discussion are the measurements made by Perry in his T-burner and which were directed at finding whether the vent had any effect on the limiting amplitudes. He reports using a vent with an opening larger than that in most tests (but apparently with the same length), and observing no change. These results are in complete agreement with the model described here, since outside the vent's resonant condition the model predicts that the vent has no effect on the acoustic field in the burner cavity. On the other hand, on the basis of this limited number of tests, Perry concluded that "the fact that no differences in the data were observed supports the generally held belief that the vent, located at the pressure node of the oscillation, plays little, if any, role in the T-burner tests". This is in general the case, but in view of the present results, and of the trend agreement shown in Figure 4, it appears that near IlL --- I/2 the vent plays an important role in T-burner tests. ACKNOWLEDGMENTS The author is grateful to Dr M. M. Ibiricu for many helpful discussions concerning this work. This work was supported by the U.S. A r m y Ballistic Research Laboratories under Contract DA-0341. REFERENCES
1. M. D. HoRToN1964 Journal of the American lnstitute of Aeronautics and Astronautics 2, 1112-1118. Use of the one-dimensional T-burner to study oscillatory combustion. 2. S. WrINER 1966 Journal of the Acoustical Society of America 40, 240-243. Standing sound waves of finite amplitude. 3. S. TEMK1N ~969 Journa~of the A~ustica~ Society ~f America 45~224-227. Pr~pagating and s~anding sawtooth waves. 4. F. E. C. COLICK1971 Combustion Science and Technology 3, 1-16. Non-linear growth and limiting amplitude of acoustic oscillation in combustion chambers. 5. F. E. C. CULICK 1972 American Institute of Aeronautics]Society of Automotive Engineering 8th Joint Propulsion Specialist Conference, New Orleans, Louisiana, November 29-December I, Paper No. 72-1049. Research on combustion instability and application to solid propellant rocket motors, II. 6. E. PERRY1970Ph.D. Thesis, California Institute of Technology, University Microfibns No. 70-24577. Investigation of the T-burner and its role in combustion instability studies. 7. L. D. LANDAUand E. M. LtrsmTz 1959 Fluid Mechanics. London: Pergamon Press, Ltd. 8. S. TEMKINand R. A. DoanINs 1966 Journal of the Acoustical Society of America 40, 1016-1024. Measurements of attenuation and dispersion of sound by an aerosol. 9. S. TEMKIN1968 The Physics of Fluids 11,960-963. Non-linear gas oscillations in a resonant tube.
LIMITING ACOUSTIC OSCILLATIONS IN T-BURNERS S. TEMKIN
Department of ~lechanical, Industrial and Aerospace Enghteerhlg, Rtttgers University, New Brunswick, New Jersey 08903, U.S.A. (Received 25 June 1973, and in revisedform 23 October 1973) A simple model is used to analyze finite-amplitude standing modes in T-burners. Mean flow effects are neglected in the model and it is assumed that a limiting condition results owing to non-linear energy transfer from the more unstable fundamental mode of the burner cavity to its harmonic. Results for the limiting pressure of the fundamental mode and its harmonic are found in terms of the growth constant, and of the damping constant of the second mode. This quantity is calculated on the assumption that the second mode losses are mainly caused by wall effects, and by radiation from the vent. It is shown that, in general, the vent does not result in large amounts of energy losses, except when a resonant field is excited inside the vent. In that condition, the amplitude of the first mode can be quite large. The theoretical prediction for the limiting amplitudes is compared with available experimental data, and it is found that while the experimental data display a resonant-like behavior about the theoretically predicted condition, the width of the resonant peak is much broader than that predicted theoretically. 1. INTRODUCTION T-burners are devices which are used to test the stability of solid propellants for rocket propulsion [1]. Basically, a T-burner is a cylindrical cavity closed at both ends by rigid caps and vented at its middle section as shown schematically in Figure I. The vent is connected to a pressurized tank of large dimensions. Samples of propellants to be tested are placed at one or both ends of the cavity, and are ignited by external means. I f the propellant is unstable, acoustic waves will be excited in the burner with frequencies corresponding to the acoustic modes 'of the cavity. For some geometries, radial and tangential modes are excited, but in most studies attention is focused on the fundamental longitudinal mode of the cavity. In order to determine how unstable a given propellant is at a frequency to, the time history ofthe acoustic pressure of the fundamental mode in a burner of length L = nao/to is recorded. (Here ao is the ambient speed of sound in the burner.) The initial growth and eventual decay of this pressure record are exponential, and yield data that can be used to compute the propellant's response function at that frequency. Although the method is based on the assumption that losses in the burner during the growth and decay stages are the same, its simplicity has proved very useful in determining the Ptopellcmt
Figure 1. Schematic diagram of a T-burner. 305
306
s. TEMKIN
intrinsic instability of solid propellants. However, as presently used, the procedure ignores one of the main characteristics of the pressure oscillations, namely, the limiting state they reach during burning. This limiting state is clearly important because it affords another measure of the propellant's instability, and because it shows the importance of non-linear mechanisms. Furthermore, since elimination of acoustic instability has not yet been accomplished, the most important questions that arise in practice relate precisely to the existence of this limiting state and to the values of the oscillatory pressure amplitude in this condition. Recently, several studies have appeared in the literature which consider the limiting amplitude state in T-burners. These studies include works by Weiner [2], Temkin [3], Culick [4, 5] and Perry [6]. Culiek's works are the most general from a theoretical point of view, since they include coupling effects between the burning propellant and the gas in the cavity. However, his results are difficult to interpret as they depend on several parameters which must be evaluated separately. Also, his results are based on the assumption that the limiting factor is the non-linearity of the damping mechanisms. This assumption may, of course, be correct, but there exist other non-linear effects which ought to be considered. Weiner [2] and Temkin [3] have studied the limiting condition using simple models for the acoustic field in the burner cavity. Thus, Temkin assumed that, in the limiting condition, the amplitudes of the waves would be so large that shock waves would be present in the cavity. However, in most tests in T-burners no shock waves are observed. Weiner assumed that in the limiting condition the wave profile is a sawtooth with superpose d small-amplitude even modes. That is, the profile he assumed had relative amplitudes P~, 1"2, Pd3, P4, P~/5, etc., for the first few modes in the cavity. The amplitudes of the first and second modes were found by computing the work done b y t h e harmonics on the fundamental. The results were given in terms of easily measured quantities, but the procedure used in the derivation and the severe assumption made about the profile have limited their usefulness. In this paper the limiting-amplitude condition in a T-burner is described by a model based on the assumption that this condition results from non-linear transfer of energy from the fundamental mode in the burner cavity to its harmonic. The amplitude of this harmonic mode is first calculated from an approximation to the equations of motion for the gas in the cavity. The limiting amplitude of the first mode is then found by means of a simple energy balance in terms of the growth constant of the fundamental mode and of the damping coefficient of the second longitudinal mode of the cavity. This damping coefficient is then calculated for a centrally vented burner, and the theoretical prediction for the limiting amplitude is compared with the experimental results of Perry. 2. ENERGY CONSIDERATIONS Consider the fundamental longitudinal mode in a cylindrical cavity of length L filled with a perfect gas. If purely longitudinal oscillations are assumed, the acoustic pressure, p~, and velocity, ul, at time t are given by
p~(x, t) = P~(t) sin (cot) cos (kx), u~(x, t) =
Pl(t) po ao
cos (cot) sin (kx),
(1) (2)
where Po is the mean gas density, co is the circular frequericy of the fundamental mode, and k = 7t/L is the wave number of that mode. The amplitude P~ is typically a slowly varying function of time. Initially, P~ grows exponentially with time. Its growth then slows gradually, and eventually a limiting state is reached. It is of interest to find the limiting value, P~r., of Pt. In some cases this can be done by simple energy-balance arguments. That is, the acoustic
OSCILLATIONS IN T-BURNERS
307
energy of the fundamental mode increases at a rate which is determined by the energy addition and removal from the oscillatory motion of the gas in the cavity. The limiting state, if it exists, is clearly obtained when these two energy rates are equal. The problem, therefore, reduces to calculating these quantities separately. Now, the instantaneous energy of the fundamental mode, per unit cross-sectional area of the cavity, is [7, see p. 250] L
L
Ex(t)=~po(u2dx+21~fp~dx. a
Substitution ofp~ and
(3)
~'poJ
0
0
ux from equations (1) and (2) yields L
Ex(t)=--~----poP~(t).
(4)
Now, since Px initially grows in an exponential manner, the acoustic energy of the first mode also grows exponentially. Therefore, the rate at which energy is transferred from the burning propellant to the fundamental mode is ~?,.,o = 2~(L/4Wo),%
(5)
where ~ is, approximately, the fractional energy added per cycle. Equation (5) is, in principle, valid only for small values of Px. However, since the energy transferred from the propellant to the acoustic waves is a small fraction of the total energy released by the combustion, the rate of energy addition to the waves would not be expected to change significantly owing to their growth. One may therefore provisionally assume that equation (5) is applicable even when the amplitude of the waves is not infinitesimal. That is, of course, an assumption which may be invalid during the growth o f the oscillations. The energy losses associated with the first mode are due to linear and non-linear mechanisms. The linear type consists mainly of viscous and thermal losses at the walls of the burner, and leads to an energy-loss rate which is also proportional to the instantaneous energy: i.e., El,out =
2fl~(L[4rpo)P?,
(6)
where fl~ is the linear damping coefficient for the first mode. If there are no other dissipative mechanisms, the energy of the first mode would change at a rate L',.,,, = 2~,(L]4wo)/2,
(7)
where ctg = e - fix. Clearly, ifcto > 0, the amplitude of the first mode will increase continuously until the assumed linear model ceases to be valid. What may happen, physically, is that if the fundamental mode reaches a sufficiently large amplitude, it will start to transfer energy to its harmonic. Some of this non-linearly transferred energy will then be dissipated by linear mechanisms associated with the second mode. If this energy loss is not sufficient to limit the growth of the fundamental mode, more modes will appear, and ifthe process is continued indefinitely, a weak shock wave will appear. This weak shock wave will result in a rate of energy dissipation proportional to Pax (as opposed to p2 produced by linear mechanisms), and will therefore result in a limiting condition [3]. However, in most T-burner tests, shock waves are not observed. This indicates that in those instances the limiting effects arise because of energy transfer to the first few harmonics. Furthermore, since a T-burner has one or more vents at its middle section, and this is a pressure maximum for the second mode, this mode may lose a great deal of energy by linear mechanisms such as radiation through the vent's opening. In that case, the first mode may not scatter energy to higher modes. Therefore, it is reasonable to consider here the ease where only one harmonic mode is excited non-linearly, and proceed to calculate its amplitude in the limiting state.
308
S. TEMKIN
3. AMPLITUDE OF SECOND MODE 3.1. BASICEQUATIONS The transfer of energy from the first longitudinal mode to its harmonic occurs because the amplitude of the fundamental mode is not infinitesimal, so that the non-linear convective terms in the gas equations of motion cannot be neglected. It is therefore necessary to consider the complete set of equations. However, in the acoustic approximation, the dissipative ,terms in these equations simply slow the energy-transfer process and limit the-growth of the highestorder modes. Therefore, consider first the gas equations of motion for an inviscid, nonconducting medium. Dissipative effects will be included at a later stage. In one spatial dimension, the governing equations are Op' 0
+
a
(p' .) = o,
(8)
0 (p' u) + ~ x (p' + p' u2) = 0,
(9)
p'(p')-' = constant.
(10)
Herep' is the pressure, p' .is the density, ~, is the ratio of specific heats of the gas, and u is the gas velocity. Equations (8) and (9) can be combined to yield a2p '
at 2
t92p '
a2
02
a~ ax---T = a~-~xZ( p' z,z) + - ~ (p'
.
alp').
(11)
This is an exact equation, and can be simplified to some degree since it contains terms which in the acoustic approximation are of third order. One first writes P'=Po+P,
P'=Po+p,
(12)
so that p and p represent the deviations of pressure and density from their mean values. To simplify the second term on the right-hand side ofequation (11), one expandsp' in terms ofp' about p' = Po, and uses equation (10) to evaluate the expansion coefficients, thus obtaining, to second order, ( ? - l)aZoP 2 p' = po + a2op q (13) 2po Similarly, to second order, p'U 2 ~ poll 2, SO that equation (l l) can be written as 02P 0I 2
2 azp
2 02 u2
(y-- 1) 2 02p2
ao 0 x 2 = poao-~x 2 + " " ~ p o a0 . -0-x-2 .
(14)
The significance ofthe two terms on the right-hand side ofthis equation is clear. They represent non-linear acoustic distortion arising from different sources. The first is simply due to nonlinear convection, while the second is caused by "medium non-linearity": i.e., the pressure and the density are not linearly related. The order of magnitude of these two terms is also different; thus, ifuo is a typical acoustical velocity and ao/co is a typical length, the magnitude of the first term is of the order ofpou2~ 2, while that of the second is (~ - l)pou2to2/2, so that the magnitude of the second term divided by that of the first is (V - 1)/2. For propellant gases this ratio is about 0.1. One can therefore neglect the medium non-linearity compared to the convective, and equation (14) reduces to a2 P Ol 2
2 a2 p a2 u2 a 00x---"S = poa2o Ox---T .
(15)
OSCILLATIONSIN T-BURNERS
309
3.2. APPROXIMATESOLUTION
Equation (15) can be solved, approximately, by the usual perturbation procedure. Let P = P l +P2 + "", u = ul + u2 + "", wherep2 and u2 are assumed to be much smaller thanpl and ul respectively. Substituting in equation (15) and collecting terms of equal order of magnitude yields a2Pt
2a2pt -----0, ao Ox'---~
~t2
~2P2 + 2fl2 ap2 2 azp2 2 02 hq2 Ot 2 - ~ - - a 0 0x----T=poao ax 2 .
(16)
(17)
The quantity//2 appearing in equation (17) is the temporal damping coefficient associated withp,, and the term 2f12(Op2/Ot) has been added, as usual, to include the effects of dissipation. Tile term cannot be derived from first principles as it accounts for dissipation in a global manner while it retains a one-dimensional geometry. The solution to equation (16) is given by equation (1) and represents the fundamental mode of the cavity. Equation (17) clearly shows how the fundamental mode drives its harmonic, for its solution represents acoustic waves due to a source of strength poa2(O 2 ll2[OX2). Now, what is of interest here is the solution forp2 when a limiting condition has been established. Thus one substitutes ul from equation (2) with PI = Ptr. (the limiting value of P~), and sets P2 = q~(t)cos(2kx) in equation (17) to obtain o;P~L[l + cos (2~ot)] dr(t) + 2fl2 dp'(t) + (209)2 ~b(t) = (i 8) )'Po The solution to this equation can be written as ~b= ~b~+ ~b2,where 41 ~b2 =
=
??,.
, 4~,po
.o,??L 4~'po b'2
sin (2~ot).
(19) (20)
The quantity q~l is associated with a small increase ofthe mean pressure in the cavity, and has no bearing on the computation ofthe acoustic energy dissipation rate associated withp2. Thus
toP?L
P2 = 4ypo fl, sin (2cot) cos (2kx).
(2 I)
This represents the second longitudinal mode of the cavity. The corresponding velocity and energy per unit cross-sectional area of the cavity are, respectively, u, ---
o~P?L
cos (2tot) sin (2kx),
(22)
~ 4)'po f12
(23)
4~po Po ao flz E2 = ~ 3.3. LIMmr~GCONDmON Equations (21)-(23) are applicable provided that P2L = [P21 '~ P~L, and can he used to measure the distortion o f the sound waves in the limiting condition. However, the main interest here is in the value of P~L, and this is obtained by equating the net energy input rate of the first mode to the energy dissipation rate of the second mode. This last quantity is given by /~2.,ut =
2fl2 E2.
(24)
310
s. TEMKIN
Thus, making use of equations (7), (23) and (24), one finds that the limiting amplitude of the fundamental mode of the cavity is Pit-
-
47 (~g --//2)1t29
/~0
(25)
O9
Weiner's result [2] is equivalent to equation (25) multiplied by 27/(7 + 1). For propellant gases the value of this ratio is about 1-1. The amplitude of the second mode, relative to that of the first, can be obtained from equations (21) and (25), the result being P2~. (agy/2 e l l = \fl---2] "
(26)
Also, from equations (25) and (26),
P2L 47%
(27)
Po
It can be seen from equation (26) that since in the perturbation procedure it was assumed that
P2, ~ PaL,the results are applicable only iffl2 >>%. That is, the second mode must be heavily damped if the results are to be valid. Equation (25) is one of the main findings of this paper; it gives PIL in terms of quantities which can be easily measured, or, in simple cases, obtained analytically. Thus, the growth constant % can be obtained from the early stages of the growth curve, the frequency is determined by the length of the burner, and the temporal damping Coefficient of the second mode can be determined from simple experiments, or from approximate analytical considerations. 4. ENERGY LOSSES OF THE SECOND MODE Now consider the energy dissipation rates of the second longitudinal mode in a T-burner with the purpose of evaluating the second-mode temporal damping coefficient, f12. Now, when several dissipation mechanisms are present, the total rate of energy dissipation of the second mode is the sum of the rates of energies dissipated separately by each mechanism. Since each of these is ofthe form 2fl2~//2, where the sub-index i represents the type of loss, it is clear that f12 is simply the sum of the separate damping coefficients: i.e., = 1
Also, since the losses occur both in the burner and in the vent, it is convenient to compute these contributions separately. 4.1. BURNER CAVITY LOSSES The losses in the burner cavity are due to dissipation in the gaseous phase, and dissipation at the walls of the burner. When there are no solid or liquid particles in the gaseous phase, and when the frequencies are not too large, one can assume that the dominant losses will be due to viscous friction and heat transfer at the walls. The dampin.g coefficient associated with this loss is the "wide-tube" damping coefficient given by
(28) where R is the cavity radius, v is the kinematic coefficient of viscosity of the propellant gases, and Pr is their Prandtl number. The notation fl2wbmeans that this is the contribution to the temporal damping coefficient of the second mode due to the wails of the burner. Equation
OSCILLATIONS IN T=BLrRNERS
311
(28) has received much attention in the past, and has been shown to give relatively good agreement with experimental results for low frequencies and large tube radii in cold cavities (see, for example, reference [8]), provided also that absorption at the closed ends is negligible. However, its application to T-burners is questionable mainly because the walls of the burner 9 are not at the same mean temperature as the combustion gases in the cavity. Also, losses at the closed ends of the cavity are not negligible. The temperature problem has been discussed in some detail by Perry [6], who heuristically modified equation (28) to account for the temperature difference. The modification changes the magnitude of the first bracket, but leaves the dependence on frequency and on cavity radius unchanged. It is therefore assumed here that the wall damping coefficient, fl2wb, is given by fl2wb = Aco 1/2,
(29)
where the quantity ,4 depends on gas properties, has the dimensions of (time) -1/2, and is inversely proportional to the cavity radius. 4.2. VENTLOSSES Vent losses are sometimes calculated on the assumption that the vent length is negligible. This reduces the problem to finding the energy radiated at the mouth of the vent. However, in some cases the vent's length is not negligible, so that viscous effects at the walls of the vent must be included. Further, at some frequencies the (fixed) length of the vent is comparable to the length of the burner. When this is the case, the pressure oscillations at the mouth of the vent are amplified by the standing waves that are set up in the vent, and this may result in a considerable increase of the energy loss, especially in the case of the second mode. Now, in order to find these losses, one must first obtain the acoustic field inside the vent. This is not easily done, for there are effects which are difficult to handle. However, considerable simplifications can be made when the vent is located at the middle section of the burner, has a small diameter, and the radiation losses are not too large. 4.2.1. Field inside vent The vent may be treated as a pipe of length I driven at one end, say z = 0, by the second mode in the burner cavity, and open at the other at z = I. The acoustic pressure field inside the vent, P2v will satisfy the linear wave equation together with some boundary conditions. These conditions are of the impedance type and involve quantities which are not known. However, if the vent diameter is small compared to the burner diameter, and the vent is centered along the burner, the boundary condition at z = 0 will be, approximately, P2v(0, t) = P2t. sin (2090,
(30)
where P2L = ogp2d4~'poflz [see equation (21)]. The boundary condition at z = 1 is more complicated since some acoustic energy is radiated from the open end of the pipe. That is, the acoustic pressure there is not identically zero as it would be in the absence of radiation. Furthermore, the vent is connected to a surge tank of finite size, so that reflected waves with unknown amplitude and phase will reach the vent. However, ifthe dimensions of the tank are very large, one can neglect the reflected waves. Also, if the energy radiated per cycle is a small fraction o f the total energy in the vent, one can neglect it in the first approximation, and write the boundary condition at z = I as p2v(l,t)~O. As stated above, P2~ is assumed to satisfy a linear wave equation, so that if wall damping in the vent is taken into account, P2v = ,4 exp (ko 2 t - iK2 z) + Bexp (i~o2t + iK2 z),
312
s. TEMKIN
where K2 = k2 + i/J2~. ao Here k2 = 2k, co2 = 2co, and f12~is the wall damping coefficient for the field in the vent (not to be confused with fl2w~+,the damping coefficient of the second mode in the burner due to wall effects in the vent). The quantity f12, is, like fl2wb,inversely proportional to the.radius, so that
f12~ = [32~b(R/a),
(31)
\vhere a is the radius of the vent. Applying the boundary conditions, one obtains, approximately, sin k2(l - z) sin (co2t - 0) P2v -----P2/., " 2 [sin k21 + (f12~l/ao) 2 cos2 k21] 1/2' P2L
u2~ = - -
COSk2(l - z) cos (092 t - 0)
Po ao [sin 2 k2 ! -t- (fl2v l/ao) 2 cos2 k2 l] i/2,
(32)
(33)
where tan0 = (ao/fl2vl)tank2l. The average acoustic energy in the vent associated with P2~ and u2~ is s ~
l-'+~fl~~"
(34)
4.2.2. Radiation losses t
The average power lost from the vent's termination by radiation to an infinite medium is given by [7, see p. 293]
~2, =
Po na +
4ao
(z~L(l, t)>,
(35)
where the brackets represent time average. Substituting u2~ from equation (33) and dividing the result by the cross-sectional area of the burner cavity yields
(coL/arpo) E2"=zrr/L'---R ] ~47pofl------~2}sin22kl+([32vllao)2COS22kl" [ a 2 ~2[ (.OP1L ~2
(36)
Now, E2, = 2f12,E2, and E2 is given by equation (23). Comparing these expressions one finds that the radiation damping coefficient of the second mode due to radiation to the outside is co(a2[LR) 2 f12, = n sin2 2kl + (f12~l/ao) 2 cos 2 2kl" (37) Now, this value of f12, was obtained on the assumption that radiation losses were relatively small. This is in fact the case, for the ratio of energy lost in one period to the energy in the vent is, from equations (34) and (36), of the order ofa2/Ll, and this is small. Similar remarks apply to the following computation of wall losses in the vent. 4.2.3. Wall losses in the vent Here one needs to compute the damping coefficient, f12,~, of the second mode in the burner cavity that results because of dissipation in the vent. Now, the average of the rate at which energy is dissipated in the vent by wall effects is d2wo = 2f12v e2v ,
(38)
O S C I L L A T I O N S IN T - B U R N E R S
,313
where fl~ is given by equation (31). Since the losses of the second mode in the burner cavity due to wall effects in the vent can be written as
F.2wv= 2fl2wvE2 = ~2w~./r~Rz,
(39)
one has, from equations (31), (34), (38) and (39),
(a/R) (ilL) [1 - sin ( 4kl)/4kl] fl, w -- fl2w, sin2 2kl + (f12J/ao) 2 c~ 2kl
(40)
Thus, the wails in the vent result in a damping coefficient for the second mode in the cavity which differs from that due to the walls in the burner. Two factors account for this difference: first, the expected geometrical factor appearing in the numerator ofequation (40), and second, the amplification factor in the denominator. 5. RESULTS 5.1. TOTALDAMPINGCOEFFICIENT If the predominant dissipation mechanisms of the second mode in the burner are those considered above, the total temporal damping coefficient of the second mode, fl,, can be obtained by adding the results given in equations (29), (37) and (40). However, in order to display the effect of the vent, one can compute from that result the ratio fl2/fl2wb and obtain Sin (4hilL)
1 + n(a3/LRl)(o)/fl2~,.b)
4rcl/L (40
s i n 2 ( 2 n l ) + ( nfl'wbl/LI2c~ r
(2 n-~) ' "
It is evident from this equation that one effect of the vent is, as expected, to increase the magnitude of the damping coefficient of the second mode in the burner cavity. The increase, as given by the second term on the right-hand side of the equation is, in general, very small. However, when ilL = N/2, where Nis a positive integer, the acoustic field in the vent is largely amplified, and this may result in a significant increase for f12. For example, when the first E)O
i
i
I
/A,0.25
IO
I0
Or2
J
04
06
----708
I
I0
l/L
Figure 2. Variations of second mode with vent's length for two values of wall damping.
314
s. TEMKIN
resonant condition is met in the vent (N-- 1), the leading term of equation (41) gives
B2
~ 1+ 8~
fl2wb
(aiRy
(fl2wb/f)2'
(42)
wherefis the frequency of the first longitudinal mode in the burner. For typical values of a, R and f, the ratio (a]R)3]([32wb/f)2 is, at least, of order one, so that the damping coefficient of the second mode is increased in this condition by at least one order of magnitude. Furthermore, in view of equation (25), it is to be expected that when this resonant condition in the vent is satisfied, the limiting pressure amplitude may reach large values (for constant %). The dependence of fl,/fl2,~b on IlL and on fl2wbis displayed graphically in Figure 2, where fl2/fl2,~b is plotted versus 1/L for constant air but for various values of A in equation (29). 5.2. FIRST-MODEDECAY
An important quantity in T-burner experimental work is the temporal damping coefficient of the first mode--fit in this work. This quantity is usually obtained from the decay records by approximating the envelope of the decaying oscillations by several exponential portions, each one y!elding a different value of fl~. There are at least two effects which may produce departures from a single exponential decay. First, as the propellant is extinguished, the temperature changes. This results in different values of other thermodynamic properties. In particular, the speed of sound decreases with temperature, and this results in a lower frequency ofoscillation. Second, non-linear effects, which may be present at larger oscillatory amplitudes, decrease in importance as the amplitudes diminish. Consider, for example, the non-linear effect described in this paper: i.e., the build-up of the second mode in the cavity by non-linear energy transfer. This mode has an amplitude proportional to P~ (where P1 is the amplitude of the first mode), and dissipates energy at a rate proportional to P~(dP,/dt). Immediately after extinguishment, this may be an important fraction of the total energy dissipation rate, but as P, decreases, it becomes less important. The effects of the second mode on the decay of the first can be studied by determining PI as a function of time after burn-out. Assume that, at t = to, the propellant is suddenly extinguished. Further, assume that the relationship between PI and P2 obtained for the limiting condition is applicable at all instants during the decay. Now, since e in equation (5) is identically zero after extinguishment, an energy balance yields d E , = ~ , out - E2 out.
dt ' ' Substitution from equations (4), (6), (23) and (24) yields
d---7-= - 2 fl, P? - f12 k @Po ] P•"
(43)
(44)
This differential equation can be integrated easily by writing it in terms of lIPS. The solution, in terms of P,, is Pi
P'< :[
,
e-Bj i
-e-:"Tl
(45)
where P,L, the limiting pressure amplitude, is the value of P, at t = to when extinguishment takes place. Note that ifP,L/Po ~ 1, there is an exponential decay with damping coefficient fix. However, for any finite value of P~L the initial decay is not exponential. Further, the asymptotic decay as t --~ oo is exponential and the damping coefficient is independent of P,L.
OSCILLATIONS IN T-BURNERS
315
IC
go (c
00~ OO
OI
02
03
04
05
06
t (secmds) Figure 3. D e c a y o f pressure oscillations for three values o f B =
(toPIJ4ypo)'[flafl2.
Equation (45) is displayed graphically, in Figure 3, in the usual semi-logarithmic plot, for several values of the parameter B = (o~Plt./4~po)Z/fllfl2, and for fll = 5 s -1. The value B = 0 corresponds to linear decay, and the slope of that line is the damping coefficient fit. For nonzero values of B, non-linear effects are present. These produce a faster decay, and delay the occurrence of the exponential-decay stage. It is clear that if non-linear effects are present, the decay constant should be obtained only from the slope of the eventual exponential portion of the decay curve.
6. COMPARISON WITH EXPERIMENTAL RESULTS It is of interest to compare the results for f12 (equation (41)) and for P,L (equation (25)) with experimental data. However, the author is not aware o f a n y systematic measurements of damping of even modes in centrally vented tubes, so that a comparison of f12 alone is not possible. On the other hand, Perry [6] has recently reported extensive measurements of limiting acoustic oscillations in T-burners, so that an overall comparison with experiments is possible. Perry's measurements were made at different mean pressures and with different propellants in T-burners with different diameters. The most complete set of measurements was made with A-13 propellant at a mean pressure of 21.4 atm (300 psig) in a 3.81 cm-diameter T-burner. The length, L, of the burner could be adjusted so that longitudinal waves with frequencies in the range 300-6400 Hz could be excited in the burner. The length of the vent, l, was about 12.7 cm for the low-frequency runs (up to 2600 Hz) and about 15-4 cm for the higher frequencies, and its diameter was 1.017 cm. From Perry's data, one can obtain the values of
3 !6
s. TEMKIN
P1L and % for various frequencies. Now, according to equation (25),
(%lf)"=
n\f ]
where f12 is predicted by equation (41). In order to provide a comparison of the analytical results with Perry's data, the experimentally determined values of (P~L/Po)[(%lf) 1t2 versus IlL are plotted in Figure 4. The quantity IlL is equivalent to a non-dimensional frequency.
I
i
i
I
I
I
I
f 06
i 07
i
I
//-
2C
~
L s
t y
-
o
I 00I
I 02
1
O-:3
I
04
\ I
0.5
i 0 8
iff.. 09 I0
t/L
Figure 4. Comparison of equations (25) and (41) with Perry's experimental data.
The open circles shown in the figure represent the average of several tests, and the length of the bars represents the spread of the data. The solid curve shown in Figure 4 represents the quantity 27(f12]f)t12/rc from equation (41) with values of a, R and I corresponding to those of Perry's burner. Also used in the reduction of equation (41) was equation (29), with A 2 = 0.60 s-1. This value of A nearly corresponds to Perry's decay data, and is about two times larger than that predicted by equation (28). The quantity A controls the height and width of the solid curve. Thus, a higher value of A would yield a broader curve of lower maximum. However, this maximum would still be centered at ill = 1/2. The reason for this maximum is that for that value of IlL, the field in the vent is a resonant one, so that in such a condition dissipation is increased as explained above. The experimental data shown in the figure also display a typical resonant behavior with a maximum of the same order as that predicted by equations (25) and (41), and also centered at about IlL = 1/2. This indicates that dissipation in the vent is in fact responsible for the limiting condition when ilL ~.. 112. However, it can also be seen that while the theoretical curve is very sharp, the experimental one is not. Several factors may account for this behavior, such as kinetic energy losses due to turbulence, mean flow losses, etc., but it appears that the dominant one is that outside a narrow frequency band near 1/L = 1/2, the vent does not dissipate much energy. That is, outside that narrow band f12 ~ fl2,,.b.Therel'ore, the assumption (fl2/~g)u2 < 1 used in deriving equation (25) is not satisfied, because % is of the same order of magnitude as fl2wb.This would imply, for example, that higher-order modes are present which may account for significant energy dissipation outside the resonant condition. In other words, the assumption that only one harmonic is excited is not valid outside that condition.
OSCILLATIONSIN T-BURNERS
317
It is of interest to point out that there are data published in the literature which seem to indicate that the vent's role is, in general, a sinall one. For example, Temkin [9] in a study of non-linear gas oscillations in a cylindrical tube observed that a small orifice ( i l l = 0"02), located near the middle section of the tube, had little effect on the total pressure amp!Rude. More relevant to the present discussion are the measurements made by Perry in his T-burner and which were directed at finding whether the vent had any effect on the limiting amplitudes. He reports using a vent with an opening larger than that in most tests (but apparently with the same length), and observing no change. These results are in complete agreement with the model described here, since outside the vent's resonant condition the model predicts that the vent has no effect on the acoustic field in the burner cavity. On the other hand, on the basis of this limited number of tests, Perry concluded that "the fact that no differences in the data were observed supports the generally held belief that the vent, located at the pressure node of the oscillation, plays little, if any, role in the T-burner tests". This is in general the case, but in view of the present results, and of the trend agreement shown in Figure 4, it appears that near IlL --- I/2 the vent plays an important role in T-burner tests. ACKNOWLEDGMENTS The author is grateful to Dr M. M. Ibiricu for many helpful discussions concerning this work. This work was supported by the U.S. A r m y Ballistic Research Laboratories under Contract DA-0341. REFERENCES
1. M. D. HoRToN1964 Journal of the American lnstitute of Aeronautics and Astronautics 2, 1112-1118. Use of the one-dimensional T-burner to study oscillatory combustion. 2. S. WrINER 1966 Journal of the Acoustical Society of America 40, 240-243. Standing sound waves of finite amplitude. 3. S. TEMK1N ~969 Journa~of the A~ustica~ Society ~f America 45~224-227. Pr~pagating and s~anding sawtooth waves. 4. F. E. C. COLICK1971 Combustion Science and Technology 3, 1-16. Non-linear growth and limiting amplitude of acoustic oscillation in combustion chambers. 5. F. E. C. CULICK 1972 American Institute of Aeronautics]Society of Automotive Engineering 8th Joint Propulsion Specialist Conference, New Orleans, Louisiana, November 29-December I, Paper No. 72-1049. Research on combustion instability and application to solid propellant rocket motors, II. 6. E. PERRY1970Ph.D. Thesis, California Institute of Technology, University Microfibns No. 70-24577. Investigation of the T-burner and its role in combustion instability studies. 7. L. D. LANDAUand E. M. LtrsmTz 1959 Fluid Mechanics. London: Pergamon Press, Ltd. 8. S. TEMKINand R. A. DoanINs 1966 Journal of the Acoustical Society of America 40, 1016-1024. Measurements of attenuation and dispersion of sound by an aerosol. 9. S. TEMKIN1968 The Physics of Fluids 11,960-963. Non-linear gas oscillations in a resonant tube.