mean flow approximations for solid rocket motors with...

Mean Flow Approximations for Solid Rocket Motorswith Tapered Walls

Oliver C. Sams IV,∗ Joseph Majdalani,† and Tony Saad‡

University of Tennessee Space Institute, Tullahoma, Tennessee 37388

DOI: 10.2514/1.15831

Current ballistics analyses require detailed information of the internal flowfield forming in the combustion

chamber of solid rocket motors. In this study, we develop a model applicable to the circular-port motor with slightly

tapered grain. A combined geometric configuration is considered in which a straight cylinder is connected to a

tapered cone. The analysis is based on the vorticity-stream function approach, allowing us to resolve the problem

under incompressible, isothermal, inviscid, rotational, axisymmetric, and steady-state conditions. Additionally,

Clayton’s procedure is implemented with the aim of producing an approximate solution that reflects the behavior of

the flowfield at higher orders in the taper angle (Clayton, C. D., “Flowfields in Solid Rocket Motors with Tapered

Bores,” AIAA Paper 96-2643, July 1996). Given a nonlinear governing equation, a solution is sought with the use of

regular perturbations. A numerical simulation is also carried out, based on a finite volume solver, to verify the

asymptotic approximations. Analytical results are discussed and their range of validity is determined. The pressure

recovery due to taper is calculated and shown to be important in most practical applications.

Nomenclature

F = momentum thrustp = normalized pressure, �p=�V2b�p = dimensional pressureR0 = dimensional radius of cylindrical motorr = normalized radial coordinate, �r=R0�r = dimensional radial coordinateu = normalized velocity, �u=Vb�u = dimensional velocity, � �ur; �uz�Vb = injection velocity at propellant surfacez = normalized axial coordinate, �z=R0�z = axial coordinate� = velocity ratio, umax=uave�z�� = coordinate transformation, � � �0r2=r2s� = density = normalized stream function� = normalized vorticity

Subscripts

b = burning surfacer, z = radial or axial component� = azimuthal component0 = leading-order, parallel chamber quantities1 = first-order correction

Superscript

� = dimensional quantity

I. Introduction

T HE extraction of critical design parameters for solid rocketmotors (SRMs) through internal ballistics analysis constitutes

an integral part of modern propulsion system design anddevelopment. The flowfields generated in combustion chambersand their performance comprise the framework necessary to evaluatespecific motor requirements, candidate grain geometries, motorconfigurations, and propellant formulations [1]. Ballistics analysesare available to predict chamber pressures, combustion instabilities,thrust, mass flux, and grain burn rate histories based on the velocityand pressure profiles provided by the mean flowfield [2–9]. Anincrease in the accuracy of predictions provided by ballisticsanalyses invariably requires enhanced flowfield models.

In the past, several researchers have developed mean flowmodelsthat have set the standard forflowfield investigations. Among them is

Culick [10], whose original rotational, inviscid solution remains at

the foundation of many studies. These include those by Majdalani

and coworkerswho have extendedCulick’s solution by capturing the

effects of viscosity, wall regression [11,12], compressibility [13],

and arbitrary headwall injection [14,15]. Other related research

involves the evaluation of unsteady wave motions [16–21], acoustic

interactions with propellant burning [12], velocity and pressure

coupling [22], nonlinear (DC) pressure shift [23], triggering

amplitudes [24], limit cycle amplitudes [25], as well as particle-mean

flow interactions [26–30], two-phase flow effects [31], distributed

combustion, parietal vortex shedding [32–34], and both spatial and

temporal instabilities of the mean flow [35–38].It is imperative that the rocket propulsion community continues to

seek refinement in current analytical, numerical, and experimental

flowfieldmodelingmethods to avoid economic losses resulting from

adjusted mission requirements, overdesign, mission failures, and

catastrophic accidents due to under and overprediction of the

pressure load in the motor case.The search for enhanced analytical models has led one to consider

the mean flowfield characteristics of SRMs with tapered bores.Examples of tapered propellant surfaces can be observed in theTitan IV, Ariane, Castor, solid rocket boosters, interceptor vehicleswith fast burning propellants, and other moderate size launchvehicles requiring thrust curve modifications. Most modern SRMsare manufactured with small tapers to facilitate the removal of thecasting mandrel after the curing process is completed. It is impliedthat these small tapers help to reduce the contact between thepropellant surface and themandrel, thereby reducing the possibilitiesof propellant tear, cracking and debonding. Additional tapering ofpropellant bores has served two functions: 1)minimization of erosiveburning and 2) implementation of boost-sustain motor configura-tions. Tapering or coning of the propellant surfaces can also help to

Received 23 March 2005; accepted for publication 16 June 2006.Copyright © 2006 by the authors. Published by the American Institute ofAeronautics andAstronautics, Inc., with permission. Copies of this papermaybe made for personal or internal use, on condition that the copier pay the$10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 RosewoodDrive, Danvers, MA 01923; include the code 0748-4658/07 $10.00 incorrespondence with the CCC.

∗Graduate Research Associate. Member AIAA.†Jack D. Whitfield Professor of High Speed Flows, Department of

Mechanical, Aerospace and Biomedical Engineering. Member AIAA.‡Doctoral Research Assistant. Member AIAA.

JOURNAL OF PROPULSION AND POWERVol. 23, No. 2, March–April 2007

445

http://dx.doi.org/10.2514/1.15831

maximize the volumetric loading fraction by increasing the port-to-throat area ratio.

Aside from the three-dimensional internal ballistics codes such asthe standard stability prediction [39] and the solid propellantperformance [2–4], there are many industrial codes that still refer toparallel profiles (such as Culick’s [10]) as a basis for design andanalysis [40].When parallel profiles are applied to tapered SRMs, theresult is overprediction of the pressure drop by as much as 50 to 85%for taper angles as small as 1 deg. Such discrepancies can drasticallyalter motor case design requirements to the extent of compromisingthe structural integrity of the case or its components.

II. Mathematical Model

For academic reasons, the SRM is characterized as a cylindrical,circular-port duct with a circumferential porous surface canted at anangle �. Our model in Fig. 1a incorporates both the nontapered andtapered geometries. This enables us to account for the bulk floworiginating from the nontapered section of the motor. The origin forthe coordinate system is placed at the interface where �z and �r denotethe axial and radial coordinates, respectively (see Fig. 1b). Thenontapered section of the cylindrical motor has dimensions of lengthL0 and radius R0. The gases are injected perpendicularly to thetapered surface. To satisfy mass conservation, the injected gases areforced to turn and merge with the bulk flow emerging from theparallel segment. The streamline behavior can be seen in Fig. 1a.

A. Governing Equations

In this problem, vorticity is produced at the surface as a result ofthe interaction between the injected fluid and the axial pressuregradient; one may begin by obtaining the required form of thevorticity. Furthermore, the flow can be taken to be 1) axisymmetric,2) inviscid, 3) incompressible, 4) rotational, and 5) nonreactive. Inaccordance with the stated assumptions, the kinematic equations ofmotion can be written in scalar notation. In the interest of clarity,these are

�u r@ �ur@ �r� �uz

@ �ur@ �z�� 1

�

@ �p

@ �r(1)

�u r@ �uz@ �r� �uz

@ �uz@ �z�� 1

�

@ �p

@ �z(2)

�� @ �ur@ �z� @ �uz@ �r�� 1

�r

@2 �

@ �z2� @@ �r

�1

�r

@ �

@ �r

�(3)

where

�u r ��1

�r

@ �

@ �z; �uz �

1

�r

@ �

@ �r(4)

B. Boundary Conditions

While it is apparent that a radial velocity component exists at theinterface, one should note that it does not contribute to the masscrossing into the tapered region; hence, it is not required to obtain asolution in the tapered domain (see Fig. 2). The key constraintsconsist of the following: 1) the axial inflow condition arising frommass balance across the taper interface (accounting for the bulk flowfrom the parallel portion of the motor); 2) no flow across thecenterline; and 3) uniform, orthonormal injection at the burningsurface. Mathematically, the boundary conditions can be evaluatedand expressed as8>>><

>>>:�z� 0; 8 �r; �uz � �Vb�L0=R0� cos�12� �r2=R20��r� �rs; 8 �z; �uz � Vb sin��r� �rs; 8 �z; �ur ��Vb cos��r� 0; 8 �z; �ur � 0

(5)

where Vb is the injection velocity at the burning surface.

C. Normalization

Normalization of the variables associated with the mathematicalmodel provides a descriptive and compact solution. One may set

r� �rR0

; z� �zR0

; L� L0R0

; r � R0 �r (6)

ur ��urVb

; uz ��uzVb

; p� �p�V2b

(7)

��

R0Vb; �� R0

��

Vb(8)

Normalizing the equations and boundary conditions, it follows that

8<:ur ��

1

r

@

@z

uz �1

r

@

@r

8>>>>>><>>>>>>:

ur@ur@r� uz

@ur@z��@p

@rradial momentum

ur@uz@r� uz

@uz@z��@p

@zaxial momentum

@2

@r2� 1r

@

@r� @

2

@z2�� 0 vorticity equation

8>><>>:z� 0; 8 r; uz � L� cos�12�r2�r� rs; 8 z; uz � sin�r� rs; 8 z; ur �� cos�r� 0; 8 z; ur � 0

(9)

aft end

head

end

solid propellantnozzle

a)

streamlines

b)

z

rCL

α

02R

0L

Fig. 1 Schematic featuring a) typical cylindrical solid rocket motor

with tapered bore and characteristic streamlines; and b) coordinate

system for the mathematical model.

αbV

sinz bu V α=

cosr bu V α= −

z( )0 no crossing of axisru =

r

Fig. 2 Schematic of physical boundary conditions.

446 SAMS, MAJDALANI, AND SAAD

D. Average Velocity

The geometry is based on the cylindrical coordinate system(depicted in Fig. 3) where �z is the dimensional axial coordinate and �ris the dimensional radial coordinate. The area of the sloped surface iscalculated to be

Ab � 2��z sec��R0 � 12 �z tan�� (10)

Also, the tapered chamber cross-sectional area is

A��z� � ��R0 � �z tan��2 (11)

The inflow cross-sectional area at the interface of the tapered andstraight portions of the chamber is given by

A0 � �R20 (12)

As a result of a mass balance in the straight segment of the chamber,the average inflow velocity becomes

�u 0 � �2L0=R0�Vb (13)

Here L0 is termed the bulk flow parameter because it is directlyproportional to the average velocity at the entrance to the tapereddomain. Because of mass conservation, the cross-sectional averagevelocity of the fluid at any axial location �z may be expressed by

�u ave��z� �A0 �u0 � AbVb

A��z� (14)

By substituting Eqs. (10–13), the average normalized velocitybecomes

uave�z� � 2L� z sec��1� 1

2z tan��

�1� z tan��2 (15)

For the case �� 0, Eq. (15) reduces to

uave�z� � 2�z� L� (16)

This relation describes the bulk flow in a straight circular-port motor.To determine the behavior of the gases when sufficiently removedfrom the head end, one may manipulate Eq. (15) and compute thelimit of the resulting expression. One gets

uave�z� � 2�1

z� tan�

��2�Lz� sec�

�1

z� tan�

2

��(17)

The limit at z!1 furnishes

limz!1

uave�z� � csc� (18)

FromEq. (18), it can be seen that the velocity in a tapered chamberconverges to a constant value as themotor length becomes large. Thisbehavior is illustrated in Fig. 4. Thus, it can be seen that the flowcannot continue to accelerate due to the gradual increase in cross-sectional area. This is contrary to the case of a parallel port motorwhere the velocity continues to increase in the downstream direction.Typically, combined motor configurations have a straight cylinderimmediately followed by a tapered segment. In light of this, it is

physically impossible for the velocity of the gases to increase beyondthe limiting value prescribed by the tapered geometry. Massconservation requires that the average velocity throughout theparallel segment of the motor remains less than or equal to thelimiting value. This is equivalent to setting

uave�z�j��0 deg � uave�1� or z � 12 csc� � L (19)

With subsequent manipulation [and expressing Eq. (19) indimensional variables], one obtains

�z � 12R0=� � L0 (20)

where the small angle approximation sin� ’ � is used. Thegeometric parameters of themotorsmust be chosen such that Eq. (20)is satisfied. Compliance with this criterion ensures a valid solution.

E. Stream Function Along the Burning Surface

A crucial component to the formulation of the flowfield understudy is the development of the stream function at the simulatedburning surface. It is clear that the behavior of the stream functiondepends on the angular orientation of the propellant surface. Asshown in Fig. 5, the normalized variables along the burning surfacecan be readily determined to be

z� s cos � (21)

rs � 1� s sin�� 1� z tan� (22)

ur;s �� cos�; uz;s � sin� (23)

As a result, the directional derivative assumes the form

d sds� @ s@r

dr

ds� @ s@z

dz

ds� @ s@r

sin�� @ s@z

cos� (24)

The stream function relations at the simulated burning surface can bedefined as

@ s@r� rsuz;s;

@ s@z��rsur;s (25)

Then, insertion of the stream function definitions into Eq. (24) yields

d sds� rs�sin2�� cos2�� 1� s sin� (26)

Subsequent integration of Eq. (26) renders s�s� � s�1�

0 0u Aave( )A z u

b bAV

z

r

Fig. 3 Mass balance over a tapered segment.

0 50 100 150 2000

25

50

75

100

α = 0o

2o

1o

3o

uave

zFig. 4 Axial variation of average velocity for L� 0.

secs z

α=

z

r1 tansr z α= +α

Fig. 5 Schematic of burning surface quantities.

SAMS, MAJDALANI, AND SAAD 447

12s sin�� C, hence

s�z� � z sec��1� 12 z tan�� C (27)

where C is a constant that may be determined from the boundarycondition, s�0� � L. Application of this constraint gives

s�z� � z sec��1� 12 z tan�� L (28)

It is to be noted that with the above expressions, the average velocityreduces to

uave�z� � 2 sr2s� 2

z sec��1� 12z tan�� L

�1� z tan��2 (29)

F. Axial Pressure Gradient

The axial pressure gradient is deemed to be the primary source ofvorticity. As a result, we may begin by expressing the axial variationof pressure in the form of Bernoulli’s equation as

p�z� � p0 � 12u2max�z� (30)

It may be useful to recall that the pressure variation is sensitive tothe shape of the axial velocity profile. One should also note that theshape of the profile changes at each axial location and that themaximum velocity is unknown. It will be therefore helpful to definean expression that captures the effect of profile variation in z as thegases make their way toward the exit plane. This will beaccomplished by defining the ratio of maximum-to-averagevelocities as

��z� � umax�z�uave�z�

(31)

The form of ��z�will be later determined from imposition of the no-slip requirement at the burning surface. By substituting Eq. (31) intoEq. (30), one is left with

p� p0 � 12�2u2ave�z� (32)

The pressure gradient is determined along the simulated burningsurface by evaluating the derivative of Eq. (32). The result is

@p

@z��2�z�uave�z�

�duave�z�

dz� uave�z�

��z�d��z�dz

�(33)

Next, the axial derivative of the average velocity may be obtainedfrom Eq. (29), namely,

duavedz� 2 sec�

rs� 4 s tan�

r3s(34)

Substituting Eq. (34) into Eq. (33) and simplifying, it follows that

@p

@z��4�2 s sec�

r3s

�1 � 2 s sin�

r2s� s cos�

rs

��0

�

��;

�0 � d�dz

(35)

G. Surface Vorticity

Having determined the relationship between stream function andpressure, the momentum equation for steady, inviscid flows may beevaluated at the surface. To begin, one may recall the normalizedform of Euler’s equation,

12r�u � u� � u �r u� � �rp or u ��r�p� 1

2u � u�

(36)

The normalized velocity at the simulated burning surface can beexpressed in terms of the taper angle via

u s �� cos�er � sin�ez (37)

Evaluating the expression at the surface by substitution of Eq. (37)into Eq. (36) gives

u ��s�sin�er � cos�ez� � ��sŝ (38)

Here ŝ represents the unit vector parallel to the burning surface. Thecorresponding pressure gradient can be expressed as

dp

dsŝ�

�@p

@r

dr

ds� @p@z

dz

ds

�ŝ�

�@p

@rsin�� @p

@zcos�

�ŝ (39)

At the surface, r�u � u� � 0; this enables us to equate Eqs. (38) and(39). At the outset, we collect

�s ��@p

@rsin� � @p

@zcos�� @p

@zcos��O�sin�� (40)

Note that, with values of � between 1 and 3 deg, the term containingsin� has been suppressed. Finally, the surface vorticity can beexpressed as

�s � 4�2 sr3s

�1 � 2 s sin�

r2s� s cos �

rs

��0

�

��(41)

At this juncture, one may seek a relationship that permitssatisfying the steady vorticity transport equation given by the curl ofEq. (36), specifically, r �u �� 0. This is identically satisfiedwhen the vorticity and stream function are proportional via�=r / . Knowing that the value of must remain constant along astreamline, onemay equate to s, its value at the tapered surface; itfollows that

�

r��srs

or �� rrs�s (42)

Substituting Eq. (41) into (42) yields

��r; z� � 4�2r r4s

�1 � 2 sin�

r2s� cos�

rs

��0

�

��(43)

This key relation is shown on Fig. 6.

III. Approximate Solutions

With a complete formulation for the chamber vorticity, a partial,nonlinear differential equation that governs the flow is at hand. Thiscan be obtained by substituting Eq. (43) into Eq. (9); the resultingvorticity equation turns into:

z

r

( )212sinL rψ π= Lψ =

s

()

12ec 1

tan

szs

zL

ψα

α

=+

+

( ) ( )22 4 1 2( / sin ( / )cos4 ( / ) )s ss r rr rψ ψ α ψ α β ββ ψ ′Ω = − +[ ]

0ψ =

Fig. 6 Mathematical model depicting surface vorticity and stream function.


@2

@r2� 1r

@

@r� @

2

@z2��4�2r2

r4s

�1 � 2 sin�

r2s� cos�

rs

��0

�

��(44)

By numerical methods, Clayton [40] notes that�0 and @2 =@z2 aresmall quantities. Clayton’s observations may be verified using ascaling analysis. Considering that

@2

@z2� @�rur�

@z(45)

onemay recall that ur is independent of z in the straight circular-port,thus causing Eq. (45) to vanish at leading order. It is clear that thepresence of small taper will not affect the size of this term, given themagnitude of sin�. Henceforth, it is posited that the axial derivativesare of small magnitude and can be neglected in the tapered domain.Using Clayton’s arguments, Eq. (44) is reduced to

@2

@r2� 1r

@

@r��4�2r2

r4s

�1 � 2 sin�

r2s

�(46)

The two boundary conditions that are required to solve Eq. (44) aregiven by

�0; z� � 0; �rs; z� � s (47)

With a nonlinear partial differential equation at hand, the method ofregular perturbations may be applied. In concert with such amethodology, the stream function and velocities may be expanded inthe form

� 0 � " 1 �O�"2� (48)

u � u0 � "u1 �O�"2� (49)

�� 0 � "�1 �O�"2� (50)

where the perturbation parameter is due to the small taper angle,namely,

" � sin�� (51)

A solution may be obtained by substituting Eqs. (48–52) intoEq. (46). The resulting expressionmay then be expanded into a linearsequence of ordinary differential equations.

A. Leading-Order Solution

At leading order, one obtains

@2 0@r2� 1r

@ 0@r� 4�

20r

2

r4s 0 � 0 (52)

This is a simple, second-order, linear differential equation with thegeneral solution

0�r; z� � C1 cos��0r2

r2s

�� C2 sin

��0r2

r2s

�(53)

Straightforward evaluation of Eq. (53) at the assigned boundaryconditions gives

0�r; z� � s sin��0r2

r2s

�(54)

where

s�z� � z sec��1� 12z tan�� L and �0 � 12� (55)

It should be noted that at L� �� 0, one recovers

0�r; z� � z sin��0r2� (56)

Equation (56) reproduces Culick’s profile [10] for flow in aninternal burning cylinder. The leading-order solution expressed byEq. (54)may be referred to as an extended version ofCulick’s profile.This form is a result of the additional bulk flow caused by theincreased surface area.

B. First-Order Solution

At first order, one obtains the following ordinary differentialequation:

@2 1@r2� 1r

@ 1@r� 4�

20r

2

r4s 1 �

8r2�0�1r4s

0 �8r2�20r6s

20 � 0 (57)

with

1�0� � 0; 1�rs� � 0 (58)

Applying these boundary conditions, one obtains

1�r; z� � s3r2s

�3 s � s cos

�2�0

r2

r2s

�� 2 s sin

��0r2

r2s

�

� 4 s cos��0r2

r2s

�� 3�1r2 cos

��0r2

r2s

��(59)

The first-order velocity ratio �1 must be determined such that the no-slip condition is satisfied along the tapered surface. This can bewritten as

r s � ŝ�r � ŝ� 0 (60)

therefore,

@

@rcos�� @

@zsin�� 0 � " 1�r cos�

� � 0 � " 1�z sin�� 0 (61)

Noting that 0 already satisfies the no-slip condition at the wall, werecover

@ 1@r

cos�� @ 1@z

sin�� 0 (62)

Again, it can be seen that the term containing sin� is negligiblysmall, being ofO�"�. SettingEq. (62) equal to zero and evaluating theresulting expression at the tapered surface, one obtains

�1 �4 s3r2s

(63)

The required forms of the leading and first-order velocity ratios, �0and �1, are presently known.

C. Velocity, Pressure, and Vorticity

With the proper form of the stream function available, it is nowpossible to evaluate the physical characteristics of the flowfield. Forsimplicity, we define

�� 0r2

r2s(64)

The leading and first-order stream functions become

0�r; z� � s sin � (65)

and

1�r; z� � s3r2s

3 s � s cos�2�� 2 s sin � � 4 s cos �

� 3�1r2 cos �� (66)


The radial and axial velocities are�ur � ur;0 � "ur;1 �O�"2�uz � uz;0 � "uz;1 �O�"2�

(67)

Using the relationship between the velocity and stream function, theleading and first-order axial velocity profiles are expressed by

uz;0 �1

r

@ 0@r� 2�0 s

r2scos �� 0uave cos � (68)

and

uz;1 �1

r

@ 1@r� s

3r4s

h�6�1r

2s � 4�0 s

�cos �� 0�8 s

� 6�1r2� sin � � 4�0 s sin�2��i

(69)

which can be written very conveniently in terms of the averagevelocity as

uz;1 � 13u2ave�2 � �0� cos �� 2��0 � �� sin � � �0 sin�2�� (70)

Because the variation of the chamber radius is ofO�sin��, rs, albeitfunction of z, is treated as a constant. With this in mind, the leading-order radial velocity can be expressed as

ur;0 ��1

r

@ 0@z�� rs

rsec� sin � (71)

The first-order radial velocity component becomes

ur;1 ��1

r

@ 1@z�� 1

r

@ 1@ s

d sdz� 1r

@ 1@rs

drsdz�� 1

r

@ 1@ s

d sdz

� tan�r

@ 1@rs�� @ 1

@ s

d sdz�O�"� (72)

At first order, the O�"� terms in ur;1 become O�"2�; therefore, weonly need to evaluate the first term in Eq. (72), specifically,

ur;1 ��2 s sec�

3rrs

�3� cos�2�� 2 sin �� 4 cos �� 4 r

2

r2scos �

�

�O�"�(73)

�� rs3ruave sec�3� cos�2�� 2 sin � � 4 cos �� 8=�� cos ��

�O�"�(74)

From the velocity profiles, the spatial variation of the pressure canbe ascertained. The substitution of the velocity components into thez- and r-momentum equations yields

@p0@z��

2

4

s sec�

r3s(75)

and so, by keeping rs constant,

@p0@r�� sec

2�

r3

nr2s cos�2�� 1� � 2�0r2 sin�2��

o(76)

By integrating and combining Eqs. (75) and (76), one is able toproduce the spatial variation of the pressure that satisfies bothmomentum equations. By inspection, one concludes that the radialcomponent of pressure is of marginal importance, being ofO�sin2��.At leading order, Eq. (75) may be integrated to obtain

p�z� � ��2

2

z sec�

r4s�z sec�� 2L� � pCulick (77)

where p�0; 0� � pCulick �� 12L2�2 stems from the head-endboundary condition. Setting �p� p � pCulick, one may expressthe leading-order expression for the pressure drop as

�p0 ��2

2

z sec�

r4s�z sec�� 2L�

� ��2

2

z sec�

�1� z tan��4 �z sec�� 2L� (78)

Knowing that rs is treated as a constant for first-orderapproximations, it may be argued that some error may be incurred asa result of performing multiple differentiation and integrationoperations that neglect the axial dependence of the radius. Also, theevaluation of the derivatives and subsequent integration of theresulting expression produce higher-order terms that, for purposes ofretrieving a leading-order solution, may be dismissed. In an attemptto improve accuracy, one may seek a higher-order correction for thepressure drop. This higher-order correction accounts for the second-order axial derivative of the stream function aswell as the variation ofrs with axial distance. Evidently, this variation becomes moresignificant at larger taper angles. Our second-order accurateapproximation for the pressure drop can be expressed as

�p1�z� � ��2

2

z sec�

r4s�z sec�� 2L� � " z sec�

18r8s

f4�� 4�2z3sec3�� z2sec2�h16�� 4�2L� 9�2r4s

i� 3Lz sec�

h8�� 4�2L� 3�2r4s

i� 16�� 4�2L3g (79)

For L� 0, we recover

�p1�z� � ��2

2

z2sec2�

r4s� " z

3sec3�

18r8s

h4�� 4�2z sec�� 9�2r4s

i(80)

In addition to the formulation of the velocity and pressuregradients, one may evaluate the vorticity. This can be accomplishedvia

��r; z� �1

r

�sec� tan�

r� 4r s

r4s

�sin

��0r2

r2s

�(81)

Each of the required flowfield characteristics particular to taperedcylindrical motors is now at hand. The effect of the bulk flowparameter will be examined along with the taper angle. One shouldnote that this parameter is also the normalized chamber lengthL� L0=R0.

D. Ideal Momentum Thrust

In pursuit of an expression that reveals the response of the thrust tothe taper angle, the axial momentum force, at any station z, may becalculated viz.

�F� 2��V2bR20Zrs

0

u2zr dr (82)

Following the convention adopted in Eqs. (6) and (7), one can put

F�z� � 2�Zrs

0

u2zr dr; F��F

�V2bR20

(83)

It is informative to evaluate Eq. (83) using the average velocity. Thecorresponding thrust is simply

Fave�z� ��2L� z sec��2� z tan��2

�1� z tan��2 (84)

The actual thrust may be estimated from F�z� � 2�R rs0 �u2z;0�

2"uz;0uz;1�r dr� F0�z� � "F1�z� � . . .; to first order, this expan-


sion yields

F�z� � 12�3�L� z sec��2 � 1

2�2"�L� z sec��4

9�3� � 10�

�L� z sec��2 � �z sec��2L� z sec�� . . . (85)

Subsequently, for L� 0, one retrieves F� 12�3z2sec2��

16�2�7� � 40

3�"z3sec3��O�"2�. The companion momentum cor-

rection coefficient may be calculated from �� F=Fave��0 � "�1 �O�"2�. Based on Eqs. (84) and (85), one collects

8>>><>>>:�0 �

�2

8� �21� � 40��z sec��2L� z sec�� 4L

2�3� � 10�72�L� z sec��

�1 ��2z�2L� z sec��8�z� L cos��

�z�z� 2L cos��sec4��21� � 40�z2 � 2L�21� � 40�z cos�� 4L2�3� � 10��72�2L� z sec��2

(86)

Finally, for L� 0, simple expressions are recovered, namely,(�0 � 18�2 � 172��21� � 40�z sec��1 � 18�2z sec�� 172��21� � 40�z2sec2�

(87)

The momentum correction coefficient reduces to a constant ��18�2 ’ 1:2337 for the no taper case. It reaches a maximum value of

1.23546 at

z� 200 � 105��27200 � 31440�� 9009�2p

2�21� � 40� cot�

’ 0:0351868 cot� (88)

The behavior of the thrust force is more fully described in Sec. V.B.

IV. CFD Confirmation

A perturbation technique has enabled us to obtain closed-formapproximations that describe the mean flowfield for the circular-portmotor with tapered bore. A numerical simulation is now invoked tovalidate the analytical results and verify their assumptions and rangesof applicability which, so far, could only be estimated withasymptotics. In the interest of simplicity, a numerical simulation isperformed with the use of a commercial code [41]; the purpose is torecreate the simulated environment evoked in deriving the analyticalsolutions.

A. Geometry and Meshing Scheme

The geometric models are created for taper angles ranging from 1to 3 deg. The dimensions are chosen in accordance with theparameters used by the analytical model. The cylindrical motor isrepresented in three-dimensional space, and the gaseous mixture(assuming single phase) is injected across its simulated burningsurface. A standard meshing scheme is employed with an intervalsize of 0.1.

B. Boundary and Operating Conditions

The reference pressurewe use is the stagnation pressure at the headend of the chamber, specifically, the pressure at the origin. Theworking fluid is injected at velocities ranging from 0.1 to 1 m=s.These values of injection velocity ensure that the Mach numberremains less than 0.3, hence justifying the incompressible flow

condition used in the analytical treatment. The selection of theinjection velocity follows from the experimental work of Brown,Dunlap, and coworkers [42,43]. This range may be sufficient in viewof Clayton’s CFD results [40]. These confirm that the taper profileschange minimally with increasing injection velocity provided thatthe wall injection Reynolds number, Re� �VbD=�, falls between102 and 104.

To capture coherent structures that correspond to our analyticalsolution, the numerical scheme that we adopt is based on the laminarmodel. We recognize, however, that the flowfield inside an actualSRMwill be turbulent, particularly, in the downstream portions of an

elongated chamber; this idea is nicely covered by Apte and Yang[44].

V. Results and Discussion

This section seeks to examine the dissimilarities that exist betweenthe numerical and analytical cases presented earlier. With availableinformation, one is able to determine the level of accuracy that isrequired andwhich physical parameters aremost important. As notedby Clayton [40], the axial velocity profiles and the pressure drop areof paramount importance. However, the pursuit of a generalexpression that describes the flowfield in tapered geometry requiresthat certain terms be neglected during the derivation process. Tovalidate the dismissal of these terms, the same problem is now solvednumerically. In the forthcoming sections, the numerical results arecompared to those found analytically.

A. Pressure Approximations

The slow increase in cross-sectional area in the axial direction actsto decrease the pressure drop by allowing a build up in local staticpressure with the accompanying decrease in dynamic pressure. Bycomparison toCulick’s solution in a straight circular-portmotor [10],Figs. 7a and 7b illustrate that there are substantial decreases in thepressure drop at higher taper angles taken at a length-to-radius ratioof z� 10. The value of z corresponds to the maximum length of thetapered segment for a bulk flow value of zero. Recovering the errorthat has accrued as a result of the constant radius assumption andneglecting the second-order axial derivative requires the addition of ahigher-order correction.

The behavior of the higher-order solution can be examined inFig. 7b. The addition of the higher-order correction seems to havemuch more of an impact at higher taper angles and larger tapereddomains; this trend suggests that the corrections are a requisite foranalytical pressure predictions for these cases. Compared with thehigher-order solution, the leading-order solution underpredicts thepressure drop at larger taper angles. In later portions of this work, thehigher-order correctionwill prove to be essential for the validation ofthe mathematical model. It is clear that there is an overprediction inpressure drop. The error increases as the gases head toward the aftend.At the edge of the solution domain, the total pressure drop can beoverpredicted by as much as 48 to 75% using the leading-orderexpression. The natural range for percent overprediction using thehigher-order correction varies here between 24 and 52%. Obviously,


use of the leading-order solution to predict the total pressure drop isnot sufficient.

B. Momentum Thrust Behavior

In Fig. 8, F, Fave, and � are shown at several taper angles and z;they are truncated in the streamwise direction when crossing theirrange of validity. The sensitivity of the momentum thrust to � seemsto be consistent with that of the pressure drop. As the taper angleincreases, continuity requires the average velocity to diminish alongwith the momentum thrust. It must be borne in mind, however, thatthis behavior corresponds to a steady-state, nozzleless, cold-flowsolution that does not account for neutral, regressive, or progressiveburning effects. The actual momentum thrust is slightly larger thanthe average value, as shown in Fig. 8b. Using L� 0, the momentumcorrection coefficient starts at 1

8�2 ’ 1:23370, following

Eq. (87); it increases to a maximum of 1.23546 at preciselyz� 0:0351868 cot�, then decreases back to unity. The descent ismore pronounced at higher taper angles for which the averagevelocity reaches its asymptotic limit more rapidly.

C. Axial Velocity

One should recall that rs was treated as a constant during theevaluation of the axial and radial velocity components.Consequently, it was expected that some error would be introducedinto the solution. In Fig. 9, it can be observed that there exists nodiscernible pattern for the analytical profiles and their numericalcounterparts. Thismay be attributed to the effects of curvature aswellas the nonuniform 3-D computational grid used for the numericalmodel. As a result, the numerical solution is seen to slightlyovershoot or undershoot the numerical solution at various axiallocations. This rather random discrepancy is more noticeable furtherdown themotor chamber; it may become non-negligible in very longmotors.

One should note that the analytical solutions shown in Fig. 9include higher-order corrections. Neglecting these higher-orderterms leads to a grossly underpredicted maximum centerlinevelocity. From the graph, it can be inferred that the leading-ordersolution lacks valuable flow information and that the higher-orderterms appear to be a requisite for accurate predictions in longchambers with large taper angles. The higher-order corrections seemto slowly recover the second-order axial derivative that was omittedin the basic solution. Clearly, Fig. 9 displays very good agreementbetween the analytical and numerical solutions.

D. Axial Derivatives

Our quest for an analytical, closed-form solution has requiredseveral assumptions. These assumptions have targeted severalderivatives, namely 1) @2 =@z2 and 2) d�=dz. Themagnitudes of thesecond-order axial derivatives of stream function have beenextracted from the numerical solution and quantified along the axis.The second-order radial derivatives were also plotted for the purposeof comparing the two quantities. Figure 9a clearly depicts anoscillatory behavior as is typical of a small fluctuating quantity.These fluctuations remain insignificant, namely, of O�10�4� downthe length of the chamber. In contrast, the radial derivatives shown inFig. 10b exhibit magnitudes ofO�1�, thus justifying their retention inthe analytical model.

-600

-400

-200

0

Culick, –π2z2/2∆p

0

a)

Ο (1)

-600

-400

-200

0

Culick, –π2z

2/2

L = 0

α = 0o

α = 1o

α = 2o

α = 3o

∆p1

b)

Ο (ε )

0

25

50

75

100O (1)O (ε )

3o

2o

L = 0

δ%

c)

1o

0 2 4 6 8 10-600

-400

-200

0

FLUENTTM

α = 0o

α = 1o

α = 2o

z

∆p1

d)

Fig. 7 Plots of a) leading-order and b) first-order approximations for

several values of�. In c) the percent overprediction at several values of �and d) numerical vs first-order approximation are shown.

0 2 4 6 8 100

200

400

600

800

1,000

L = 0

α = 0o

α = 1o

α = 2o

α = 3o

α = 0oF

a)

0 2 4 6 8 100

200

400

600

800

1,000

α = 0oFave

b)

0 5 10 15 20 25 301.00

1.05

1.10

1.15

1.20

1.25

L = 0

α = 0o

α = 1o

α = 2o

α = 3o

218 1.2337λ π=

z

λ

c)Fig. 8 Plots of a) actual thrust, b) average thrust and c) momentum

correction coefficient, F=Fave at several values of �.


Similarly, the magnitudes of the axial derivatives (described inFig. 10c) for the velocity ratio are small enough that they would haveno appreciable effect on the analytical solution. This plot shows theaxial variation of d�=dz at several radial locations for �� 3 deg.From the graph, it can be inferred that the axial variation of eachderivative dictates the shape of the velocity profile. For example, atr� rs, the axial derivative slowly increases. One may recall that thevelocity profile must adjust itself at each axial location to satisfy theno-slip requirement at the tapered surface. Bearing this inmind, it canbe realized that the rate of change at the wall must increase due to theincreased axial variation as the gases propagate down the chamber.At the axis of the chamber, one notices a decreasing derivative. Thisalso indicates that the centerline velocity must decrease in the axialdirection. From a physical standpoint, these changes must occur tosatisfy mass conservation. The increasing rate of change at the wallworks in conjunction with the decreasing rate of change at the axis tocause the profile to slowly relinquish its radial dependence with axialdistance. Hence, the profile may evolve into a near constant shapeover the cross-section for sufficiently long tapered domains.

VI. Limitations

For practical applications of the analytical solutions presentedheretofore, one may be concerned with their parametric limitations.Previously, such limitationswere explored by analytically predictingthe behavior of the gases at an infinite distance from the head end.The results of this inquiry suggest that some relationships, guided bymass conservation, must be maintained between the geometricparameters. As shown in Sec. II.D, the pertinent relationship may beexpressed as

�z � 12

R0�� L0 (89)

This criterion was obtained using the average value of the velocity asopposed to the maximum centerline velocity. To determine amaximum range for which the analytical solutions remainapplicable, one must calculate the maximum relative error betweenasymptotic predictions and numerical solutions. To do so, it isexpedient to examine the asymptotic limit where the velocities ineach chamber are at their maximum values; this can beaccomplished, for instance, by comparing the centerline velocitiespredicted by numerics and those by asymptotics. For the nontaperedsegments, the relationship between centerline and average velocitiescan be easily found to be

umax � 12�uave (90)

By translating this result to the centerline velocity, the criteria thatestablish the upper limit of the solution domain may be extrapolated.One finds

�z � 1�

R0�� L0 (91)

Equation (91) can now be solved for L0 � 0 to obtain the maximumconservative domain aspect ratio for a given taper angle. One finds

zcons ��zconsR0� 1��

(92)

It is our observation that, so long as z � zcons, the percent errorbetween numerics and asymptotics remains less than 1%. Theconservative range represents a domain of asymptotic validity in

0

0.5

1

r

α = 1o

a)

0

0.5

1

r

b)

α = 2ο

0

0.5

1

0 8 16 24 32

z = 2.5z = 5z = 7.5z = 10 FLUENT

TM

r

c) uz

α = 3o

Fig. 9 CFD versus analytical velocity profiles at various axial locationsfor select taper angles.

-0.0050

0.0000

0.0050

0.0100

0.0150 2 2zψ∂ ∂ α = 1o

α = 2o

α = 3o

a)

0

1.0

2.0

3.0

4.0

5.0 2 2rψ∂ ∂

α = 1o

α = 2o

α = 3o

b)

0 2 4 6 8 100

0.025

0.050 r = 0r = 0.25r = 0.50r = 0.75r = rs

z

dβ/ dz

c)

Fig. 10 Numerical axial derivative approximations from FLUENTTM

shown at several taper angles.


which the accrued error is virtually insignificant. By requiring aminimum chamber aspect ratio of 4 (lest edge effects becomeimportant), the maximum conservative taper angle for which anasymptotic solution would exhibit a smaller than 1% error can bereadily calculated from Eq. (92). One finds the maximumconservative taper angle to be 4.5 deg. Therefore, a suitable range oftapers would be 0< � � 4:5 deg.

Shown in Figs. 10a and 10b are the centerline velocities for �� 1and 4.5 deg, respectively. The behavior at�� 1 deg is quite similar,although the range is different (due to the difference in geometries).In Fig. 11a, it is seen that the maximum percent deviation is about

16.4%. Here, the conservative range is 0< z � 18:2. As predicted,the two solutions (numerical and analytical) begin to diverge outsideof the conservative domain.

For the case of the maximum taper angle, �� 4:5 deg, theconservative range is reduced to 0< z � 4. The location at which a20% discrepancy accrues between the two solutions is found to beapproximately 17.0. For values greater than 17.0, the deviationbegins to increase until it reaches a maximum value of 42.2%. Thenumerical cases shown here clearly support the analytical criterionwithin the conservative range. It may be inferred that the analyticalsolution exhibits a larger range of applicability for smaller taperangles.

For the idealized circular-port motor with very small taper(Fig. 11a), the maximum percent deviation is about 16.0% (atwhich point the error between numerics and asymptotics remainsapproximately constant as the solution levels off to a constant value).For such small taper, one may apply the analytical solution withminimal error, at least in theory, over an infinitely long domain. Therange of aspect ratios for which relative discrepancies remain under20% is generally much longer than those currently used in practice.In cold-flow models for SRMs, the present solution will remainwithin 20%of the numerical cases up to z=zcons ’ 4. In the interest ofclarity, the key components of the solution are summarized inTable 1.

VII. Conclusions

In this study, we have presented an approximate solution for themean flowfield in circular-port rocket motors with tapered bores. Inpursuit of expressions that characterize the chamber pressure,mathematical operations were performed with the assumption thatthe distance from the centerline was constant, given its variation ofO�sin��. Higher-order corrections were needed and these werefound to compensate for the assumption of axial invariance of thechamber radius. Later, the higher-order corrections helped to recoverthe effect of the second-order axial derivative of the stream function.

Given the methodology to obtain the desired flowfield, thenumerical simulation has been instrumental in substantiating theassumptions carried throughout the analytical derivation. Boththeoretical avenues concur in that:

0

30

60

90

Analytical FLUENT

TM

δ = 0%

umax

α = 1o

a)

δmax

= 16.4%

xcons

= 18.2,

0 10 20 30 40 50 600

10

20

30

40

x = 17,δ = 0% δ = 20.0 %

z

umax

α = 4.5o

b)

δmax

= 42. 2%

xcons

= 4

Fig. 11 Numerical and analytical approximations of the maximum

(centerline) velocity with attendant relative error for two taper angles.

Table 1 Components of the canted Taylor–Culick profile

Variable Equation

Stream function at canted surface s�z� � z sec ��1� 12 z tan�� LRadius at canted surface rs � 1� z tan�Vorticity at canted surface

�s � 4�2 sr3s

�1 � 2 s sin �

r2s� s cos�

rs

��0

�

��Vorticity along streamline

�� 4�2r r4s

�1 � 2 sin�

r2s� cos�

rs

��0

�

��Average velocity at flow cross section uave � 2 sr2s � 2

z sec��1�12z tan��L�1�z tan ��2

Maximum-to-average velocity ratio �0 � 12�, �1 �4 s3r2s� 2

3uave

Independent similarity variable �� 0 r2

r2s� �r2

2�1�z tan��2

Stream function 0 � s sin �, 1 � 16uave s3� cos�2�� 2 sin � � 4 cos �� 8=�� cos ��Axial velocity uz;0 � 12�uave cos �, uz;1 � 13 u2ave�2 � 12�� cos �� 2�� sin � � 12� sin�2��Radial velocity ur;0 �� 1�z tan��r sec� sin �, ur;1 ��

uave3sec� �1�z tan��

r

3� cos�2�� 2 sin �� 4 cos �� 8=�� cos ��

Leading-order axial pressure p�� 28z sec� �z sec ��2L��1�z tan��3 �

�2L2

2

Leading-order axial pressure drop �p0 �� 2

2

z sec��z sec��2L��1�z tan��4

First-order axial pressure drop �p1 �� 2

2z sec �r4s�z sec �� 2L� � " z sec�

18r8sf4�� 4�2z3sec3�� z2sec2�16�� 4�2L� 9�2r4s �

�3Lz sec�8�� 4�2L� 3�2r4s � � 16�� 4�2L3gAverage momentum thrust Fave � �2L�z sec��2�z tan ��

2

�1�z tan ��2

Actual momentum thrust F�z� � 12�3�L� z sec��2 � 1

2�2"�L� z sec ��4

9�3� � 10��L� z sec��2 � �z sec��2L� z sec��

Momentum correction coefficient for L� 0 �0 � 18�2 � 172��21� � 40�z sec �, �1 � 18�2z sec �� 172��21� � 40�z2sec2�Limit on range of applicability z � 1

2csc� � L


1) The incorporation of the taper is required to avoidoverpredicting the pressure drop.

2) The taper effect is more pronounced as the gases move awayfrom the head end due to the increasing cross-sectional area.

3) Long motors with L � 4 experience reduced sensitivity to thetaper angle. Smaller motors experience increased sensitivity.

4) The mean flow approaches its asymptotic limit in sufficientlylong motors.

5) When modeling short motors or those with very small taperangles, (0:5 deg

“Coupling betweenAcousticVelocityOscillations and Solid PropellantCombustion,” Journal of Propulsion and Power, Vol. 2, No. 5, 1986,pp. 428–437.

[43] Dunlap, R., Blackner, A. M., Waugh, R. C., Brown, R. S., andWilloughby, P. G., “Internal Flow Field Studies in a SimulatedCylindrical Port Rocket Chamber,” Journal of Propulsion and Power,Vol. 6, No. 6, 1990, pp. 690–704.

[44] Apte, S., and Yang, V., “Effect of Acoustic Oscillation on Flow

Development in a Simulated Nozzleless Rocket Motor,” SolidPropellant Chemistry, Combustion, and Motor Interior Ballistics,Vol. 185, edited byV.Yang, T. B. Brill, andW.-Z. Ren,AIAAProgressin Astronautics and Aeronautics, Washington, DC, 2000, pp. 791–822.

J. OefeleinAssociate Editor


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