NASA Technical Reports Server (NTRS) 20020031713: Hydrodynamic Instability in an Extended Landau/Levich Model of Liquid-Propellant Combustion

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/ ,_ Y i 't f)
AIAA 98-0248
Hydrodynamic Instability in an
Extended Landau/Levich Model of
Liquid-Propellant Combustion
Stephen B. Margolis
Combustion Research Facility
Sandia National Laboratories
Livermore, California 94551-0969 USA
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AIAA-98-0248
HYDRODYNAMIC INSTABILITY IN AN EXTENDED LANDAU/LEVICH
MODEL OF LIQUID-PROPELLANT COMBUSTION*
STEPHEN B. MARGOLIS
Combustion Research Facility
Sandia National Laboratories
Livermore, California 94551-0969
Abstract
of Landau instability in which one or more of these
effects were neglected. In contrast, the pulsating
The classical Landau/Levich models of liquid- hydrodynamic stability boundary is found to be in-
propellant combustion, which serve as seminal exam- sensitive to gravitational and surface-tension effects,
ples of hydrodynamic instability in reactive systems, but is more sensitive to the effects of liquid viscosity,
have been combined and extended to account for a
which is a significant stabilizing effect for O(1) and
dynamic dependence, absent in the original formu- higher wavenumbers. Liquid-propellant combustion
lations, of the local burning rate on the local pres- is predicted to be stable (i.e., steady and planar)
sure and/or temperature fields. The resulting model only for a range of negative pressure sensitivities
admits an extremely rich variety of both hydrody- that lie between the two types of hydrodynamic sta-
namic and reactive/diffusive instabilities that can bility boundaries.
be analyzed in various limiting parameter regimes.
In the present work, a formal asymptotic analysis,
based on the realistic smallness of the gas-to-liquid 1. Introduction
density ratio, is developed to investigate the com-
bined effects of gravity, surface tension and viscos- The burning of liquid propellants is a funda-
ity on the hydrodynamic instability of the propa- mental combustion problem that is applicable to var-
gating liquid/gas interface. In particular, a com- ious types of propulsion and energetic systems. The
posite asymptoticexpression,spanning threedistin- deflagration process is often rather complex, with va-
guishedwavenumber regimes,isderivedforboth cel-
porization and pyrolysis occurring at the liquid/gas
lularand pulsatinghydrodynamic neutralstability interface and distributed combustion occurring ei-
boundaries Ap(k), where Ap isthe pressuresensi- ther in the gas phase 1 or in a spray. 2 Nonetheless,
tivityof the burning rateand k isthe disturbance
there are realistic limiting cases in which combus-
wavenumber. For the caseofcellular(Landau) in-
tion may be approximated by an overall reaction at
stabilityt,he resultsdemonstrate explicitlythesta-
the liquid/gas interface. In one such limit, the gas
bilizingeffectofgravityon long-wave disturbances,
flame occurs under near-breakaway conditions, ex-
thestabilizingeffectofviscosityand surfacetension
erting little thermal or hydrodynamic influence on
on short-waveperturbations,and theinstabilitays-
the burning propellant. In another such limit, dis-
sociatedwith intermediatewavenumbers forcritical
tributed combustion occurs in an intrusive regime,
negativevaluesofAp. In thelimitingcaseofweak
the reaction zone lying closer to the liquid/gas in-
gravity,itisshown thatcellularhydrodynamic insta-
terface than the length scale of any disturbance of
bilityinthiscontextisa long-waveinstabilitpyhe-
interest. Finally, the liquid propellant may simply
nomenon, whereas atnormal gravity,thisinstability
undergo exothermic decomposition at the surface
isfirstmanifestedthrough 0(i) wavenumber distur-
without any significant distributed combustion, such
bances. Itisalsodemonstrated that,inthe large-
as appears to occur in some types of hydroxylam-
wavenumber regime,surfacetensionand bothliquid
monium nitrate (HAN)-based liquid propellants at
and gasviscosityallproduce comparable stabilizing
low pressures) Such limiting models have recently
effectsinthelarge-wavenumber regime,therebypro-
been formulated, 4,s thereby significantly generaliz-
vidingsignificanmtodificationstopreviousanalyses
ing earlier classical models 6,7 that were originally
introduced to study the hydrodynamic stability of a
*This paper isdeclared a work of the U.S. Government and reactive liquid/gas interface. In all of these investi-
isnot subject to copyright protection in the United States. gations, gravity appears explicitly and plays a sig-
nificantrole,alongwithsurfacetension,viscosity,
reduced gravity cases. In addition, the pressure cou-
and,in themorerecentmodels,certainreaction-
pling in the model predicts not only the cellular type
rateparameterasssociatewdith thepressuraend
of hydrodynamic instability attributed to Landau,
temperaturseensitivitieosfthereactionitself. In
but also a pulsating instability associated with suf-
particular,theseparameterdseterminethestabil-
ficiently negative burning-rate pressure sensitivities.
ityofthedeflagratiownithrespecttonotonlyclas-
Both forms of hydrodynamic instability are analyzed
sicalhydrodynamidcisturbancebs,utalsowithre-
here, leading to explicit asymptotic representations
specttoreactive/diffusivinefluenceasswell.Indeed,
of the neutral stability boundaries.
theinversFeroudenumberr,epresentinthgeratioof
buoyanttoinertialforces,appearsexplicitlyin all
gas/liquid interface:
ofthesemodels,andconsequentliyn,thedisper-
sionrelationthatdeterminetsheneutralstability
boundariebseyondwhichsteadyp,lanarburningis
unstabletononsteadya,nd/ornonplana(rcellular)
modesofburnings,,9Theseinstabilitiesthuslead
toanumbeorfinterestingphenomensau,chasthe
sloshingtypeofwavesthathavebeenobserveidn
mixturesofHANandtriethanolammoniunmitrate
(TEAN)withwater3.AlthoughtheFroudenumber
wastreatedasanO(1)orlargerquantityinthese
studiest,helimit ofsmallinverseFroudenumber
correspondintogthemicrogravitryegimeisincreas-
inglyofinteresatndcanbetreatedexplicitlyl,eading
tovariouslimitingformsofthemodelst,heneutral
stabilityboundariesa,nd,ultimately,theevolution
equationtshatgovernthenonlineadrynamicosfthe
propagatinrgeactionfront.
In thepresentwork1, 5'16weformallyinvesti-
gateboththenormal-andreduced-graviptyarame-
terregimesinthecontextof
a theory that exploits
the realistic smallness of the gas-to-liquid density ra-
tio p. The resulting analysis and results thus facili-
tate a comparison of some of the features of hydrody-
2. Mathematical Model
namic instability of liquid-propellant combustion at
reduced gravity with the same phenomenon at nor-
mal gravity. In addition, the small-p limit allows a The starting point for the present work is our re-
tractable synthesis of the separate models described cent model 4,s that generalizes classical models 8,7of a
in previous analyses to simultaneously account for reactive liquid/gas interface by replacing the simple
the effects of viscosity (both liquid and gas), sur- assumption of afixed normal propagation speed with
face tension and pressure sensitivity of the burning a reaction/pyrolysis rate that is a function of the lo-
rate on this phenomenon. In contrast, the original cal pressure and temperature. This introduces im-
analysis of Landau 6 neglected viscosity, while that portant new sensitivity parameters that couple the
of Levich 7 retained liquid viscosity, but neglected local burning rate with the pressure and tempera-
surface tension and gas viscosity. Neither of these ture fields. Thus, it is assumed, as in the classical
classical studies accounted for any variation of the models, that there is no distributed reaction in ei-
local burning rate due to pressure perturbations, ther the liquid or gas phases, but that there exists
which has recently been introduced into this type of either a pyrolysis reaction or an exothermic decom-
analysis. 4,5The latter study, in particular, presented position at the liquid/gas interface that depends on
a highly tractable analysis of the inviscid case, thus local conditions there. In its most general form, the
effectively generalizing Landau's model to account model includes full heat and momentum transport,
for a pressure-dependent burning rate. By develop- allowing for viscous effects in both the liquid and
ing a perturbation analysis in the limit of small p, gas phases, as well as effects due to gravity and sur-
we show how all the phenomena just described may face tension. For additional simplicity, however, it
be simultaneously included, for both the normal and is assumed that within the liquid and gas phases
separatelyth,edensityh,eatcapacityk,inematicvis-
liquid and gas-phase kinematic viscosities. We note
cosityandthermadliffusivityareconstantsw,ithap-
that the Froude number Fr represents the ratio of
propriatejumpsinthesequantitiesacrostshephase
inertial to buoyant (gravitational) forces, and, for fu-
boundary.
ture reference, that p)_Prg = I_Pq, where # = _g/]2l
Thenondimensionlaolcationofthisinterface
is the gas-to-liquid viscosity ratio. In addition, we
asafunctionofspaceandtimeisdenotedby
x3 = introduce the nondimensional mass burning rate
_,(xl,x2, t), where the adopted coordinate system
is fixed with respect to the stationary liquid at x3 = A(pl==o+e,l==o)=/i(t%_:_+, _l_.=_.)/_r), (6)
-00 (Figure 1). Then, in the moving coordinate
system x = Xl, y = x2, z = x3 - _s(Zl,X2,t), in which is assumed to depend on the local pressure and
terms of which the liquidigas interface always lies temperature fields in the vicinity of the liquid/gas
at z = 0, the complete formulation of the problem is interface. Clearly, A = 1for the case of steady, pla-
given as follows. Conservation of mass, energy and nar burning, but perturbations in pressure and/or
momentum within each phase imply temperature result in Corresponding perturbations
in the local mass burning rate.
v-v=0, z#0, (1) Equations (1) - (3) are subject to the boundary
conditions
00 0¢_00 {1}v2o, zX0, (2)
Ot at Oz + v" VO= A v=0, O=O at z=-oo,
(7)
Ov 0¢ _Ov O=l at z=+oo, O],= 0- =el==0+
+ (v. V)v -- (0, O,-Fr -_)
Ot at Oz (3) and appropriate jump and continuity conditions at
the liquid/gas interface. The latter consist of con-
- p-1 Vp+ V2v, z <0,
tinuity of the transverse velocity components (no-
where v, O and p denote velocity, temperature and slip),
pressure, respectively, Pri and Prg denote the liquid fi.x v_ = ft.x v+, (8)
and gas-phase Prandtl numbers, p, A and c (used be-
where v+ = viz=o._ , conservation of (normal) mass
low) are the gas-to-liquid density, thermal diffusivity
flux,
and heat-capacity ratios, and Fr is the Froude num-
ber. Here, the nondimensional variables have been
defined in terms of their dimensional counterparts _,.(v-_pv+)= (1-p)s(¢,)-_-, (9)
(denoted by tildes) as
the mass burning rate (pyrolysis) law,
v : -=-, p=_ (9=_
v _O_' ¢o- _'
(4) fi,.v_- 3(¢,)--_7- = A(pl=:o+,O1==o), (10)
¢'= X,' t=37, ='=qT' conservation of flux of the normal and transverse
components of momentum,
where 0 is the reference propagation speed of the
interface for the case of steady, planar deflagration. pl:=o- - pl==o+= _,.(v_ - pv+)s(¢,) a__/
The nondimensional parameters, some of which first
+ ,_,.[pv+(_,.v+) - v_(_,. v_)
appear below in the conditions at the gas/liquid in-
terface (at z = 0), have been defined as -pkPrge+ •*%,+ Prle- •fi,]
p= _P-l, m, = _aa, Fr = 9-_I' 7 - _zA-IU~ ,
o_, [ fo_,__]_2o¢_o_¢,,o_, }
+'-#_--y_l+to=/J az Na-_ '
(n)
(5)
fi,x[pv+(_,- v+) - v_(_,- v_)
where 55u is the unburned (liquid) temperature at
z = -oo and T_ is the adiabatic burned (gas) tem- +(v_- = (n)
perature at z -- +oo, _ is the coefficient of surface
tension for the liquid surface, and _ and _ are the Asx(pAPr_e+ •fis - Pr_e_ •As),
and conservation of heat flux respectively. Finally, we note the influence of viscos-
ity on the jump in pressure across the liquid/gas in-
fa,.(cp_Velz=o, - vole=0-) = terface through the appearance of the Prandtl num-
-v_)el +=o -v_)] bers Prl and Prg in Eq. (11). We also remark that
the momentum equation (3) suggests, as noted by a
+ [(i - cp)el,:o + _(1 - a_p)]S('_s)-_-, reviewer, a singularity in the pressure distribution at
the interface itself due to the discontinuity in veloc-
(13)
ity there. However, this singularity does not affect
where 6 = c/(I- c,_),e isthe rate-of-straitnensor
the jump in p across z = 0 that follows from conser-
(e+ = elz=0,), 7 is the surface-tension coefficient,
vation of the normal component of momentum flux,
a_ is the unburned-to-burned temperature ratio, and
Eq. (11), which is all that is required to close the
S(@s) and the unit normal fis are defined as
present model.
A nontrivial basic solution to the above prob-
s(¢,) = [1+ (o¢,/o=) _+ (ov,/ay) _]-'/=, (14) lem, corresponding to the special case of a steady,
_. = (-a,_,/a=, -O¢,lay, 1)s(,_,). planar deflagration, is given by
Here, the factor multiplying 3' in Eq. (11) is the cur-
vo=_t eO(z)={e=, z<O
vature -V •fi of the liquid/gas interface in the mov- ' I, z>0,
ing coordinate system, and the corresponding ex-
(19)
pressions for the gradient operator V and the Lapla- v° = (0,0,v°), v0 = { 0p,-l_l, zz><0, 0
cian V2 in this system are given by
P°(z) = {-F-rp-Flzr-lz, + p-l - l' zz <> OO.
o o{, o 0 o¢,a o) (15)
v = -_z oz Oz' Oy oy 0z' oz
The linear stability analysis of this solution now pro-
ceeds in a standard fashion. However, owing to the
and
significant number of parameters, a complete analy-
v== as °2 r (0' '12 (° '121 a2 sis of the resulting dispersion relation is quite com-
plex. Realistic limits that may be exploited to facil-
oz---_+-ff_y2 + L1+ \ o= / + k Oy / j -O--z_
itate a perturbation analysis of the dispersion rela-
_ 20¢_s 05 2CO@s 05 tion include p << 1,/z << 1, and in the microgravity
cOx OxOz COyCOyOz regime, Fr -t << 1. In contrast, the earlier classi-
cal studies considered special limiting cases and/or
- k-O-_-2x + ay 2 / Oz assumptions. Thus, in the study due to Landau, °
(16) viscosity was neglected and the effects of gravity (as-
However, thevectorv stilldenotesthevelocitywith sumed to act normal to the undisturbed planar in-
respecttothe(xl,x2,x3)coordinatesystem.We re- terface in the direction of the unburned liquid) and
mark that the mass burning rate A may be typically surface tension were shown to be stabilizing, leading
decomposed as to a criterion for the absolute stability for steady,
planar deflagration of the form (in our nondimen-
A(pl,,=o+,el==o) = sional notation) 47Fr-lp2/(1 -p) > 1. In the study
due to Levich, 7 surface tension was neglected, but
FN(I-
a,,)(el,,=o- ])]
A(pl,=o+,el,=o)exp L a,,+ (1- o,,)el,=o J' the effects due to the viscosity of the liquid were
included, leading to the absolute stability criterion
(17)
Fr-lPr_(3p) 3/2 > 1. Thus, these two studies, both
where .4isa ratecoefficientand N = E/R°Ta is
of which assumed a constant normal burning rate
the nondimensional activation energy (E). However,
(A = 1), demonstrated that sufficiently large values
this more explicit representation will not be needed
of either viscosity or surface tension, when coupled
in the stability analysis that follows. Instead, re-
with the effects due to gravity, may render steady,
sults may be expressed in terms of the pressure and
planar deflagration stable to hydrodynamic distur-
temperature sensitivities, defined as
bances. In the present work, these results will be
synthesized and extended to the more realistic case
Ap = cOA/cople=l,p=o, (lS) of a nonconstant burning rate (i.e., Ap _ 0) in both
Ae = coA/Oele=tm= o = N(1 - cr,) + Ae, normal and reduced gravity regimes.
(o ,21
3. The Hydrodynamic Linear Stability Problem
pAPrg \ Oz [,=o+ Oy z---o+]
With respect to the basic solution (19), the var-
(31)
ious perturbation quantities Cs(x, y, t), u(x, y, z,t), -Prl\oz Is=0- + Oy _=o-J =
((x, y, z, t) and 0(x, y, z, t) are defined as
(o-1-
I)_ +,_2l.-=ou+2}.=o-,
• , =_°(t) + Cs, v =v°(z) + u,
dO0 (20)
cpA-z0-01 o01 - cOl,=o+ + 01,=o- =
p = p°(z) + _ , O = O°(z) + O+ ¢s--_z-z •
OZ Iz=0+ _qz z=0-
Substituting Eqs. (20) into the nonlinear model (1) epu3[==o-+(1+ e)uz]==o_+ [i + a(i -p)]_O_,
- (16) and linearizing about the basic solution (19), (32)
we obtain a problem for the perturbation quantities
where a ---c(1 - _)-l.
as
In the present work, we shall focus, using our ex-
Oul Ou2 Ou3
o-E +-g_-y+-gF = 0, z#o, (21/ tended model described above, primarily on hydro-
dynamic (Landau) instability. Thus, in the linear
{l 0u ou
stability analysis, we retain only the pressure sen-
pJOt + 0--_=
sitivity Ap in the perturbation of the pyrolysis law
(10), neglecting the temperature sensitivity Ao. The
latter assumption thus filters out reactive/diffusive
instabilities associated with the thermal coupling of
Pr, (o2u O2u o2_
the temperature field, 4'5 but greatly facilitates the
analysis of instability due to hydrodynamic effects
(22)
alone. We note that the mass burning rate of many
1 00 00
propellants has been shown empirically to correlate
(23) well with pressure.
+ + + zXo Nontrivial harmonic solutions for Cs, u and ¢,
pA _-g_z: _ Oz2/ '
proportional to ei_t+ik_z+_k2_, that satisfy Eqs. (21)
- (22) and the boundary/boundedness conditions at
u=0, 0=0 at z=-o¢, (24) z = 4-0o are given by
0=0at z=+oo, 0l==0+-0l_=o- =¢,,
0¢, (25) Cs = e_t+ik'z+i_ , (33)
_l_=o-- _1_=o+= (p-_- i)_, (26) -----eiwt+iklx+ikay I blekZ --FT-1 (34)
[ b2e-_z _ pFr -1 '
"_I_=o--_,I_=o+= (1-P)-_t'" (27) ei_t+i_,z+i_ ! b3eqz - iki(iw + k)-ibie kz
Ztl
( baerz - iki (iwp - k)-ib2e -_z,
(35)
0¢, = ApCl,=o+ + Ae0l_=o+, (28)
U3]z=0- Ot ei_+,_=+,_u _ b_eq:- ik2(iw+ k)-Ibie_
_2 tb_erz - ik2(iwp - k)-lb2e -_z,
¢[==o-- Cl.=o+= (2- P)U3[z=o+- USlz=O-
(36)
Ou3
zP'_rr9--_-zm=o+ (29) u3 = e_t+_k'_+ik_ _bTeq_ - k(iw + k)-Ible_
+2Prl-_-z]_=o -' _ _ .. Ou3
( bse + k(iwp - k)-lb2e -kz,
TM
(3_)
--(1--p)--_----7\ c0z2 -4- Oy 2 ], for z X 0, where we have normalized the above so-
lution by setting the coefficient of the harmonic de-
pendence of ¢, to unity. Here, the signs of kl and
pAPrg \ Oz .=o+ + Ox z=O+/
k2 may be either positive or negative and we have
+ _-
Ou3 _ (30) employed the definition k = _, and q and r
are defined as
k Oz I.---o- 0x ,=o- ]
(p--_i)_-'-+__,I:=-o_+,I,_-o-,
2Prz q = 1+ x/'l+ 4Prz( iw + Pr_ k2) , (38)
2#Prlr = 1 - _/1 + 41_Prl(iwp + #Pr_ k2), (39) these wavenumber regimes, which can then be com-
bined using asymptotic matching principles to pro-
where we have used the fact, noted below Eq. (5),
duce a uniformly valid composite expression for each
that pAPrg = _Prl.
boundary.
Substituting this solution into the interface con-
ditions (25) - (31) and using Eq. (21) for z < 0yields
1 NEUTRAL STABILITY BOUNDARIES
nine conditions for the eight coefficients bl - bs and O
the complex frequency (dispersion relation) iw(k). Ap unstable
In particular, these conditions are given by / (_-'.o) (O-p)_)-'.o)
iklb3 + ik2b5 + qb7 = O, (40)
iklb4 + ik2b6 + rb8 = 0, (41)
,*Y**- O,71"0
\ o.o/
ikl -b4+ ikl
53
iw + kbl iwp -
\..._"
b5 ik2 kbl _ bs+ ik2
iw + iwp - ........ _.-'.r.7..'.0............ _..'..q...........................
pk
b7 iwk+kbt-pbs iwpLk b2=(1-p)iw' (44) . o,-p_-p)-')
stable
57 k kbl_Apb2=iw_pFr_lAp, (45)
iw +
[1 + iw-_(2kPr,- 1)]bl
k O¢.
- [1+ +2- p)]b (46)
iwp -
_0
+ (1 - 2PrL q)b7 - (2 - p - 2/_Pr_ r)bs = '_o,-_x-p)-')
(1 - p)(Fr -1 - iw) + 7k 2,
unstable (pulsating)
(_Prl r - 1)b4 + (2k#Pr_ + 1). ikl , b_
zwp -
+ ikl#Prt bs + (1 - Prl q)b3 - iklPrl b7 (47) Figure 2
+ (2PrL k - 1)_--+--_Ol = P - 1 ikl, Hydrodynamic neutral stability boundaries in the
limit of zero viscosity.
(/_Prl r - 1)bs + (2k_PrL + 1) b2
Uop--
4. Hydrodynamic Instability: Zero-Viscosity Limit
+ ik2#Prl bs + (1 - Prl q)bs - ik2Prl b7 (48)
+(2Pr, k-1) /_kbl= (_-l)ik2. Although both liquid and gas-phase viscous ef-
fects will be shown to be comparable in general to
While the above problem is linear in the coefficients
those due to surface tension, the inviscid limit pro-
bl - bs, the relationship for iw is highly nonlinear.
vides a particularly tractable limiting case and moti-
Accordingly, we ultimately will exploit the fact that
vates the scalings to be used in the asymptotic anal-
the gas-to-liquid density and viscosity ratios pand #
ysis of the fully viscous problem. Thus, in the limit
are typically small, as is Fr-1 in the case of reduced
of zero viscosities (Prt = Prg = 0), our extended
gravity, and seek an asymptotic representation of the
model differs from the classical one due to Landau 6
dispersion relation in this parameter regime. As we
shall see, this distinguished limit simplifies the prob- only in the local pressure sensitivity of the normal
lem by inducing three distinct wavenumber regimes burning rate. In that limit, the dispersion relation
that can be analyzed individually. Consequently, it with respect to infinitesimal hydrodynamic distur-
is possible to develop separate perturbation expan- bances proportional to ei_t+ikx, where k and x are
sions for the neutral stability boundaries in each of the transverse wavenumber and coordinate vectors,
respectiveliys,determineads
5 existence of which was a new prediction for liquid-
propellant combustion. Zero and negative values of
App2(l - p)(iw)3- p2k[I+ p + Ap(l - p)](iw)2
Ap over certain pressure ranges are characteristic of
+ pk {Ap [p(1 + p)Fr -i + pk2-y - (I- p)k] the so-called "plateau" and "mesa" types of solid
-2ok} propellants, l° as well as for the HAN-based liquid
propellants mentioned above. 3
+ Apk2[(l - p)(2- p)k+ p2(3 - p)Fr -1 + p2k2_,]
Of particular interest in the present work is
+ pk2 [(I- p)k - p(l- p)Fr -1 - pk27] = 0. the cellular hydrodynamic, or Landau, instability of
(49) liquid-propellant combustion in the limit of small
The corresponding neutral stability boundaries are gravitational effects (in microgravity, for example).
easily determined explicitly and are exhibited in Fig- In this limit, the shape of the upper hydrodynamic
ure 2. Steady, planar burning is always unstable for stability boundary in Figure 2, corresponding to the
positive values of Ap, but in the region Ap _ 0, there classical type of instability that was first described
exist both cellular (w -- 0) and pulsating (w _ 0) by Landau, 6 clearly approximates the Fr -i = 0
stability boundaries Ap(k; p, % Fr -1) given by curve except for small wavenumbers, where, unless
the inverse Froude number is identically zero, the
p [p(1 - p)Fr -1 + p'yk 2 - (I- p)k] neutral stability boundary must turn and intersect
Ap = p2(3_p)Fr_ 1+ p_?k 2+ (1 -p)(2-p)k <-O, the horizontal axis. Consequently, the neutral sta-
bility boundary has a minimum for some small value
W=0,
(50) of the transverse wavenumber k of the disturbance,
and implying loss of stability of the basic solution to long
wavelength perturbations as the pressure sensitivity
Ap defined above decreases in magnitude. This, in
Ap= 1-pp , w2=k[l+PFLr__-lp +_-p'Yk+s k] ' turn, suggests a small wavenumber nonlinear stabil-
(5i)
ity analysis in the unstable regime, which generally
respectively, where k = Ik[. Steady, planar com- leads to simplified nonlinear evolution equations of
bustion is therefore stable in the region Ap < 0 the Kuramoto-Sivashinsky type for the finite ampli-
that lies between these two curves, where the pul- tude perturbations, n,i2
sating stability boundary is a straight line in the To establish the nature of hydrodynamic insta-
(Ap, k) plane and the cellular stability boundary is bility in the microgravity regime in a formal sense, as
a curve which lies above the former in the region well as to introduce a formalism whereby the anal-
Ap >_ -p/(2 - p). The shape of the latter bound- ysis of the fully viscous case may be simplified, we
ary depends on whether or not the parameters Fr -I may realistically consider the parameter regime p <<
and/or "y are zero. In the limit that the product 1, Fr-i << 1, with Fr-i _ p. For example, typical
"yFr -I approaches the value (1 - p)/4p 2 from be- values are p ,-, 10-3 - 10-4, liquid thermal diffusiv-
low, the cellular stability boundary recedes from the ity Al "" 0.1 m2/sec, and the steady, planar burning
region Ap < 0. For 7Fr -I > (1 - p)/4p 2, the sta- rate U ,-, 1 - 10 cm/sec depending on pressure. 3
ble region is the strip -p/(1 - p) < Ap < 0. Thus, Hence, from the definition Fr -1 --OAI/U 3, we con-
when Ap = 0, the classical Landau result for cellu- clude that Fr -1 _ p implies that the dimensional
lar instability is recovered. However, even a small
gravitational acceleration _ < 10-Sm/sec 2, which,
positive value of Ap renders steady, planar burning roughly speaking, marks the onset of the micrograv-
intrinsically unstable for all disturbance wavenum-
ity regime. Thus, introducing the bookkeeping pa-
bers, regardless of the stabilizing effects of gravity
rameter e << 1, we define scaled parameters g*, p*
and surface tension, since there always exists a pos-
and Ap according to
itive real root of Eq. (49) for Ap > 0 (for small p,
this root is given by iw ,,, k/Ap). This result may • {
be anticipated from quasi-steady physical consider- p= p*e ' Ap= Ape, Fr -1 = gg*e ' (52)
ations. That is, a burning velocity that increases
with increasing pressure is a hydrodynamically un- with 7 remaining an O(1) parameter. Here, the
stablesituation, since an increase in the burning ve- lower scaling on Fr -i corresponds to the reduced
locity results in an increase in the pressure jump gravity limit, whereas the upper definition indicates
across the liquid/gas interface, and vice-versa. How- the normal gravity case. In both regimes, it is read-
ever, a sufficiently large negative value of Ap re- ily seen from Eq. (50) that there are three distinct
sults in a pulsating hydrodynamic instability, the wavenumber scales that must be considered in order
toproperlyaccounfotrtherelativemagnitudoefthe
with a single local minimum at
varioustermsinthenumeratoarnddenominatoinr
theexpressiofnor
Ap. In particular, in addition to
k=k~{v (60)
v 74 o(v )'
the O(1), or outer, scale k, there is an inner scale k_
and a far outer scale kI that are respectively defined
corresponding to the critical value
as
_k/e, Fr -1 ,_ O(1) Ap. = A; ,,, -2p1*_ + { eP*2v,'_ (61a)
I
k,= [.k/e2 ' Fr_ 1,,_O(e) ' kf=ke. (53)
Thus, in terms of the unscaled parameters, instabil-
In the thin inner and thick far outer regions, we thus
ity first occurs for disturbances at the critical value
readily obtain
• {
Ap = f_p ,_ _lp + p2x/_Fr_I . (61b)
Ap _ A_ (i)_., p*(p*g - ki)/2k, (54)
p*(p'g*- k,)/2k'_
The essential difference, as illustrated by Figure
and 3, between the normal and reduced gravity limits in
• 1.-. k ,
Ap .._A;(])..._p (p 7 / - 1) (55)
HYDRODYNAMIC STABILITY BOUNDARIES (p << I)
respectively• Each of these two expansions may be "O_._A_ Inviscid Case (P = 0)
matched to the O(1) outer expansion, which is given
(cfer-'.o)unstable k ((,dT}-'.o)
by i i i _ i i l i i i
. (o) 1 , t 2 3 4 5 6 "} _0 tt
Ap,_A; ,-_-_p , (56) _.
?-
and thus a uniformly valid composite expansion, de-
noted by A_(C)(k), may be constructed as unstable (eellular_#
A_( c),,_ _?". .__, "1
A; (0 + A; (°) + A; (1)- klilnooA; (')- lim A; (/)
kl---*O
........... Pr" _0.7.,0 _=0
,..-2pl. + lep.27k + { eep2*p2*2gg/*2/k2k , ,9_"
,,_,¢o.-p'm .",--'=:,- o ,.,- o
(57a)
where the definitions of k_ and kI have been used to
express the final result in terms of k (Figure 3). In
terms of the original unscaled parameters, Eq. (57a) o7_
beomes
stable
P p27k p2Fr-1 (57b)
Ap ,., -2 + T + 2_'
?-
which eliminates the small scaling parameter e from
the composite expansion• Noting the simplicity of
(O,-p') r_# 0
the asymptotic result (57) relative to the exact ex-
pression given in Eq. (50), it is easily seen that the unstable (pulsating)
7
hydrodynamic stability boundary in the regime con-
Figure 3
sidered here lies in the region Av _<0, intersecting
the Ap = 0 axis at
Asymptotic representation of the cellular hydro-
1 dynamic neutral stability boundary in the limit of
k= k_,-.,-- >>1 (58) zero viscosity. The upper (lower) solid curves cor-
p*Te
respond to the two cases described by Eqs. (57) for
and at normal and reduced-gravity, respectively (curves are
drawn for the case e = 0.04, p* = 1.0, g = 2.5,
(59)
k = k2 ,_ pp*.gg.ee 2 << 1, g* = 1.0).
the
realistic parameter regime considered here isthat Substituting these expansions and the scalings (52)
in the latter instance, gravity is only capable of sta- and (62) into Eqs. (40) - (48), and collecting coef-
bilizing disturbances whose wavenumbers are very ficients of like powers of e, we obtain a sequence of
small, O(e2), whereas in the former case, gravity algebraic problems for the coefficients in Eqs. (65)
is sufficiently strong to stabilize disturbances whose - (67). In particular, from either of Eqs. (42) and
wavenumbers are O(e). As a consequence, hydro- (43), we obtain
dynamic instability becomes a long-wave instability
phenomenon in the reduced gravity regime consid- b_-_) =-k/p', (68)
ered here, since the most unstable wavenumbers are
O(V_ ), rather than O(1), in that case. We note that and the leading-order versions of Eqs. (44) - (46)
the role of surface tension, on the other hand, serves are given by
to stabilize only short-wave disturbances, which are,
go)_blo)_p'b_-')+p'b_-_)=0, (69)
to this order of approximation, unaffected by the
magnitude of the gravitational acceleration.
b?)- bl°)- A;(%-') =0, (70)
hi-')- 2hi-'): 0 (7_)
5. Hydrodynamic Instability: Yhlly Viscous Case
Subtracting Eqs. (69) and (70), and using Eqs. (68)
Guided by these results for the inviscid case, the and (71) for b(-D and b_-D, we obtain the result
linear stability analysis may be extended to include
the effects of viscosity as follows. Retaining the scal-
A_ (o) = -_p'. (72)
ings (52), we have noted that pAPrg = #Prl, where
# = #9/#1 is the gas-to-liquid viscosity ratio. Thus,
Equations (67) and (72) are equivalent to the invis-
it is reasonable to treat Prt - P as an O(1) param-
cid result (56) in the outer wavenumber regime, and
eter, and to consider the limit
thus, to the leading order of approximation, viscos-
= #*e <<1. (62) ity does not affect the neutral stability boundary for
O(1) disturbance wavenumbers.
Introducing the scalings (52) and (62) directly into
A corresponding result is obtained in the inner
Eqs. (40) - (48), approximate solutions for the coef-
wavenumber regime defined by the first of Eqs. (53).
ficients may be sought in the form of appropriate ex-
In particular, the expansions for q and r are now
pansions in powers of e. As suggested by the inviscid
given by
analysis, it is necessary to construct separate expan-
sions for the inner, outer and far outer wavenumber
regimes defined above. Since we are interested in ob- r._#.pk_{e3+... es+ ' q"_ _1 + Pk 2{_2e4++... '
taining the hydrodynamic cellular stability bound-
(73)
ary, we may set iw equal to zero directly and seek
and the expansions for the coefficients bl - b8 and
solutions for bl - bs and A_, where the latter ex-
Ap may be sought in the form
pressed as a function of k will describe the stabil-
ity boundary. Forms for the various expansions are
suggested in part by the corresponding solutions for bi ~ .o,((i1o)) _t_o,.((2ii)) e _"2r-" " i=2,8, (74)
these coefficients in the inviscid limit (for which ex- O_ _ "-_ 0i _ "_- '''
act solutions are analytically tractableS). .( ) _.(2) 2-
{0,i"_ _ 0i _ Jr''"
In the O(1) wavenumber regime, the expansions bi _ (1).2 -- _(3).3 -- , i = I, 3, 4, 5, 6_T_)
bi ¢ -r% ¢ -t-. •
for q and r are, from Eqs. (38) and (39),
r,..rte+..., rt=_*Pk 2, (63) Ap = A;(0e ~ e(A; (0 + AT(0e +-..). (76)
1+ _/1 + 4P2k 2
Substituting these expansions and the scatings (52)
q_q0+'", q0= 2P , (64)
and (62) into Eqs. (40) - (48) as before, we again
and appropriate expansions for the coefficients bl - obtained a much-simplified set of algebraic equations
bs and Ap are given by at each order. In the reduced-gravity regime, for
example, we obtain from Eq. (40) and either of Eqs.
b,~ b_-1),-1+b?)+b?)_+.., i =2,8_65)
(42) or (43) that
bi~b_ °)+b?)e+''', i=1,3,4,5,6,7,(66)
_,_=2_o, C =-,=k,, (77)
Ap = A_(°)e ,-. e(A_ (°) + A*l(°)e +...). (67)