Periodic propellant ames and uid-mechanical e�ects

T.L.Jackson�, J. Buckmastery, A.Hegabz

University of Illinois at Urbana-Champaign

Urbana IL 61801

Nomenclature

D Damk�ohler numbere internal energyh enthalpyL 1/2-period of the grainM mass uxM Mach numberp pressureRR reaction rateT temperatureTBS Burke-Schumann ame temperatureu; v velocity componentsX mass fraction of oxidizer (gas phase)Y mass fraction of fuel (gas phase)� supply stoichiometric ratio� scaled Damk�ohler number� viscosity()o surface values

Abstract

We construct the combustion �eld generated by a periodic reactant supply that models thesupply from a heterogeneous propellant. A full accounting of the uid mechanics is incorporated.However, solutions di�er little from those generated asssuming a uniform mass ux (the constantdensity model). On the other hand, allowing for temperature dependent transport (power law orSutherland's law) strongly a�ects the reaction rate and the heat ux to the propellant surface.The e�ects of nonuniform mass ux at the propellant surface are discussed, such nonuniformityarising, for example, because of the di�erent densities of the oxidizer and binder. And we discussthe e�ects of hills and valleys in the propellant surface.

�Senior Research Scientist, Center for the Simulation of Advanced Rockets, 1304 West Spring�eld Avenue;

tlj@csar.uiuc.eduyProfessor, Department of Aeronautical and Astronautical Engineering, 104 South Wright Street; limey@uiuc.eduzPost-Graduate Associate, Department of Aeronautical and Astronautical Engineering, 104 South Wright Street;

hegab@uiuc.edu

1

Introduction

There has long been an interest in understanding the nature of the combustion �eld supported bya heterogeneous propellant, and there have been many important contributions to this subject.Notable is the ame structure proposed by Beckstead, Derr, and Price (the BDP model1), andvarious improvements of this, e.g. Refs[2,3], and a number of 1D models, e.g. Refs[4,5,6].

The BDP model is characterized by a de agration sitting over the AP, a 'primary' di�usion ame supported by AP and binder gases, and the '�nal' di�usion ame supported by AP de- agration products and binder gases. A signi�cant correction to this picture is implicit in therecognition by Price7 of what he calls the 'leading-edge- ame', and what one of us, in a generalcombustion context, calls the edge- ame8;9. This is the edge structure - a region where reactantmixing is important, neither a de agration nor a di�usion ame - that exists when a di�usion ame has an edge. In the propellant context this edge occurs because the di�usion ame can notextend all the way to the surface, the temperature there being too low. The maximum reactionrate within the edge- ame is greater than in the trailing di�usion ame.

A second correction to the BDP picture is implicit in the results of elementary amemodeling10;11. The use of ux conditions at the propellant surface (rather than Dirichlet data)together with the stoichiometric requirement for large amounts of oxidizer, causes the stoichio-metric level surface (SLS) in a mixture-fraction formulation to intersect the surface on the AP,not at the AP/binder boundary, and the di�usion ames must lie on the SLS. (This assumesthat the oxidizer and fuel Lewis numbers are equal.)

A third correction, long recognized, is apparent when one examines the nature of a typicalAP mix. Miller12 describes the propellant SD-1-88-17 for example, consisting of 30% by weightof 90 micron AP particles and 40% by weight of 20 micron particles. Of course, the distributionis not bimodal. In a typical 20 micron AP batch examined at the Thiokol Corporation13, 30%by weight is smaller than 7 microns; overall there is a heavy bias in the distribution towards �nepowder. This powder, mixed with the binder, will generate a fuel-rich mixture near the surfacethat can support a de agration, rather than individual di�usion ames. (The disappearance ofthe SLS for small Peclet number particles is discussed in Ref[11]).

What these various corrections or modi�cations suggest is that real propellant ames arecomplex creatures whose structure can not reliably be guessed, even with substantial physicalinsights. There is no substitute for numerical simulations.

We have published a number of papers11;14;15 (see also Ref[16]) that examine simple modelsrelevant to the combustion �eld that is supported by a heterogeneous propellant, part of asystematic study of this subject, of ever increasing complexity. Our goal is to describe the3D combustion �eld, accounting for AP de agration and AP/binder reaction, the nonuniformregression of the propellant surface, nonuniform mass ux from the surface (particularly thatarising from the fact that the AP density is roughly double that of the binder), the hills andvalleys in the propellant surface arising from the nonuniform regression, heat conduction withinthe solid, the e�ects of pressure and velocity uctuations in the chamber gases, etc. Amongstother things, this requires the numerical description of the random propellant morphology (apacking model) something that has not yet been addressed by the community. (The work ofRef[17] is an exception, but morphologies are not explicitly described.)

The general problem is a di�cult one, and we are engaged in an incremental approach partlydesigned to explore the various physical ingredients, to gain some sense of what is importantand what might not be. An example is the analysis of Ref[14] in which the e�ects of a shear owon an isolated AP/binder ame are examined, and it is shown that there is a signi�cant e�ecton the heat ux to the surface. It could be argued that this is relevant to the heat ux near theboundaries of large AP particles randomly distributed in a 3D propellant. On the other hand,similar calculations (not reported) carried out for a periodic propellant of the kind discussed inRef[15] (and one that we examine in the present paper) reveal a negligible e�ect. Which resultis relevant to 3D random propellants is not clear at this time. Should shear not be signi�cant

2

in the �nal analysis, it is unlikely, in our opinion, that erosive burning can be attributed to thepenetration of turbulence to the wall vicinity with a concomitant enhancement of di�usivity (themixing length phenomenon). The ame structure sits very close to the surface and it is di�cultto see how eddies, small on this scale, could penetrate undissipated against the convective owfrom the surface. A more likely scenario is that large eddies drive the ame towards the surfacein some regions, pull it back in others, and the net e�ect on the heat ux averaged over an areagreater than that characteristic of a single eddy is positive. These eddies might be generatedby uctuations in the ux from the surface associated with small scale casting irregularities, orby the roll-up of acoustically generated wall shear layers18;19. The calculation of the e�ect ofvarious ow-�elds on heterogeneous propellant ames is incomplete.

All of the work that we have done so far adopts the constant-density model. (This is standardterminology in the combustion community but perhaps misleading in the present context - seethe later remarks.) In this model a solenoidal velocity �eld is assigned that a�ects the thermal�eld and the reactant �elds, but fails to account for the e�ect of thermal expansion on the ow-�eld. This model is a useful one, of proven e�cacy, but can not be used in a true simulation,our ultimate goal, and so here we describe the �rst results obtained when a complete couplingbetween the combustion processes and the ow �eld is accounted for. Speci�cally, we examinea 1D-periodic propellant that supports a 2D combustion �eld, a sequel to the constant densitystudy of Ref[15].

As we have already noted in our discussion of the BDP model, even in the absence of metalparticles there are several kinds of gas-phase ames in the combustion �eld of a heterogeneouspropellant. One of these is the AP decomposition ame, essentially a 1D de agration that sitsclose to the AP surface. Its importance lies in the fact that much of the AP surface receiveslittle heat from the other two ames15, and so it strongly controls the surface regression.

A second ame is the di�usion ame that is supported by binder gases and the products ofthe AP decomposition ame. Signi�cant heat can be generated in this ame but little, if any,is conducted back to the propellant surface, as its elements stand o� too far from the surface.More precisely, the Peclet number based on the stand-o� distance of any element is large, sothat the convective heat ux directed away from the surface is much greater than the conductiveheat ux directed towards the surface.

A third type of ame is the edge- ame de�ned by the edge of the di�usion ame. Un-bounded edge- ames (also known, unfortunately, as triple ames, a locution that should beabandoned) have a well-de�ned propagation speed (positive or negative, according to the valueof the Damk�ohler number20;21) and so it is useful, perhaps, to think of the edge- ame as thecomponent of the di�usion- ame/edge- ame duo that maintains the position of this duo in thewind from the propellant surface, although there are dangers in this view. The importanceof the edge- ame is that it provides the heat to the part of the surface that is not heated bythe AP decomposition ame, and so provides heat to the fuel binder. It can only perform thisnecessary service if the Peclet number de�ned by its stand-o� distance has order of magnitude1 or smaller.

Our concern in the earlier work and in the present paper is with the edge- ame, with itsstructure, shape, and position, and the surface heat ux that it generates. For that reason we donot include the model ingredients necessary to support the AP decomposition ame. Explicitjusti�cation for this comes if we assume that the AP ame is so close to the surface that itcan be simply be regarded as a surface ingredient. And then, since we shall only examine thegas-phase, imposing simple boundary conditions at the propellant surface, speci�cation of thesurface temperature being one of them, the role of the AP ame is merely implicit. It generatesheat, but the sole impact of this heat is to sustain the prescribed surface temperature.

Coupling of the gas-phase/solid-phase processes is a crucial ingredient in the calculation ofthe surface regression, but is something that we defer to a later study. In its stead, we specifythe mass ux (not necessarily uniform) at the surface, and we explore a number of issues. We are

3

concerned with the error introduced by using the constant density model, as it is an attractivetool for exmining the 3D problem; with the role played by temperature dependent transport;with the e�ects of varying surface mass ux arising from the di�erent densities of AP and binder;and with the e�ects of surface irregularites (hills and valleys) on the combustion �eld.

The constant-density model

It is useful to summarize the formulation of the constant density model before addressing thecomplete problem, as this enables us to introduce most of the model ingredients together withvarious convenient scalings in the context of a modest set of equations.

The speci�c details of the constant-density model for our problem are as follows: the densityis set equal to a constant (so that the equation of state, Charles's law, is jettisoned); and auniform velocity �eld u = 0; v = constant is adopted, which satis�es both the continuity equationand the momentum equation. Then the mass ux �v is constant, and the model is merely avariation on that proposed by Burke & Schumann in 192822. (An alternative and essentiallyequivalent formulation is to permit density variations via the equation of state, but set the mass ux equal to a constant, satisfying continuity but jettisoning the momentum equation). Then,with the adoption of 1-step Arrhenius kinetics, the equations for the temperature (T ) and theoxidizer (X) and fuel (Y ) concentrations are

M@

@x[CpT;X; Y ] = �Dr2 [CpT;X; Y ] + [Q;��X ;��Y ]BXY e�E=RT ; (1)

where we have assumed that both Lewis numbers are equal to 1.These equations are nondimensionalized using L, the 1/2-period of the periodic grain con-

�guration, and T0, the surface temperature (the temperature at y = 0). At the same time wewrite X = �X �X, Y = �Y �Y , and then

@

@�x

��T ; �X; �Y

�=

1

Pe�r2

��T ; �X; �Y

�+D

��Q;�1;�1

��X �Y e��=

�T ;

where Pe =ML

�D; �Q =

Q

CpT0; D =

�X�YBL

M; � =

E

RT0:

(2)

It is convenient to write

D = �e20: (3)

In all calculations of the present type, the Peclet number Pe plays a vital role, and it isnoteworthy that this role is not discussed in the most recent compilation of solid propellantcombustion theory23.

The periodic propellant con�guration is shown in Fig.1. Boundary conditions are applied atthe surface �y = 0 and are

�T = 1; �X �1

Pe

@ �X

@�y= �Xs or 0; �Y �

1

Pe

@ �Y

@�y= �Ys or 0; (4)

where �Xs = Xs=�X , �Ys = Ys=�Y , and Xs is the fraction of AP that is X , Ys is the fractionof binder that is Y . Equations (4b,c) are ux conditions and the choices on the rhs are madeaccording to whether the surface is AP (� < �x < 1) or binder (0 � �x < �).

An important parameter is

� =�Ys�Xs

=�XYs�YXs

(5)

4

the ratio of the solid AP to solid binder required for stoichiometric combustion. To see this,note that an amount of solid AP equal to �XYs generates �XXsYs of X , and an amount ofbinder equal to �YXs generates �YXsYs of Y . � is an assigned parameter, closer to 10 than to1 for real propellants, and we have typically made the choice � = 7.

The actual supply ratio of AP to binder is 1��� and this is equal to � if the supply is

stoichiometric, corresponding to a value of �

�s =1

1 + �: (6)

Should � be larger than �s the supply will be fuel-rich, should � be smaller than �s the supplywill be fuel-lean.

It is clear that an arbitrary choice can be made for �Y since only the ratios �Q=�Y and�X=�Y control the solution. Thus �Ys may be arbitrarily assigned, and for consistency withRef[15] we make the choice

�Ys = 0:4(1 + �): (7)

This arises in a natural way in Ref[15] because of the way the equations are nondimensionalizedthere, di�erent from what has been done here.

With �Ys assigned, �Xs is determined by the choice of �, and �Q is chosen to generate asuitable ame-temperature. The Burke-Schumann ame temperature de�ned by 2 1/4-planesof propellant11 is

�TBS = 1 +�Q �Xs

�Ys�Xs + �Ys

(8)

and this can be used to select a suitable �Q.Figures 2,3 show typical solutions, obtained for parameter choices � = 30; �Q = 5; � =

7; � = 0:1; P e = 5; � = 1=8: The ame is characterized by two strong mixing structures eachcentered at j�xj � 0:2 and surface heat ux concentrated at �x = 0, the center of the fuel-binder,with little at �x = 1, the center of the AP. As we noted earlier, this makes clear the importanceof the AP decomposition ame in determining the overall regression rate.

Variable density model

When full uid-mechanics coupling is accounted for, the system of equations (2) is replaced by(with the checkmark dropped):

@ ~F

@x+@ ~G

@y=

@ ~Fv@x

+@ ~Gv

@y+ ~S ; (9)

where the convective ux vectors are given by

~F =

��u; �u2 +

p

M2; �uv; u

��e+

� 1

p

�; �uY; �uX

�T;

~G =

��v; �uv; �v2 +

p

M2; v

��e+

� 1

p

�; �vY; �vX

�T;

(10)

the di�usive ux vectors by

~Fv =

�0; �xx; �xy; ( � 1)M2 (u�xx + v�xy) +

�Sc

PrPe

@T

@x;

�

Pe

@Y

@x;

�

Pe

@X

@x

�T;

~Gv =

�0; �xy; �yy; ( � 1)M2 (u�xy + v�yy) +

�Sc

PrPe

@T

@y;

�

Pe

@Y

@y;

�

Pe

@X

@y

�T;

(11)

5

����������������������������������������������������������������������

����������������������������������������������������������������������

x = 1

=αx

Periodic boundary

x y

x

AP

condition

condition

Symmetric boundary

= 0

^

^

^

^

^

FUEL-BINDER

Figure 1: Periodic propellant con�guration.

and the source term by

~S = [0; 0; 0; 0; Q; � 1; � 1]T : (12)

The normal and shear stresses are

�xx =2�Sc

Pe

�@u

@x�1

3

�@u

@x+@v

@y

��;

�yy =2�Sc

Pe

�@v

@y�1

3

�@u

@x+@v

@y

��;

�xy =�Sc

Pe

�@u

@y+@v

@x

�;

(13)

and the internal energy equation and the equation of state are

e =T

+ � 1

2M2

�u2 + v2

�; P = �T: (14)

Finally, the reaction rate is

= DXY e��=T ; D = �e20: (15)

In addition to the scales used for the constant-density model, we use the normal velocity atthe surface V0, the surface density �0, the pressure p0 = �0RT0, and the viscosity �0. Nowthe Peclet number is Pe =ML=�0D0 (constant) and other nondimensional parameters are theSchmidt number Sc = �=�D (constant), the Prandtl number Pr = �Cp=k (constant), and the

6

0 0.5 1 1.5 2−1

−0.5

0

0.5

1

0.0001

0.005

0.10.4

0.30.2

0.2

Oxid (X)x

0 0.5 1 1.5 2−1

−0.5

0

0.5

1

0.01

0.10.20.5

1

Fuel (Y)

0 0.5 1 1.5 2−1

−0.5

0

0.5

1

2.32.8

1.8

1.3

Temp

x

y0 0.5 1 1.5 2

−1

−0.5

0

0.5

1Rate

y

Figure 2: Constant density solution: X; Y; T; RR contours; � = 30; �Q = 5; � = 7; � = 0:1; P e =5; � = 1=8; RR contours: 3.5, 3, 2, 1, 0.5.

7

−1 −0.5 0 0.5 10.2

0.4

0.6

0.8

1

x

Sur

face

hea

t flu

x

Figure 3: Surface heat ux �@ �T=@�y(�x; 0), constant density solution (see Fig.2).

8

0 1 2−1

−0.5

0

0.5

1

2.5

2

1.5

Vx

0 1 2−1

−0.5

0

0.5

1

−0.1

−0.1

−0.03

−0.030.03

0.03

0.1

0.1

U

0 1 2−1

−0.5

0

0.5

1

−0.5−0.2

0.20.5

Vorticity

0 1 2−1

−0.5

0

0.5

1

1.3

1.8

2.3

2.8

Temp

x

0 1 2−1

−0.5

0

0.5

10.769

0.556

0.435

0.357

ρ

0 1 2−1

−0.5

0

0.5

1

0.387

0.774

1.16

1.55

1.94

2.32

p1

0 1 2−1

−0.5

0

0.5

1

1

0.5

0.2

0.1

Fuel (Y)

y0 1 2

−1

−0.5

0

0.5

10.4

0.4

0.3

0.3

0.10.005

Oxid (X)

y

x

0 1 2−1

−0.5

0

0.5

1Rate

y

Figure 4: Solution contours, variable density, � = 1; M = :01; = 1:2; P r = Sc = 1; otherparameters as in Fig.2. RR contours: 3.3, 3, 2, 1, 0.5.

9

−1 −0.5 0 0.5 10.2

0.4

0.6

0.8

1

1.2

x

Sur

face

hea

t flu

x

Figure 5: Surface heat ux for Fig.4 solution. The broken line is the constant density solution andis reproduced in later �gures for comparative purposes.

10

0 1 2−1

−0.5

0

0.5

1

2.5

2

1.5

Vx

0 1 2−1

−0.5

0

0.5

1

−0.01

−0.01

−0.04

−0.04

−0.07

−0.070.07

0.07

0.04

0.04

0.01

0.01

U

0 1 2−1

−0.5

0

0.5

1

−0.5−0.2

0.2

0.5

Vorticity

0 1 2−1

−0.5

0

0.5

11.3

1.8

2.3

Temp

x

0 1 2−1

−0.5

0

0.5

10.769

0.435

0.556

ρ

0 1 2−1

−0.5

0

0.5

11.95

1.95

2.35

2.35

0.391

0.782

1.17

1.56

p1

0 1 2−1

−0.5

0

0.5

1

1

0.5

0.20.1

0.01

0.01

Fuel (Y)

y0 1 2

−1

−0.5

0

0.5

10.4

0.4

0.3

0.30.1

0.005

Oxid (X)

y

x

0 1 2−1

−0.5

0

0.5

1Rate

y

Figure 6: Solution contours: variable density, power-law viscosity, other parameters as in Fig.4.RR contours: 2, 1, 0.5.

11

0 1 2 3−1

−0.5

0

0.5

1Rate

y

x

Figure 7: Reaction-rate contours: variable density, power-law viscosity, Pr = 0:7, other parametersas in Fig.4; RR contours: 1.5, 1, 0.5.

12

−1 −0.5 0 0.5 10.2

0.6

1

1.4

1.8

x

Sur

face

hea

t flu

x

Pr=0.7

Pr=1.0

Figure 8: Surface heat ux: variable density, power-law viscosity, Pr=0.7,1.0; other parameters asin Fig.4.

13

0 1 2−1

−0.5

0

0.5

1

2.251.8

1.4

Vx

0 1 2−1

−0.5

0

0.5

1

−0.01

−0.01

−0.03

−0.03

−0.05

−0.050.05

0.05

0.03

0.03

0.01

0.01

U

0 1 2−1

−0.5

0

0.5

1 −0.05

−0.05

−0.25

−0.25

−0.5

−0.5

Vorticity

0 1 2−1

−0.5

0

0.5

11.3

2.3

1.8

Temp

x

0 1 2−1

−0.5

0

0.5

10.769

0.435

0.556

ρ

0 1 2−1

−0.5

0

0.5

11.71

1.71

2.05

2.05

0.33

0.675

1.021.36

p1

0 1 2−1

−0.5

0

0.5

1

1

0.50.2

0.1

0.0

0.01

Fuel (Y)

y0 1 2

−1

−0.5

0

0.5

1 0.4

0.4

0.3

0.3

0.1

0.0050.0001

Oxid (X)

y

x

0 1 2−1

−0.5

0

0.5

1Rate

y

Figure 9: Solution contours: variable density, � = :5, power-law viscosity, Pr=0.7, other data as inFig.4.

14

−1 −0.5 0 0.5 10.5

1

1.5

2

2.5

3

x

Sur

face

hea

t flu

x

δ=0.5

0.2

0.15

0.1

Figure 10: Surface heat ux: variable density, power-law viscosity, various �, Pr = 0:7, other dataas in Fig.4.

15

Mach number M = V0=c0. M is the dimensional mass ux �0V0, and c0 is the speed of soundp p0=�0.In what follows we assume that the viscosity can have one of three forms:

constant: � = 1;

power law: � = T 0:7;

Sutherland's law: � =aT 3=2

b+ T; a = 1 + b:

(16)

We shall compare results for all three cases, and for each case we shall take the Prandtl numberto be either 1 or 0.7. The Schmidt number is always 1. The boundary conditions at the surfaceare as before, together with

u = 0; v = 1 at y = 0 (17)

in the case of uniform mass injection, or

u = 0; v = 1 at y = 0; � < jxj < 1

u = 0; v =1

RMat y = 0; jxj < �

(18)

when the injection velocity at the binder surface is di�erent from the injection velocity at the APsurface. The rationalization for the latter choice originates in the fact that the speci�c gravityof AP is 1.95, that of a typical fuel binder 1.0124 so that, if we assume a at regressing surface,the mass ux through the AP surface is nearly double that through the fuel surface, and RM

should be assigned the value 1.95/1.01. In practice, of course, the surface will only be at onaverage, and in due course we will need to account for surface irregularities, but a nonuniformsurface mass ux is clearly a component of a complete simulation. Note that with the choice(18), the stoichiometric value of � is

�s =1

1 + �RM

: (19)

For the far-�eld condition as y ! 1 we assume that u vanishes, the pressure p = 1, and they�derivatives of all other quantities vanish.

Numerical method

Since we are interested in low Mach number ows, we follow the numerical strategy outlined inTseng and Yang25. To be more precise, we �rst de�ne a reduced pressure

p = 1 + M2p1(x; y; t); (20)

and substitute into the equations of motion given above. The equation of state now becomes

� =1 + M2p1

T; (21)

and is used to update the density. Next, we add a pseudo-time derivative term of the form

�@ ~Q

@�; where ~Q = [p1; u; v; h; Y;X ]T ; (22)

to the governing equations, where � is the pseudo-time, � is a preconditioner matrix added toaccelerate convergence, and h = T + 1

2( � 1)M2(u2 + v2) is the enthalpy. We then march

16

the solution to steady-state (transient behavior has no meaning in pseudo-time). The code isstopped when the relative di�erence between solutions at two di�erent time values is less thansome prescribed tolerance, taken here to be 10�6.

All numerical calculations were performed on a 160 � 100 grid, uniform in the x-directionand stretched in the y-direction. Convergence tests were carried out and it was determined thatany further re�nement resulted in less than 1% relative error.

Results

The �rst results we show are a test of the constant density model, and so we have taken � = 1,rather than one of the more realistic choices. The general expectation is that the constant densitymodel does not lead to qualitative anomalies, but quantitative accuracy is not expected. Forthe present problem, however, a comparison of Fig.4 with Fig.2 reveals substantial qualitativeagreement for T;X; Y and the reaction-rate RR. This is due to the fact that u, the transversevelocity, is everywhere small, so that the continuity equation forces a nearly uniform mass uxin the y�direction. Absent some introduction of signi�cant u variations by an applied shear14,signi�cant deviations of the surface normal from the y�direction, etc., the constant densitymodel is accurate. Heat ux predictions, for example, are in error by less than 10%, Fig.5. Inaddition, the maximum temperature for variable � is 2.86 compared to 2.88 for the constantdensity solution, and the maximum RR (RRmax) is 3.36 compared to 3.60 for the constantdensity solution. Thus the qualitative conclusions reached in Ref[15] can not be doubted merelybecause the constant density model was used.

An important corollary of these results is that we can, with reasonable con�dence, explorecertain aspects of the 3D problem using the constant density model without fear of being ledastray on this account.

The adoption of a more realistic viscosity law a�ects the solution of course, as there are thenstrong variations in all the transport coe�cients (�; k; �D) between the propellant surface andthe main body of the ame. Figure 6 shows results which can be compared to Fig.4 except thata viscosity power law is adopted. The di�erences appear modest for most variables, but now themaximum reaction rate is 2.12 (cf. 3.36 for Fig.4). The di�erences in the maximum temperatureare less sharp, 2.77 for Fig.6 compared to 2.86 for Fig.4. The RRmax ratio is 2.12/3.36=0.63,and the Arrhenius factor ratio is exp[� 30

2:77 + 30

2:86 ] = 0:7, suggesting that the di�erences inmaximum reaction rate arise because of di�erences in temperature.

Adjusting the Prandtl number has little a�ect on most of the contours, or on the maximumtemperature, but a reduction of Pr to 0.7, all other parameters �xed at Fig.6 values, reducesRRmax to 1.62, Fig.7.

Despite the signi�cant reduction in RRmax in these calculations compared to that for � = 1,and therefore a reduction in the maximum heat generation, the heat ux to the surface isincreased by 50% or more, Fig.8. An e�ect of decreasing Pr is to atten out the distribution ofthis ux. These results make it clear that careful attention to modeling di�usive tranport willbe important in detailed simulations, and is more important for the present geometry than anaccounting of 2-way uid coupling.

An increase in the Damk�ohler number � to 0.5 increases the vigor of the reaction and pushesthe ame closer to the surface, Fig.9. Now RRmax is 2.27 and Tmax is 2.49. There is acorrespondingly strong increase in the heat ux, as shown in Fig.10, which also shows curvesfor � = 0:2; 0:15; and 0:1.

Evidence that all of these solutions properly capture the low Mach number ow�eld, thatthe computational strategy permits calculations for arbitrarily smallM, comes from comparingresults forM = 0:01 (Fig.9) with those forM = 0:001, all other parameters �xed:

17

M = :01: RRmax = 2:26896, Tmax = 2:48630, p1max = 2:39787;

M = :001: RRmax = 2:26841, Tmax = 2:48626, p1max = 2:40483.

(Recall that p = 1 + M2p1 de�nes p1).

Calculations using Sutherland's law do not di�er signi�cantly from those using the powerlaw, and we merely note the following values of RRmax and Tmax: 2.29 and 2.79 when Pr = 1;1.73 and 2.69 when Pr = 0:7. (All other parameters as in Fig. 4.)

constant density, temperature-dependent transport

Our results suggest that for the periodic propellant the use of the constant-density model doesnot introduce serious errors. The comparisons we have made, however, are for constant transportcoe�cients, Fig.5. It is of interest, then, to see if the same conclusion can be drawn for themore physically relevant case of temperature-dependent transport. Figure 11 shows contourscalculated when the mass ux is taken to be uniform, but a power-law is adopted for thetransport coe�cients, results which are to be compared directly to Fig.6. The correspondingresults for the heat ux are compared in Fig.12. For the uniform mass ux, RRmax is 2.2491and the maximum temperature is 2.7951; for the full equations the corresponding results are2.1175 and 2.7727. The di�erences are small.

nonuniform injection

We have carried out calculations similar to those described above but with the choice (18) forwhich the normal velocity at the fuel surface is � 0:52 (but remains equal to 1 at the AP surface).Results for � = 1 (Fig.13) show that the slow ow accelerates and v becomes more or less uniformin the neighborhood of the ame, with a positive bulge at the center-line downstream of the ame. The ame itself is substantially broadened compared to the uniform injection case (Fig.4)but the surface heat ux is sharply enhanced to the fuel binder only (Fig.14) because of thelocally reduced Peclet number. RRmax is 3.13, compared to 3.60 for uniform injection, andTmax is 2.82 compared to 2.88 for uniform injection.

A power-law viscosity weakens the ame and increases the heat ux, Figs.15,16. Moreover,the ame more closely resembles a plane de agration here than it does for any of the othercomputations. (A plane de agration is characteristic of small Peclet number calculations15).RRmax is 1.25, Tmax is 2.53.

hills and valleys

The �nal ingredient that we examine is that of a non- at surface. Speci�cally, we suppose thatthe surface is located at

y = �� cos(�x) (23)

the plus sign corresponding to a hill centred on the binder at x = 0, the minus sign to ahill centred on the AP at jxj = 1. Calculations have been done for the parameters used inFigure 2 for two choices of mass ux at the surface: (a) Mn, the normal mass ux, equalto 1; (b) Mn = cos(�) on the AP, Mn = 1

RMcos(�) on the binder, where � is the surface

slope, tan(�) = �� sin(�x). In both cases the tangential velocity is zero. For case (a), massconservation at the surface implies that the shape is changing. Case (b), on the other hand,corresponds to a constant regression of the surface at constant speed with unchanging shape.This is the probable late-time scenario for the periodic propellant, albeit with a shape that isdetermined by the gas/solid coupling and the surface physics, not assigned.

18

Figures 17,18 show results for case (a), for di�erent values of �. Note that the vertical scaleis greater than the horizontal scale. Figure 17 reveals not only signi�cant e�ects on the shape ofthe ame, but also on the maximum reaction rate. When � = 0 all 8 contour values are realizedso that RRmax > 3; but when � = 0:3 only 4 are realized so that 1:5 > RRmax > 1. Signi�cante�ects are also seen in Figure 18. The general trend is similar for case (b), and we show theimpact on the surface heat ux in Figure 19. Our focus in the variety of calculations reportedhere has been on the surface heat ux, since this drives the regression. Figure 19 suggestsstrongly that a complete simulation of heterogeneous propellant burning will not be accurate ifsurface unevenness is ignored.

Conclusions

There has been a great deal of interest in the past three decades and more in the details ofheterogenous propellant combustion, but most modeling e�orts have been one-dimensional innature, and those that are not have not been examined using numerical tools. Here, for the �rsttime, we present 2D calculations for a model that incorporates key ingredients of the combustion�eld with a full accounting of the uid mechanics.

This is a sequel to earlier work15 in which the Navier-Stokes equations were not used, anda constant mass ux was assigned. Important qualitative conclusions were drawn in the earlierwork, and what we have shown here is that allowing for the in uence of the thermal expansionon the ow �eld has little e�ect on the solution when the surface is at and the surface uxis uniform. We are presently engaged in calculating the 3D combustion �eld supported by apropellant with 2D surface variations, and the present results suggest that we will not have toinclude the uid mechanics coupling to draw useful qualitative conclusions. In this case thereduction in required computing resources is substantial.

Far more important than an accounting for the uid mechanics coupling is an accounting oftemperature-dependent transport. If a power law or Sutherland's law is adopted for the viscosity-temperature dependence, with corresponding variations of the other transport coe�cients, thereis a signi�cant increase in the heat ux to the propellant surface, a reduction in the maximumtemperature, and a decrease in the maximum reaction rate.

It is tempting to consider modeling the propellant burning by assuming a at surface and auniform mass ux from that surface, as this is a substantial simpli�cation. However, our resultssuggest that this could lead to serious quantitative error. Variations in mass ux associatedwith the density di�erences in the constituents (which therefore do not have a zero average) canstrongly a�ect the combustion �eld. And surface unevenness likewise can have a large impact.Models which do not account for these phenomena might be useful, but results obtained fromthem should be interpreted with care.

Acknowledgement

The work of JB and AH is supported by AFOSR and by the U.S. Department of Energy throughthe University of California under Subcontract number B341494. TLJ is supported by the latteragency.

19

0 1 2−1

−0.5

0

0.5

12.05

1.6

0.8

Vx

0 1 2−1

−0.5

0

0.5

1−0.1

−0.18−0.035

0.035

0.180.1

U

0 1 2−1

−0.5

0

0.5

1

−0.5−0.2

0.2

0.5

Vorticity

0 1 2−1

−0.5

0

0.5

1

1.3

1.8

2.3

Temp

x

0 1 2−1

−0.5

0

0.5

1

0.769

0.556

0.435

ρ

0 1 2−1

−0.5

0

0.5

1

0.655

0.655

1.47

1.47

p1

0 1 2−1

−0.5

0

0.5

1

2

1

0.5

0.2

0.1

0.1

0.1

0.1

Fuel (Y)

y0 1 2

−1

−0.5

0

0.5

1 0.4

0.4

0.3

0.30.1

0.005

0.0001

Oxid (X)

y

x

0 1 2−1

−0.5

0

0.5

1Rate

y

Figure 11: Solution contours: uniform mass ux, power-law transport; RR contours: 2, 1, 0.5. cf.Fig.6.

20

−1 −0.5 0 0.5 10.2

0.6

1

1.4

1.8

x

Sur

face

hea

t flu

x

Figure 12: Surface heat ux: A, full uids model; B, constant density but temperature-dependenttransport.

21

0 0.5 1 1.5 2−1

−0.5

0

0.5

1

1.3

2.3

1.8

Temp

x

y0 0.5 1 1.5 2

−1

−0.5

0

0.5

1Rate

y

0 0.5 1 1.5 2−1

−0.5

0

0.5

1

0.01

0.10.2

0.51

Fuel (Y)

0 0.5 1 1.5 2−1

−0.5

0

0.5

1

0.005

0.1

0.2

0.2

0.3

0.3

0.4

0.4

Oxid (X)x

Figure 13: Solution contours: non-uniform injection, � = �s (Eq.(19)), � = 1, other parameters asin Fig.4, RR contours: 3, 2, 1, 0.5.

22

−1 −0.5 0 0.5 10.2

0.6

1

1.4

1.8A

B

x

Sur

face

hea

t flu

x

Figure 14: Surface heat ux for the solution of Fig.13.

23

0 1 2−1

−0.5

0

0.5

1

2.31.60.8

Vx

0 1 2−1

−0.5

0

0.5

1

−0.1

−0.18

−0.035

−0.0350.035

0.035

0.18

0.1

U

0 1 2−1

−0.5

0

0.5

1

−0.5

−0.2

−0.2

0.2

0.2

0.5

Vorticity

0 1 2−1

−0.5

0

0.5

11.3

1.8

2.3

2.6

Temp

x

0 1 2−1

−0.5

0

0.5

10.769

0.556

0.435

0.385

ρ

0 1 2−1

−0.5

0

0.5

1

1.77

0.990.99

0.207

p1

0 1 2−1

−0.5

0

0.5

1

2 1 0.5

0.2

0.1

0.1

Fuel (Y)

y0 1 2

−1

−0.5

0

0.5

1 0.4

0.4

0.3

0.3

0.1

0.005

0.0001

Oxid (X)

y

x

0 1 2−1

−0.5

0

0.5

1Rate

y

Figure 15: Solution contours: non-uniform injection, power-law viscosity, � = �s, Pr = 0:7; RRcontours: 1.2, 1, 0.8, 0.5.

24

−1 −0.5 0 0.5 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

Sur

face

hea

t flu

x

Figure 16: Surface heat ux for Fig. 15.

25

−1 0 10

0.5

1

1.5

2

2.5

3R with ε=0.0

−1 0 1

0

0.5

1

1.5

2

2.5

3R with ε=0.2

−1 0 1

0

0.5

1

1.5

2

2.5

3R with ε=0.3

−1 0 1

0

0.5

1

1.5

2

2.5

3R with ε=0.35

Figure 17: RR contours, hills and valleys, case(a).

26

−1 0 10

0.5

1

1.5

2

2.5

3R with ε=0.0

−1 0 1

0

0.5

1

1.5

2

2.5

3R with ε=0.1

−1 0 1

0

0.5

1

1.5

2

2.5

3R with ε=0.15

−1 0 1

0

0.5

1

1.5

2

2.5

3R with ε=0.2

Figure 18: RR contours, hills and valleys, case(a).

27

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

x

dT/d

n

Valleys

ε=0.0 ε=0.1 ε=0.15ε=0.2

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

x

dT/d

n

Hills

ε=0.0 ε=0.1 ε=0.2 ε=0.25

Figure 19: Surface heat ux, hills and valleys, case(b).

28

Bibliography

1Beckstead,M.W., Derr,R.L., Price,C.F., "A Model of Composite Solid-Propellant CombustionBased on Multiple Flames," AIAA Journal Vol. 8 No.12, 1970, pp.2200-2207.2Cohen,N.S., Strand,L.D., "An Improved Model for the Combustion of AP Composite Propel-lants," AIAA Journal Vol.20 No.12, 1982, pp.1739-1746.3Glick,R.L., "Distribution Functions for Statisitical Analysis of Monodisperse Composite SolidPropellant Combustion," AIAA Journal Vol.14 No.11, 1976, pp.1631-1633.4Cohen,N.S., "Review of Composite Propellant Burn Rate Modeling," AIAA Journal Vol.18No.3 1980, pp.277-293.5Kerstein,A.R., "Percolation Model of Polydisperse Composite Solid Propellant Combustion,"Combustion & Flame Vol.69, 1987, pp.95-112.6Murphy,J., Krier,H., "Linear Pressure Coupled Frequency Response of Heterogeneous SolidPropellants," Proceedings of the Combustion Institute Vol.27, 1998, pp.2343-2350.7Lee,S.T., Price,E., Sigman,R.J., "E�ect of Multidimensional Flamelets in Composite Propel-lant Combustion," Journal of Propulsion & Power Vol.10 No.6, 1994, pp.761-768.8Buckmaster,J., "Edge-Flames," Journal of Engineering Mathematics Vol.31, 1997, pp.269-284.9Buckmaster,J., Zhang Yi., "Oscillating Edge-Flames," Combustion Theory & Modelling Vol.3,1999, 547-565.10Jia,X., Bilger,R.W., "The Burke-Schumann Di�usion Flame with Zero Net Flux BoundaryConditions," Combustion Science & Technology Vol.99 No.2, 1994, pp.371-376.11Buckmaster,J., Jackson,T.L., Yao,J., "An Elementary Discussion of Propellant Flame Geom-etry," Combustion & Flame Vol.117, 1999, pp.541-552.12Miller,R.R., "E�ects of Particle Size on Reduced Smoke Propellant Ballistics," AIAA paper82-1096, June 1982.13Bennett,R. private communication.14Buckmaster,J., Jackson,T.L.,"The E�ects of a Time-Periodic Shear on a Di�usion Flame An-chored to a Propellant," Combustion & Flame Vol.120, 2000, pp.211-221.15Jackson,T.L., Buckmaster,J., "Nonpremixed periodic ames supported by heterogeneous pro-pellants," Journal of Propulsion & Power, Vol.16, No.3, 2000, pp.498-504.16Buckmaster,J., Jackson,T.L., "Response of Propellant Flames to Unsteady Flows, and Re-lated Questions," AIAA paper 99-0323, Jan. 1999.17Davis,I.L., Carter,R.G., "Random Particle Packing by Reduced Dimension Algorithms," Jour-nal of Applied Physics Vol.67 No.2, 1990, pp.1022-1029.18Roh,T-S., Apte,S., Yang,V., "Transient Combustion Response of Homogeneous Solid Propel-lant to Acoustic Oscillations in a Rocket Motor," Proceedings of the Combustion Insitute Vol.27,1998, pp.2335-2341.19Lee,Y., Beddini,R., "Acoustically Induced Turbulent Transition in Solid Propellant RocketChamber Flow�elds," AIAA paper 99-2058, June 1999.20Dold,J.W., Hartley,L.J., Green,D., "Dynamics of Laminar Triple-Flamelet Structures in Non-Premixed Turbulent Combustion," article in Dynamical Issues in Combustion Theory, eds. PaulC. Fife, A. Li~n�an, F. Williams. Springer-Verlag (New York), 1991, pp. 83-105.21Kioni,P.N., Rogg,B., Bray,K.N.C., Li~n�an,A., "Flame-Spread in Laminar Mixing Layers: theTriple Flame," Combustion & Flame Vol.95 No.2, 1993, pp.276-290.22Burke,S.P., Schumann,T.E.W., "Di�usion Flames," Industrial and Engineering Chemistry

Vol.20, 1928, pp.998-1004.23Nonsteady Burning and Combustion Stability of Solid Propellants eds. Luigi De Luca, EdwardW. Price, Martin Summer�eld. Progress in Astronautics and Aeronautics, vol.143, AmericanInstitute of Aeronautics and Astronautics (Washington DC), 1992.24Rocket Propulsion Elements Sixth Edition, George P. Sutton. John Wiley & Sons (New York),1992, p.450.25Tseng,I.S., Yang,V., "Combustion of a Double-Base Homogeneous Propellant in a Rocket Mo-

29

tor," Combustion & Flame Vol.96, 1994, pp.325-342.

30