NASA Technical Reports Server (NTRS) 20020023132: On Pulsating and Cellular Forms of Hydrodynamic Instability in Liquid-Propellant Combustion

12 Pages·0.69 MB·English

by  NASA Technical Reports Server (NTRS)

Twtmty-Seventh Symposium International) on Combustion/The Combuslion_nstitute, 1998/pp, 237,5-2-386
ON PULSATING AND CELLULAR FORMS OF HYDRODYNAMIC
INSTABILITY IN LIQUID-PROPELLANT COMBUSTION
STEPHEN B.MARGOLIS
Combustion Research Facility, MS 9052
Sandia National Laboratories
Livermore, CA 94551-0969, USA
An extended Landau-Levieh model of liquid-propellant combustion, one that allows foralocal depen-
dence of the burning rate on the (gas) pressure at the liquid-gas interface, exhibits not only the classical
hydrodynamic cellular instability attributed to Landau but also apulsating hydrodynamic instability asso-
ciated with sufficiently negative pressure sensitivities. Exploiting the realistic limit of small values of the
gas-to-liquid density ratio p, analytical formulas for both neutral stability boundaries may be obtained by
expanding allquantities in appropriate powers ofp ineach of three distinguished wave-number regimes.
In particular, composite analytical expressions are derived forthe neutral stability boundaries Ap(k), where
At, isthe pressure sensitivity of the burning rate and kis the wave number of the disturbance. For the
cellular boundary, the results demonstrate explicitly the stabilizing effect of gravity on long-wave distur-
bances, the stabilizing effect of viscosity (both liquid and gas) and surface tension on short-wave pertur-
bations, and the instability associated with intermediate wave numbers fornegative values of Ap,which is
characteristic of many hydroxylammonium nitrate-based liquid propellants over certain pressure ranges.
In contrast, the pulsating hydrodynamic stability boundary isinsensitive togravitational andsurface-tension
effects but is more sensitive to the effects of liquid viscosity because, for typical nonzero values of the
latter, the pulsating boundary decreases tolarger negative values ofAj_askincreases through O(1)values.
Thus, liquid-propellant combustion ispredicted to be stable (that is, steady and planar) only fora range
of negative pressure sensitivities that lie below the cellular boundary that exists for sufficiently small
negative values ofAp and above the pulsating boundary that exists forlarger negative values of thisparam-
eter.
Introduction results of Landau's study, along with a subsequent
paper by Levich [3] that replaced the effects due to
The stability of liquid-propellant deflagration is a
surface tension in the earlier study with those due
fundamental problem that was first treated by Lan- to (liquid) viscosity, have been widely quoted and
dau in a classical study [1] that introduced the con-
offer much in terms of physical insight into the na-
cept of hydrodynamic instability in a combustion
ture of this type of instability, which, as in the case
context. Referred to as the "slow combustion of liq-
of gaseous combustion, is associated with the density
uids," that analysis is most applicable to certain re- change across the reaction front. However, because
alistic, limiting cases in which combustion may be
these models assumed a constant normal burning
approximated by an overall reaction at the liquid- rate, it has proven useful to improve upon them by
gas interface. For example, the gas flame may occur incorporating amore realistic coupling oft.he normal
under near-breakaway conditions, exerting little propagation speed with the local pressure and tem-
thermal or hydrodynamic influence on the burning perature fields, thereby allowing for alocally varying
propellant; or distributed combustion may only oc- burning rate [4,5]. One result that has emerged from
cur in an intrusive regime such that the reaction this generalization is that, in addition to the classical
zone lies closer to the liquid-gas interface than the Landau (cellular) type of instability, the models now
length scale of any disturbance of interest; or the predict apulsating hydrodynamic instability as well.
liquid propellant may simply undergo exothermic The latter arises specifically from the local pressure
decomposition at the surface without any significant coupling and thus may be physically achievable be-
distributed combustion, such as appears to occur in cause the mass burning rate of many propellants has
some types of hydroxylammonium nitrate (HAN)- been shown empirically to correlate well with
based liquid propellants at low pressures [2]. The pressure. Thermal coupling, on the other hand,
2375
2376 PROPELLANTS
introduces additional thermal/diffusive instabilities p_ -p. = f,,.[pv,(t,_.v_) - v (f_.v_)
[5] that will not be considered here. In the present
work, we shall consider both types of hydrodynamic - p;tPrge. •fi_ + Prte_"fis]
instabilities, but particular focus will be placed on
the pulsating stability boundary, because this type of
+ fis.(v_ - av+) S(,/'2 o¢,
instability isabsent from the earlier models that ne- ot
glected the pressure coupling previously indicated.
Asinacompanion study that focused onthe classical
(cellular) hydrodynamic instability [6],we develop a lax 2 \ Oy] J
formal asymptotic theory byconsidering the realistic
limiting case in which the gas-to-liquid density ratio
+ 0_" 1 + -
p issmall.
(4)
f
)
Mathematical Model fi_ × [pv+ (fi_.v÷) - v_ (fi,.v
The governing hydrodynamic equations consist of
+(v_- pv+)S(_O -_]
mass and momentum on 6ither side ofthegas-liquid
interface, supplemented by a pressure-dependent
burning-rate lawand associated continuity andjump = 6s X (p2Prge._ "fis - Prte-'fas) (5)
conditions across the interface [4--6]. Thus, it is as-
where the latter represent continuity of transverse
sumed, as in the classical models, that there is no
velocity components (no slip), conservation of (nor-
distributed reaction in either the liquid or gasphases
mal) mass flux, the mass burning-rate law,and con-
but that there exists either a pyrolysis reaction or an
servation of normal and transverse components of
exothermic decomposition at the liquid-gas inter-
momentum flux, respectively. Here, vand pdenote
face that depends on the local pressure. For sim-
velocity [with respect to the original (xl, x2,x3) co-
plicity, it is assumed that within the liquid and gas
ordinate system] and pressure, and the + subscripts
phases separately, the density and other fluid prop-
denote evaluation atz = 0-*.The parameters Pr_and
erties are constants, with appropriate jumps across
Prg denote the liquid and gas-phase Prandtl num-
the lphase boundary. The nondimensional location of bers, p and ).are the gas-to-liquid density and ther-
the latter isdenoted byx3= ¢is(xi,x2,t), where the
mal diffusivity ratios, Fr isthe Froude number, e is
adopted coordinate system is fixed with respect to
the rate-of-strain tensor, and ?isthe surface-tension
the stationary liquid at x3 -- - ao.Then, inthe mov-
coefficient. All of these quantities are defined in
ing coordinate system x = xl, y = x._,z = x3 -
terms of their dimensional counterparts in the no-
¢_s(Xl,x2, t), in terms of which the liquid--gas inter-
menclature. In addition, the factor S(¢_,) and the
face always lies at z = 0, the complete nondimen-
unit normal fi, are given by S(¢_) -_ [1 + (OCPs/i}'"x2)
sional formulation of the problem in the absence of
+ (0¢,_/aS)2]-t_andfi_= (-0¢,/_x,-o_J_y,
thermal coupling isgiven by
1)S(¢_), while the expressions for the gradient
operator V and the Laplacian V°-in the moving
V.v = 0, z#0 (1)
coordinate system are given by V = [0/_ -
(O¢,/Ox)d/Oz, O/Oy - (OCJOy)O/Oz, O/Oz] and
Ov 0_ Ov
+ (v.V)v = (0,0, - Fr-_) V2 = O_lo_ex + ff21Oy_+ [1 + (O¢_./0x)e +
d.t Ot dz (OCJOy)2]02/0z2 - 2(O_,lOx)O21OxOz -
2(O¢,lOy)O21OyOz- (02¢,/0x 2 + 02¢_JOy2)OlOz.We
{pl} ! l remark that the factor multiplyin_y in equation 4is
-- 1 Vp + [_rgJ V2v' z X 0 (2) the curvature -V" r, and note that p2Prg = 12Prl,
where txisthe gas-to-liquid viscosity ratio.
subject to v = 0at z = -oo and the interface con-
Finally, we observe that the burning rate A(p) in
ditions
the lastofequat/ons 3isassumed, inaccordance with
numerous experimental correlations for both solid
fis × v_ = !_s )< V+ and liquid propellants [7],todepend onthe local gas
pressure at the interface, where A isnormalized to
unity for the case of steady, planar burning. Indeed,
_,.(v_ - av+) = (1 - p)S(,,) -_ in the linear stability analysis that follows, the pres-
sure sensitivity A_ = 0A/_lp =o(where, as indicated
fas.v_ - S(¢_) _ = A(p+) (3) in equation 6 in t'he following, p = 0 isthe unper-
O$ turbed gas pressure at z = 0*)of the local burning
HYDRODYNAMIC INSTABILITY IN LIQUID-PROPELLANT COMBUSTION 2377
rate will emerge as an important parameter. In fact, ¢(x, y, z, t) are defined as _, = _'(t) + 6, v =
A,, isthe only information about A(p) that will appear v°(z) + u, and p = p°(z) + if,respectively. Substi-
e_plicidy, although in a nonlinear stability analysis, tuting these definitions into the nonlinear model de-
more detailed information in the form of higher de- fined earlier and linearizing about the basic solution,
rivatives would enter into the calculations (cf. Ref. the perturbation problem becomes
[8]). We remark that, depending on the particular
propellant ,Apis not necessaril yp ositive and mayin OuI + au_ + _u3 O, z # 0 (7)
Ox Oy Oz
tact take on zero and negative values over certain
pressure ranges [2,7].
Anontrivial basic solution to the foregoing prob- plljO-ug + a_u- = _ (_ + {_1 Fr_] O-idfp,s
lem, corresponding to the special case of a steady,
planar dettagration, is given by
_= -t, v° = (0,0, v°) 0..(_+ {_} Fr_I 06, 0_)
IP ,I
v0 = (0p-I _ 1, zz<>00, + LoXt,dr 2u 02U _--_2) (8)
Oatz = -ooand
subject to u =
p°(z) = I-Fr-lz + p-t _ 1, z < 0
[- pFr-lz, z > 0 (6) 06,
Ullz= O- -- Ullz.o* = (p-I _ 1) --_--
The linear stability analysis of this solution now pro-
ceeds in a standard fashion. However, owing to the
significant number of parameters, acomplete anal- _ uel:.o , = (p-I _ 1)86, (9)
ysis of the resulting dispersion relation is quite com-
plex, and we follow our previous approach [6] by 06,
restricting further consideration to the realistic pa- - pu31...o. = (1 - p) _-
rameter regime p _ 1,/_ _ 1 and, in the ease of
microgravity, Fr-1_ 1. In contrast, the earlier clas-
06,
sical studies only considered special limiting cases (10)
u31:.o- - _ = Avfl:.o.
and/or assumptions. Thus, in the study due to Lan-
dau [1], viscosity was neglected and the effects of _1:. 0- - _1=.0. = 2(ual..0. - u31=.0- )
gravity (assumed to act normal to the undisturbed
+
planar interface in the direction of the unburned liq-
Ou.3 _ _u3
uid) and surface tension were shown to be stabiliz-
ing, leading to a criterion for the absolute stability
for steady, planar deflagration of the form (in our
nond/mensional notation) 47Fr- lp2/(l - p) > 1.In {a26_ a2_,]
- e\-_-T + -_T/ (Ii)
the study due to Levich [3], surface tension was ne-
glected, but the effects due to the viscosity of the Oul Ou3
liquid were included, leading to the absolute stability
criterion Fr- ]Prl(3p)3/2 > 1,Thus, these two studies,
both of which assumed a constant normal burning Ou] &t3
rate (A = 1), demonstrated that sufficiently large
values of either viscosity or surface tension, when
coupled with the effects due to gravity, may render
steady, planar deflagration stable to hydrodynamic
disturbances. In our recent study [6], these results
were synthesized and extended to the more realistic - er_ -_-_-0- + _-:-o- = o (13/
case of anonconstant burning rate (i.e., Ap # 0) in
the limiting parameter regime identified earlier. In where equations 8and 10have been used to simplify
the present work, we summarize these results for the equations 11-13.
classical cellular boundary and use the resulting scal- Nontrivial harmonic solutions for 6. u, and _,pro-
portiona] to eia+'k_x+ik,, that satisfy equations 1and
ings to derive an expression for the pulsatin_hydro-
dynamic stability boundary that arises from the pres- 2 and the boundary and boundedness conditions at
= +__aregivenby
sure dependence of the local burning rate.
_b,_ = eh"C'*'tklx+tk._l
Linear Stability Problem
With respect to the basic solution, equation 6, the ( = d,a÷,j,,,+a,._ _btek= - Fr-1, z < 0 (14)
perturbation quantities 6s(x, y, t), u(x, y, z, t), and [b2e -k: - pFr-] z > 0
2378 PROPE LLANTS "
It I = _d,,_*iklx+tk_l (ttPrtr - 1)b6 +
.Ib3e'lz - ikl(io9 + k)-lble k=, z < 0 (2k/xPrt + 1)(imp - k)-lik._b2
[b4e'=- ikl(iogp - k)-lbze -k:-, z > 0 (15)
+ ik,2ttPrlb8 + (1 - Prtq)bg
U2 = el,Ot+iklx+ik_
+ (2Prtk - 1)(io9 + k)-tik2bj
._b._e't= - ikz(io9 + k)-Ible kz, z < 0 (16)
[b6e'=- ikz(icop - k)-lbz, e-k:, z > 0
_ ikzertbr = (p-l _ 1)ike (25)
U3 _ ehot÷iklx+ik._.y Although the foregoing problem is linear in the co-
efficients bl-bs, explicit expressions for the disper-
sion relation im(k) and the neutral stability bound-
• bbTseeq=:" +- kk((iioog9p) +- kk))--llbbt_eekZ,-k=, zz >< 00 (17) aries are not readily obtainable in closed form,
where the foregoing solution has been normalized except in certain special cases as noted later. How-
by setting the coefficient of the harmonic depen- ever, it is possible to develop tractable perturbation
dence of q_, to unity. Here, the signs of kt and kz expansions for these quantities in the realistic limit
may be either positive or negative, and we have em- that the density and viscosity ratios pand/_ are small,
ployed the definitions k = (k_ + k_) 1/2,and q and r as is Fr- _in the case of reduced gravity.
are defined as 2erlq = 1 + [1 + 4Prl(io9 +
Prtk_)] 1/2 and 2pPrtr -- 1 - [1 + 4#Prl(io_ p +
l_er_k2)]jl_. Asymptotic Analysis of the Cellular Stability
Substituting this solution into the interface con- Boundary
ditions 9-13 and using equation 7 for z X 0 yields
It turns out that the p ,_ i limit implies the exis-
nine conditions for the eight coefficients bt-b s and
tence of several different wave-number regimes [6],
the complex frequency (dispersion relation) too(k).
which in turn implies several different expansions for
In particular, these conditions are given by
the dispersion relation. This is motivated by consid-
iklb3 + ik_b5 + qb7 ering the solution of equations 18-25 in the limit of
zero viscosity (p = Prt -- 0), which leads to a trac-
= iklb4 + tkob_ + rbs = 0 (18) table form of the dispersion relation for arbitrary p.
In particular, the neutral stability boundaries with
b3 - ikl(ia_ + k)-tbj - b4 respect to infinitesimal hydrodynamic disturbances
proportional to e_t*-_, where k and x are the trans-
+ tkl (iogp - k)-lb_ = (p-t _ 1)ik1 (19) verse wave number and coordinate vectors, respec-
bs - ik2(i_o + k)-tbl - b_ tively, are given for Av < 0 by [5]
AP=
+ ik2 (imp - k)-lb.z = (p-t _ 1)ik2 (20)
b7 - k(io9 + k)-lbt - pbs p(1 - p)Fr -] + p_,k_ - (1 - p)k
Pf(3 - a)Fr-1 + a°-_,_k + (1 - a)(2 - a)k
- pk (io9p - k)-lb2 = (1 - p)io9 (21)
--<0, 09 =0
b7 - k(io9 + k)-lbl - Apb2
(26)
-- io9 - pFr-lAp (22) and
[1 + k(io9 + k)-l(2kPrt - 1)]bt Ap = -#/(1 - p), fob = k [(1 + p)
- [1 + k(iwp - k)-1(2kltPrt + 2 - p)]b_ •(1 - p)-_Fr -_ + k2(1 - p)-t_, + k/p] (27)
where k -- Ikl. For A, > 0 the basic solution is
+ (1 - 2,Prlq)b7 - (9 - p - 2,uPrtr)bs
y
always unstable, and thus steady, planar combustion
= (1 - p)(Fr-_ - i_o) + _,k_ (23) is only stable in the region A., < 0 that lies between
these two curves (Fig. 1). Tl_e first of these bound-
(#erlr - 1)b4 + (2kltPrt + 1)(t¢op - k)-liklb.2 aries is acellular boundary (o9 = 0) that corresponds
to the classical Landau instability. Indeed, in the
+ ikllzPrtbs + (l - Prtq)b3 + (2Prtk - 1) limit Ap = 0, corresponding to a constant normal
burning rate independent of pressure, Landau's clas-
(io9 + k)-liklbl - iklPrtb 7 =. (p-I _ 1)ikl sical result is recovered since in the limit that ),Fr- _
(24) approaches the value (1 - p)/4p _-from below, the
HYDRODYNAINMSICTABILIINTLYIQUID-PROPELCLOAMNBTUSTION 2379
N0 NEUTRAL STABILITY BOUNDARIES
unstable
°IA(i_.,,0)
.1 ((1-p)(pT)-',O)
_''_.0... Pr-t_ O,7= 0
.._J.. / J_'lr'l mZ " 0 _10
......................................................
0
stable
o
'7
_0
Co,-_l-p)")
unstable (pulsating)
F[c. 1.Hydrodynamic neutral sta-
bility boundaries in the limit of zero
viscosity.
cellular stability boundary recedes from the region Fr- lcorresponds to areduced gravity limit, whereas
Ap < 0. The pulsating stability boundary (27), on the the first definition indicates the normal gravity case.
other hand, only occurs for nonzero Ap and, hence, Equation 26 then suggests three wave-number
was not predicted by the classical theories that as- scales; an inner (small) scale ki, the outer O(1) scale
sumed aconstant normal burning rate. We note that k, and a far outer (large) scale kf = ktL where the
zero and negative values of Ap over certain pressure inner scale is defined as ki = k/e for Fr-l _ O(1)
ranges are characteristic of the HAN-based liquid and kt = k/e2for the reduced gravity limit Fr- 1
propellants mentioned earlier [2]. O(e). Thus, in each of these regions, the correspond-
The formalism necessary to analyze the fully vis- ing leading-order expressions for A_ are deduced
from equation 26 as
cous problem issuggested by the fact that, for small
p, the cellular boundary (26) has different limiting
forms dependent on the relative magnitude ofk with
Aj*01-- /P*(P*g - k_)/2k_
respect to p. Thus, based on characteristicparameter {p*(p*g* - kj)/2k_
values, we introduce abookkeeping parameter t ,_:
1 and define the scaled parameters g*, p*, p*, and
_(o) - 1_p,, _(f) - I_p.{p,rk_- _) (28)
A_ according to p = p'e, p = It*e, Ap = ApS, and
either Fr- l = g or Fr- 1 = g'e, where ? and Prl =-
Pare regarded as O(1) and the scaling for Ap_is mo- and hence a uniformly valid composite expansion
tivated by equation 26. Here, the second scaling for A_(_)(k) may be constructed as
2380 PROPELLANTS
AT,<,_-AT,_'_+ _,,A*¢+"'A*/r>
r _la* ekzi {et._5 ++ ...
- lim ATI_/ - lira A*If>
k_ k£-,0 v q - (2P) -J + ek} + ... (33)
1 P* + 1 ep*2yk + [ep-2g/2k (29) Ibl'' +bl i=2,8
bi - [bll Ie + blel e"2+ "'"
where the definitions of ki and k£have been used to
express the final result in terms of k, and the upper t,, [bl___ + b131e3 ...
and lower expressions for A*(_ correspond respec-
tively to the normal and rec_uced gravity limits de- i = 1, 3, 4, 5, 6, 7 (34)
scribed earlier. In terms of the original unscaled pa-
rameters, equation 29 becomes 2Av _ -p + 7p2k Ap = A._*(pi) e- e(a*,l + a_Ioe + ...)
+ Fr- lp2/k, which, in the parameter regime consid-
ered, isa leading-order asymptotic representation of [,o*(p*g - ki)/2ki
the exact relation 26. It is readily deduced from A_(_I _ [p*(p*g* - kt)/2k, (35)
equations 28 and 29, as discussed in further detail
elsewhere [6], that surface tension stabilizes large (ill) k = kr/e
wave-number disturbances, while gravity stabilizes
r _ r(_i) e-1 + ""
small wave-number perturbations. In the reduced-
gravity limit, the minimum in the stability boundary
is thus shifted to smaller wave numbers, and thus 1 - (1 + 4/_*_P_k]) v_
r_-_l = 2#*P
the hydrodynamic cellular instability becomes a
long-wave instability phenomenon (Fig. 2).
q _ ql_il* -_ + "", q¢_i) = kf (36)
Corresponding results may be obtained for the vis
cous case. However, to deduce the asymptotic b_-bl_-_lt-_ + bl°>+ "", i = 1,3,5,7
forms of both the cellular and the pulsating stability
boundaries, it is preferable from the standpoint of
b_ _ b}-21e-a + bj-l_ e-I + "-
tractability to introduce the scalings introduced pre-
viously and appropriate perturbation expansions for t = 2, 4, 6, 8 (37)
the coefficients b_directly into equations 18--25 for
each wave-number regime, and to obtain the neutral Ap = A_Ille_ _.(A_Il)l + A_IIIe + ""); A*_II- -p*
stabi//ty boundary in question from the expanded
form of those equations. Thus, for the cellular 2p*_*ell+kd/p*_+_*p + _o*e)]
+
boundaryA*(k) [6], we deduce the following expan- _u*e(1 + p*ekr)- [_ - (1 + 4u*_e_])_'_](p*_ + _u*e)
sions and results for each wave-number regime. (38)
(i)k _ O(1) We observe that the leading-order results 32 and
35 are equivalent to the corresponding inviscid re-
r _ rze + ..., rt = la*Pk 2 suits 28. Thus, to leading order, neither the inner
nor the outer wave-number regimes are influenced
q _ qo + "'" by viscous effects, which, to a first approximation,
are only significant for large wave-number distur-
qo = (2P) -1 [1 + (1 + 4P2k 2)1/2] (30) bances. This is reflected in the leading-order ex-
pression for the cellular stability boundary given by
b,- + + + ...
equation 38, where, among other features, it isread-
ily observed that both the liquid and the gas-phase
i = 2, 8; bi --bl°) + bt11e + "" viscosities (through the parameters P and/a'P, re-
spectively) enter into this expression, reflecting an
i = i, 3, 4, 5, 6, 7 (31) equal influence of viscous and surface-tension ef-
fects on the neutral stability boundary in the large
Ap = A_(') e-- e(A_(''_ + A"{I'')e + '") wave-number regime. The equal importance of gas-
phase viscosity relative to that of the liquid phase
A?/o_= - p*/Z (32)
stems from the fact that gas-phase disturbances are,
according to equations 37, larger in magnitude than
those in the liquid phase, such that aweak damping
of alarger magnitude disturbance is as significant as
HYDRODYNAINMSICTABILIINTLYIQUID-PROPELCLOAMNBTUSTION 2381
HYDRODYNAMIC STABILITY BOUNDARIES (p << 1)
°1A_ Inviscid Case (P = O)
l(ep'er-Lo) unstable k ((cp'_,)-'.o)
i l i i i i i _1 i !
t s a 4 s e v' __to tt
?-
unstable {eellular)yo
0'
I
.'._. t_ ...................P.r.-.t.#.O....7.=.=.0...... u,=O
o,-p'_) Pr-' ="r=0 ,-,=0
r- ,
stable
FIG.2. Asymptotic representation
of the cellular hydrodynamic neutral
stability boundary in the limit ofzero
viscosity. The upper (/ower) solid
curves correspond to the two cases
O.-p') t# _ 0 described byequations 29fornormal
and reduced-gravity regimes, re-
spectively (curves drawn for the
unstable (pulsating) caset -- 0.04, p* = 1.0, g = 2.5,
7
g* = 1.0).
an O(1) damping of a smaller magnitude distur-
A_,_> _ -p +
bance. In the limit P -->0, the inviscid expression 28
i.s... recovered. It xseasily shown that hmk_ Ao*(J) = 2p_e[1 + k(pr + _ue + 2pe)]
- p*/2, so that the far outer solution can be matched
to the outer solution 32. Indeed, a uniformly valid
composite expansion spanning all three wave-num-
ber regimes may be constructed as in the inviscid + -_Fr -I
case, giving the result (4o)
The cellular stability boundaries, based on equation
A_(c) _ -p* +
39, are reproduced in Fig. 3, where only those por-
2p.p*e[] + _k(p*r + _*e + gp'e)] tions of the curves that lie in the region A_ --<0 are
4p*/'(1 + d<p*e)- (p*), ÷ 2p*/')[] - (1 + 4p'2i_k_) =_] shown. For sufficiently small positive values of Ap, it
may be shown that there always exists a positive
(real) root ico of the dispersion relation, which im-
+ 2k [_g" plies that this region is intrinsically unstable. We
(39) note from equation 38 that as kf increases, A_(f )in-
creases, intersecting the A*(J0 = 0 axis at the value
or, reverting to unsealed parameters, ky= (p*r)-_[I - p*e/(p*_ + 2p'e)], which agrees
2382 PROPELLANTS
-. HYDRODYNAMIC STABILITY BOUNDARIES (p << I)
(tp'_-',o) unstable k ((tp'?)-'.o)
d_
11
Fxc. 3.Asymptotic representation
ofthe cellular hydrodynamic neutral
stability boundary for the viscous
case. The upper and lower sets of
curves correspond to the normal and
reduced-gravity regimes, respec-
tively, inthe asymptotic limit consid-
ered in this work (curves drawn for
the case _ -_.04, p* = 1.0,g = 6.0,
g* = 2.0). The solid curves corre-
spond to the inviscid limit (P = 0)
with nonzero surface tension (? =
2.5). The dash-dot curves corre-
spond to nonzero surface tension (?
= 2.5) and liquid viscosity (P = 1.0)
but zero gas-phase viscosity _*P --
0). The dash-dot-dot curves differ
from the dash-dot curves bythe ad-
dition ofgas-phase viscosity _*P =
1.0)and are similar to the dash-dot-
dot-dot curves, where the latter cor-
respond to larger viscosities (P =
_u*P -- 2.0). The dash-dot-dot-dot-
dot curves correspond to a viscous
case (P --/_*P = 1.0) but with zero
surface tension (? = 0), sothat, from
equation 38, the curves donot inter-
cept the A_ = 0 axis.
with the inviscid result in the limit ,u*P _ 0. It is of the hquid--gas interface but also inits formal treat-
readily seen from Fig. 3 that, as in the inviscid case, ment of those processes (surface tension, liquid and
the essential qualitative difference between the gas-phase viscosity) that affect damping of large
normal and reduced-gravity curves is the location of wave-number disturbances.
the critical wave number for instability. Specifically,
the minimum in the neutral stability boundaries oc-
curs for O(1) values of k under normal gravity, and Asymptotic Analysis of the Pulsating Stability
Boundary
at k _ O(e ]_) in the reduced-gravity limit considered
here. It is also clear from Fig. 3 that increasing the As indicated previously, the existence of anonsta-
values of any of the parameters P,p'P, or _,serves tionary pressure dependence on the burning rate
to shrink the size of the unstable domain through (i.e., Av # 0 leads to the prediction of a pulsating
damping of short-wave perturbations. The non-neg- hydroc[ynamic stability boundary that isabsent when
ligible effects of gas-phase viscosity represents an such apressure coupling isneglected, as inthe origi-
important correction to Levich's original treatment nal Landau-Levich theories. In the inviscid case,
[3] in which these effects were simply assumed to this boundary (equation 27) is astraight line that lies
be small. The results 39-40 thus synthesize and sig- below the cellular boundary discussed earlier, but
nificantly extend the classical Landau-Levich results this is modified under the influence of viscosity, as
[1,3], not only in allowing for adynamic dependence we shall demonstrate.
of the burning rate on local conditions in the vicinity For the scalings adopted in the preceding section
HYDRODYNAINMSICTABILIINTY
LIQUID-PROPELLANT COMBUSTION 2383
[inparticulafor,r
P - O(1),/z _ O(e)], it turns out equations 18--25 give redundant results. Combining
that, unlike the cellular stability boundary for which equations 46, we obtain
viscous effects only have aleading-order effect inthe
far outer wave-number regime, the effects of viscos- (itoo)2 = (k2/p*)(1 + 2A_,/p*) (47)
ity have a leading-order effect on tile pulsating and thus (ito0)2<>0 for A* ,_ -p*/2, which essen-
boundary for O(1) wave numbers as well. Thus, in tially recovers the leading-order cellular stability
the outer wave-number region, we seek a solution boundary (32) for O(1) wave numbers, but gives no
for the dispersion relation in the form ito - _- 1/2(ico0 information on the pulsating boundary because im0
+ iohe TM + io)_ l_z+ " "), where the leading-order is purely imaginary forA_ < -p*/2. Hence, stabil-
term is suggested by the explicit results for the in- ity in the latter region isdetermined byhigher-order
viscid case [5,6], and the expansion in powers ofe TM coefficients in the foregoing expansion for i_o.
is suggested by the expansions for r and q given be- Continuing with the analysis of the expanded
low equation 17, which have the form forms of equations 18-25, we obtain the second-or-
der equations and results
r _ r(i/2)t 1_ + r(.3/4)6 "3/4 + ¢-18 + --.
ik) b_-1/4) + ik2b.(5-1/4) + q(-],'4) b_7°)
r<t/z) = -io%p*, r(_4) = --icolp*
+ qo b¢7-1/4) = 0 (48)
r1= -ito2p* - (g*Pk)_ (41)
ikl b(4-v4) + ikzb(6-1/4) + rll_ )b(s-3/4)
q _ q(_l/4)e -1/4 + qoe° + ...
+ r(_4)b_s-t) = 0 (49)
q(- t/4) = (i¢°o/P) vz
J,b_-,,_>- q(__,,h_7-,,,_= ia,d,(t - A_/p')
q0 = (2P) -1 [1 + iCOl/(io_o/P)v2] (42)
b_-_ - (k/iO_o-)ab'__ = io_ (50)
Corresponding expansions for the coefficients bt in
equations 18-25 are determined as
bi = bl-1)e-I + bi-3/4)e -.3/4 = b_s-t/_) = 0 (51)
+ bl-l_)t-1/2 + ..., i = 1,2, 8 (43) where the first of equations 50 was obtained from
the sum of equation 19 multiplied by ik_ and equa-
bj = b)-tn-)t-l?z + b(-l/4) e-t/4 + ... tion 20 multiplied by ikz and the first of equations
i
45, the fact that b(-_4) = 0 follows from the differ-
i = 3,4,5,6; ence of equations 21 and 22, and the last two of
equations 51 follow from equations 24 and 25in con-
b_ = bl-V')e -_f4 + bl°>e0 + ..., i = 7 (44) junction with the result b_-'w4) = 0. From these re-
suits and the first of equations 45, we thus conclude
where the leading terms in the expansions for bi, b2, that
b4, b6, and bs are consistent with the inviseid results
[5] and the remaining coefficients appear only for b(7-1/4) = i_o_ = 0,
nonzero values of P and are conservatively postu-
lated to have the indicated expansions. Substituting bl-_/_t = imo (1 - A_/p*) (52)
2
these expansions into equations 18--25 and equating
where the fact that i_ = 0 implies the need to
coefficients of like powers of e, we obtain the lead-
ing-order equations and results calculate ico_ to determine stability in the region
A_*< -p*/2. Proceeding in this fashion, we obtain
ikj b_- l_z) + ik_ b._-l_) + q__v4) b_- v4) = 0 from the next-order versions of equations 21 and 23,
the difference of equations 21 and 22, and the sum
= ikl bC-41/2) + ik_ b_-l_z) + r()/z)bg 1 of equation 24 multiplied by iki and equation 25
(45) multiplied by ik_, the relations
b_O_- (l_/i_lb_-_) +
tC" + b(_-" - _(s-" = o
b[-" = -k/p*, b_-_>= (i%)_/k [(i¢o_+ k)/(i_oo)_lk b_-t)
- p*b[-_) + p*b(_-_) = itos (53)
b¢s-l) = _ (k/p*)(1 + A_/p*) (46)
where the last of equations 46 was obtained from bl -_,_) + (k/ia_o)(2Pk - 1)bt-_) + b[-_/e)
the leading-order difference of equations 21 and 22,
and the remainder of the leading-order versions of + (2icoop*/k)b[ -l) - 2b(8-_tu) = -i_o (54)
2384 PROPELLANTS
_. HYDRODYNAMIC STABILITY BOUNDARIES (p << 1)
°].
Ap Viscous Case (P > 0)
unstable
o, ,_'_,-,.o) k (C_l-'.o)
,,,_':,;._., .._" .
_" (o,-p'/z)
stable
. I(o,-o') pulsatinl[boundaries (co_ O)
! " "_ "*_, *', '............ P=.I
\\ \, ""--...... .....
\ \ \, ...........
\ \. \ _b,e .......£.-.25
_ \ \\ "-_ _,)_ ........
\ \ -
\ \ _P **.5 Flc. 4.Asymptotic representation
of the pulsating hydrodynamic neu-
unstable \ "_" tral stability boundary forthe viscous
'7 case (P> 0). The region between the
... \ pulsating and cellular boundaries
\ \. (the latter are shown onanexpanded
x \ scale in Fig. 3) is the stable region
with respect to hydrodynamic insta-
bility.
-p*b_s-l/21 + p*b_-l_zl + (iooop*_-/k)b_-ll i_oz--k[(A_/a*-)_ziok- I] (58)
Equation 58 isthe desiredresult,from which we
(55) conclude thatio2__ 0 for(A_/p*)"2_ 1 + 2Pk.
Thus, in the regionA_ < 0,i¢o_vanisheson the
-iO2op*bC8-1) + kb_z-l_'z) + io2op*b_-1)
boundary
+ icool-,_(oz_ek- 1)(kz/i_)b=_-0_ (56)
A_ _ -p*(1 + 2Pk) _/_ (59)
which, when combined with the expressions for which is apulsating boundary (Fig. 4) because, from
b_-l), b_-l), b_8-1),b_-l/2), and imo given earlier, con_ equation 47, ito0ispurely imaginary along this curve.
stitute four equations for the four unknowns Equation 59 is valid for O(1) wave numbers, but
b_-u_), b(0t7,b_-s_), and io)_. Solving these simulta- because it matches to the leading-order inviscid in-
neous equations, we thus obtain ner pulsating boundary A_ _ -p* as k -_ 0 and
becomes large in a negative sense as k becomes
b_O) = - 2Pk _. large, it is clear that equation 59 represents the pul-
sating boundary for arbitrary wave numbers. That is,
for P _ O(1), the effects of (liquid) viscosity on the
pulsating boundary are, to a first approximation, ab-
b_-*/_ = [1 - (A_,/p*)Z]icoo (57) sent for small wave numbers, are first felt for O(1)