Research Article Impact of the Surface Morphology on the Combustion of Simulated Solid Rocket Motor
11
Hindawi Publishing Corporation Mathematical Problems in Engineering Article ID 485302 Research Article Impact of the Surface Morphology on the Combustion of Simulated Solid Rocket Motor Abdelkarim M. Hegab, 1 Hani Hussain Sait, 1 and Ahmad Hussain 2 1 Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, Rabigh, P.O. Box 344, Jeddah 21911, Saudi Arabia 2 Department of Nuclear Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia Correspondence should be addressed to Ahmad Hussain; [email protected] Received 19 May 2014; Accepted 20 October 2014 Academic Editor: Chuangxia Huang Copyright © Abdelkarim M. Hegab et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An advanced and intensive computational solution development is integrated with an asymptotic technique, to examine the impact of the combustion surface morphology on the generated rotational flow field in a solid rocket chamber with wide ranges of forcing frequencies. e simulated rectangular chamber is closed at one end and is open at the aſt end. e upper and lower walls are permeable to allow steady and unsteady injected air to generate internal flow mimicking the flow field of the combustion gases in real rocket chamber. e frequencies of the unsteady injected flow are chosen to be very close or away from the resonance frequencies of the adapted chamber. e current study accounts for a wide range of wave numbers that reflect the complexity of real burning processes. Detailed derivation for Navier-Stokes equations at the four boundaries of the chamber is introduced in the current study. Qualitative comparison is performed with recent experimental work carried out on a two-inch hybrid rocket motor using a mixture of polyethylene and aluminum powder. e higher the percentage of aluminum powder in the mixture, the more the corrugations of the combustion surface. is trend is almost similar to the computational and analytical results of a simulated solid rocket chamber. 1. Introduction Nowadays, the designers of solid and hybrid rocket systems have a big challenge to understand the dynamics of the complex rotational fluid flow inside the rocket’s chamber, nozzle, and the exhaust plume. In order to have a stable rocket system with desirable performance, individual study for the combustion model that incorporates the burning of microscale propellant and the nonreactive model that accounts the complex fluid flow inside the chamber and the nozzles must be examined first. Some recent studies that consider the burning of microscale solid rocket propellant [1– 6] concluded that the injection conditions of the cold models in [7–14] are found to be different than that considered with the reactive models [1–6]. As a result, the current study is conducted to examine the unsteady flow field inside the solid rocket motor chamber and the complex generated wave and vorticity patterns and their impact on the burning of real solid propellant surfaces. We try in this study to assign unsteady velocity injection with axial distributions to simulate the real variation of composite propellants. Recent studies of the rotational flow field generating from transient side wall mass injection have been surveyed in detail in [12–14]. ese studies revealed that the coexistence of the acoustic wave patterns and the injection mass addition are the reason behind the generating of the vorticity dynamics in the combustion chamber. However, many researches have been directed to analyze the acoustic wave patterns including the acoustic velocity, temperature, and pressure time history without any consideration for the existence of rotational flow (vorticity) transients. A competitive linear [15–17] and nonlinear theory [11, 13, 14] are formulated to show that the precise descriptions of the fluid flow in the simulated rocket motor chamber require the presence of the acoustic and the rotational flow fields.
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Transcript of Research Article Impact of the Surface Morphology on the Combustion of Simulated Solid Rocket Motor
Hindawi Publishing CorporationMathematical Problems in EngineeringArticle ID 485302
Research ArticleImpact of the Surface Morphology on the Combustion ofSimulated Solid Rocket Motor
Abdelkarim M Hegab1 Hani Hussain Sait1 and Ahmad Hussain2
1Mechanical Engineering Department Faculty of Engineering King Abdulaziz University Rabigh PO Box 344 Jeddah 21911 SaudiArabia2Department of Nuclear Engineering King Abdulaziz University PO Box 80204 Jeddah 21589 Saudi Arabia
Correspondence should be addressed to Ahmad Hussain ahassainkauedusa
Received 19 May 2014 Accepted 20 October 2014
Academic Editor Chuangxia Huang
Copyright copy Abdelkarim M Hegab et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
An advanced and intensive computational solution development is integrated with an asymptotic technique to examine the impactof the combustion surface morphology on the generated rotational flow field in a solid rocket chamber with wide ranges of forcingfrequencies The simulated rectangular chamber is closed at one end and is open at the aft end The upper and lower walls arepermeable to allow steady and unsteady injected air to generate internal flow mimicking the flow field of the combustion gasesin real rocket chamber The frequencies of the unsteady injected flow are chosen to be very close or away from the resonancefrequencies of the adapted chamber The current study accounts for a wide range of wave numbers that reflect the complexity ofreal burning processes Detailed derivation for Navier-Stokes equations at the four boundaries of the chamber is introduced in thecurrent study Qualitative comparison is performed with recent experimental work carried out on a two-inch hybrid rocket motorusing a mixture of polyethylene and aluminum powder The higher the percentage of aluminum powder in the mixture the morethe corrugations of the combustion surface This trend is almost similar to the computational and analytical results of a simulatedsolid rocket chamber
1 Introduction
Nowadays the designers of solid and hybrid rocket systemshave a big challenge to understand the dynamics of thecomplex rotational fluid flow inside the rocketrsquos chambernozzle and the exhaust plume In order to have a stablerocket system with desirable performance individual studyfor the combustion model that incorporates the burningof microscale propellant and the nonreactive model thataccounts the complex fluid flow inside the chamber and thenozzles must be examined first Some recent studies thatconsider the burning ofmicroscale solid rocket propellant [1ndash6] concluded that the injection conditions of the cold modelsin [7ndash14] are found to be different than that considered withthe reactive models [1ndash6] As a result the current study isconducted to examine the unsteady flow field inside the solidrocket motor chamber and the complex generated wave andvorticity patterns and their impact on the burning of real solid
propellant surfaces We try in this study to assign unsteadyvelocity injection with axial distributions to simulate the realvariation of composite propellants
Recent studies of the rotational flow field generating fromtransient sidewallmass injection have been surveyed in detailin [12ndash14] These studies revealed that the coexistence of theacoustic wave patterns and the injection mass addition arethe reason behind the generating of the vorticity dynamicsin the combustion chamber However many researches havebeen directed to analyze the acoustic wave patterns includingthe acoustic velocity temperature and pressure time historywithout any consideration for the existence of rotationalflow (vorticity) transients A competitive linear [15ndash17] andnonlinear theory [11 13 14] are formulated to show that theprecise descriptions of the fluid flow in the simulated rocketmotor chamber require the presence of the acoustic and therotational flow fields
staff
Note
Dear Professor1313We cannot change Figures to be ldquofiguresrdquo or ldquoFigsrdquo as per journal style Please check all the corrections we made and confirm1313Best Regards
2 Mathematical Problems in Engineering
00
y998400
H998400
Clos
ed en
dCombustion surface
Symmetry conditions
u998400= 0
998400= 0
(u998400
998400 P
998400 T
998400 120588
998400)
P998400= P
998400o ldquoPressure noderdquo
L x998400
Solid propellant
rb(x t)
rb(x t) cong 998400|wall(x t) = minus998400rws(x) +
998400o|wall(x
998400 t
998400)
998400= 0
120597(120575998400)
120597y998400= 0 120575
998400= (u
998400 T P
998400 120588
998400)
u998400= 0 T = T
998400o
Figure 1 The physical domain and the boundary conditions for the simulated solid rocket motor chamber
1
1
09
09
08
08
07
07
06
06
05
05
04
04
03
03
02
02
01
0100
y
Asymptotic analysisComputationExperimental Rew = 1030
Rew = 1302
Rew = 1565
u(05 y)u(05 0)
Figure 2 Normalized axial velocity profiles at the midlength of thechamber for the steady state flow
Numerical approaches for the presence of vorticity in aninjected internal flow are presented in [19 20] by solvingthe Navier-Stokes equation Moreover the studies in [9 10]presented detailed comparison of asymptotic and numericalapproaches without any consideration to the thermal effectIntensive numerical procedures for the effect of exothermicreactions very close to the sidewall have been carried out in[21 22]
Finally the root mean square value for the axial distri-bution for the coexistence of rotational and irrotational flowfield is calculated in [8 23]
In the present study asymptotic and computationalresults are obtained to consider the unsteady variation ofthe vorticity and temperature dynamic in the prescribedchamber Results are given for steady state solution generatedfrom spatially uniform injection velocity The convergentsolution is oscillated with similar amplitude unsteady massadditions with wide ranges of injection wave numbers Themagnitude of the superimposed injection velocity is chosen
0 5 10 15 20 25 30 35 40Time
02
015
01
005
0
minus005
minus01
minus015
minus02
ComputationalAnalytical
T
x = 05 y = 00
Figure 3 Comparison between the computational and the asymp-totic solutions for the perturbed temperature time history with 120596 =
1 (nonresonance) 120576 = 04 119872 = 002 and 119877119890= 3 lowast 10
5 at the half-length of the chamber
to be large enough to make sure that the fluid is alwaysinjected from the side wall and reduce the probability ofsucking outside the chamber
2 The Physical and Mathematical Formulation
The present numerical study is performed to study the effectof the complicated surface morphology of the combustionsurface on the generated vorticity in a chamber of length1198711015840 and half-height 119867
1015840 The aspect ratio in this study 120575 =
1198711015840
1198671015840 is taken to be twenty Pressure node is assigned at
the chamber exit while closed end is considered at the headend Air is injected from the side wall with a characteristicvelocity 119881
1015840
119910119900
to simulate the combustion products from real
Mathematical Problems in Engineering 3
002
0406
081
0
05
10
5
10
15
20
25
xy
120596
(a)
002
0406
081
0
05
1
0
1
2
3
4
xy
minus1
minus2
120596
(b)
002
0406
08 1
0
05
1
xy
0
1
2
3
4
minus1
minus2
120596
(c)
Figure 4 Vorticity topography for 120596 = 1 (nonresonance) and 120576 = 04119872 = 002 and 119877119890= 3 lowast 10
5 at 119905 = 42 at several axial wave numbers (a)119899 = 1 (b) 119899 = 3 and (c) 119899 = 5
burning of propellant This modeling generates an axial flowcharacterized by 119880
1015840
119911119900
= 1205751198811015840
119910119900
as in Figure 1The steady and the unsteady injection velocity are denoted
by V1015840119903119908119904
(119909) V1015840119900|wall(119909
1015840
1199051015840
) while the exit pressure and thetemperature of the fluid injected from the side wall aredenoted by 119875
1015840
119900and 119879
1015840
119900
The parabolized Navier Stokes equations for unsteadytwo dimensional compressible fluid flow are solved numeri-cally in the prescribed chamber andmay bewritten as follows
120597119876
120597119905
+
120597119864
120597119909
+
120597119865
120597119910
= 0 (1)
where
119876 = [120588 120588119906 120588V 119864119879]119879
119864 = [1198721205881199061198721205881199062
+
1
1205741198722119901119872120588119906V119872 119864
119879+ (120574 minus 1) 119901 119906]
119879
119865 = [119872120588V119872120588119906V minus
1205752
119872
119877119890
119906119910119872120588V2 +
1205752
120574119872
119901
119872 119864119879+ (120574 minus 1) 119901 V minus
1205741205752
119872
119877119890119875119903
119879119910]
119879
(2)
119864119879is the total energy (120588[119879+120574(120574minus1)119872
2
(1199062
+(V120575)2]) and theequation of state for a perfect gas is
119901 = 120588119879 (3)
In this study the specific heats the viscosity and conductivitycoefficients are considered to be constant since the change
4 Mathematical Problems in Engineering
0 02 04 06 0809
095
Injection surface
x
y
20
0
minus20
(a)
0 02 04 06 08x
09
095y
10
0
minus10
(b)
0 02 04 06 08x
09
095y
10
0
minus10
(c)
Figure 5 The vorticity contours adjacent to the injection surfacefor the same parameters as in Figure 4 for several eigenvalues (a)120582119899= 1205872 (119899 = 1) (b) 31205872 (119899 = 3) and (c) 51205872 (119899 = 5)
in the temperature in the cold model is relatively smallNondimensional variables are given by
119901 =
1199011015840
1199011015840
119900
120588 =
1205881015840
1205881015840
119900
119879 =
1198791015840
1198791015840
119900
119906 =
1199061015840
1198801015840
119911119900
V =
V1015840
(1198801015840
119911119900
120575)
119909 =
1199091015840
1198711015840
119910 =
1199101015840
1198671015840 119905 =
1199051015840
1199051015840
119886
(4)
The axial flow Mach number Prandtl number flow andacoustic Reynolds numbers are written as
119872 =
1198801015840
119911119900
1198621015840
119900
119875119903=
1205831015840
1199001198621015840
119901119900
1198961015840
119900
119877119890=
1205881015840
1199001198801015840
119911119900
1198711015840
1205831015840
119900
119877119890119860
=
119877119890
119872
(5)
The characteristic Mach number is chosen to be 119874(10minus1)while the characteristic Reynolds number is119874(105-106)Theseorders of magnitudes approximate the characteristics of thefluid flow in similar real rocket engineThe nondimensional-ized time 119905
1015840
119886is defined as the axial acoustic time 119905
1015840
119886= 1198711015840
1198621015840
119900
21 The Initial and Boundary Conditions For 119905 le 0 time-independent side wall transverse velocity with 119881|wall =
minus119881119903119908119904
(119909) is used to generate steady flow solution while for119905 gt 0 119881|wall = 119881
119903119908119904(119909 119905) = minus119881
119903119908119904(119909) minus 120576120593wall(119909)(1 minus cos120596119905)
as the unsteady longitudinal variation for the injectiontransverse at resonance or away from the resonance caseThe function 120593wall represents the longitudinal variation ofthe velocity at the injection surface Moreover the specifiedconditions at the four boundaries can be written as follows
119909 = 0 119906 = 0 V = 0
119909 = 1 119901 = 1 ldquopressure noderdquo
119910 = 0 V = 0
120597119906
120597119910
=
120597119879
120597119910
=
120597119901
120597119910
=
120597120588
120597119910
= 0
119910 = 1 119906 = 0 119879 = 1
V =
minusV119903119908119904
(119909) 119905 le 0
minusV119903119908119904
(119909) minus 120576120593119903119908
(1 minus cos120596119905) 119905 gt 0
(6)
and the initial condition is
119905 = 0 119906 = V = 0 119901 = 119879 = 1 (7)
The initial conditions for the unsteady flow computations arethe convergence solutions for the steady state conditions
22 The Analytical Approach
221 Steady and Unsteady Flow Hegab [7] has concludedthat the total axial velocity component includes two mainpartsThefirst part represents the steady state component andthe other part represents unsteady state component and canbe written as follows
(119906 V 119875 120588 119879) = (119906119904 V119904 119875119904 120588119904 119879119904) + ( V 120588
119879) (8)
Using the expended variables used inHegab [7] and insertingthem in (1) the side wall steady vorticity generation (Ω
119900119904|wall)
and the accompanying heat transfer (120597119879119900119904120597119910|wall) are derived
and written as follows
Ωos1003816100381610038161003816wall = minus
1
120574V119903119908119904
120597
120597119909
(
1205741205872
4
int
1
119909
V119903119908119904
(119909) int
119909
0
V119903119908119904
(120591) 119889120591 119889119909)
+
120597V119903119908119904
(119909)
120597119909
120597119879os120597119910
10038161003816100381610038161003816100381610038161003816wall
= 0
(9)
Equations (9) show that nonzero steady vorticity and zeroheat transfer are found with the use of no-slip and constantinjection boundary conditions
The sequence of the numerical solution consists of twosteps The first one is the initial steady state computationwhich arises from constant injection with isothermal con-dition The second one is the unsteady flow computationbased on initial conditions as computed from the first stepThe perturbation theory introduced by Staab et al [11] and
Mathematical Problems in Engineering 5
(a) (b)
(c) (d)
Figure 6 The flame geometry for the burning of hybrid rocket systems using (a) pure PE fuel grain (b) 975 PE +25 Al fuel grain (c)925 PE + 75 Al fuel grain and (d) 85 PE + 15 Al fuel grain [18]
200
201
202
203
204
205
600 625 650 675 700 725 750 775 800Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
Recording rate 60 points frequency about 20 cps and about 003 bar pure PEamplitude
Figure 7The recorded pressure time history for a burning of hybridmotor using pure polyethylene (PE) fuel grain
applied by Hegab [7] leads to the following weakly rotationalequations
119905 = 0 119900= 0
120597119900
120597119905
= 0
119909 = 0
120597119900
120597119909
= 0 119909 = 1 119900= 0
119910 = 0 V0= 0
119910 = 1 V119900= minus120576120593
119903119908(1 minus cos120596119905) 119905 gt 0
(10)
200
201
202
203
204
205
500 525 550 575 600 625 650 675 700Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
about 002 bar 75 AlRecording rate 60 points frequency about 20 cps and amplitude
Figure 8The recorded pressure time history for a burning of hybridmotor using 75 aluminum (Al) additive to PE fuel grain
205
204
203
202
201
200500 525 550 575 600 625 650 675 700
Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
Recording rate 60 points frequency about 20 cps about 0018 bar 15 Aland amplitude
Figure 9The recorded pressure time history for a burning of hybridmotor using 15 aluminum (Al) additive to PE fuel grain
6 Mathematical Problems in Engineering
The final nonresonance solution for the flow in the simulatedchamber in Figure 1 is
119900(119909 119905) =
infin
sum
119899=0
2120576120574120596int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
119900(119909 119905) =
infin
sum
119899=0
2120576120574120596int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
(cos120573119899119905 minus 1) minus
120573119899
120596
(cos120596119905 minus 1)
times sin120573119899119909
V119900(119909 119910 119905) = minus119910 120593
119903119908(119909) (1 minus cos120596119905)
(11)
Equations (11) show the time history of the perturbed valuesvariables (
119900 119900 V119900) at any axial locations
Following the same policies the resonance solutions arederived and formulated as
119900(119909 119905) =
120574119862119899lowast
120573119899lowast
sin120573119899lowast119905 minus 120573
119899lowast119905 cos120573
119899lowast119905 cos120573
119899lowast119909
119900(119909 119905) = 119862
119899lowast minus
2
120573119899lowast
(cos120573119899lowast119905 minus 1) minus 119905 sin120573
119899lowast119905
times cos120573119899lowast119909
(12)
where 119862119899lowast = 2 int
1
0
120593119903119908
(119909) cos120573119899lowast119909
23 The Computational Approach In the present studyhigher order accuracy difference equations are employedto solve the parabolized Navier-Stokes equations in theinterior of the prescribed chamber and at the boundariesThe numerical multisteps scheme in [24] is used to solve the2D unsteady and compressible Navier-Stokes equations forthe points located near the boundaries while the two-fourexplicit predictor-corrector scheme in [25] is employed tosolve the adapted equations at the interior points The lattermethod is a fourth-order accuracy variant of the fully explicitMacCormack scheme Uniform grids are used The accuracycriterion employed byMacCormack [24] is used to determinethe size of the time step
Here the NSCBC technique applied by [7] is presented indetail The full derivation for the parabolized Navier-Stokesat the upper and lower boundaries and at the head and exitends is formulated to specify the nonphysical conditions atthese boundaries Moreover this method is applied to reducethe generated reflected waves arising from the numericaltechniques
Based on the idea of characteristics for Euler equationsthe first derivatives normal to the boundaries appearing inthe convective terms of the Navier-Stokes equations may beexpressed in terms of the amplitude of incoming outcoming
and entropy waves weierp119894rsquos The fluid dynamics equations can
be written in nondimensional wave modes form as follows
120597120588
120597119905
+
1
1198622[weierp2+
1
2
(weierp4+ weierp1)] + 119872
120597120588V120597119910
= 0 (13)
120597119906
120597119905
+
1
2120588119862
[(weierp4minus weierp1)] + 119872V
120597119906
120597119910
=
120594119909
120588
(14)
120597V120597119905
+ weierp3+ 119872V
120597V120597119910
+
1198602
120574119872
1
120588
120597119901
120597119910
=
120594119910
120588
(15)
120597119901
120597119905
+
1
2
[(weierp4+ weierp1)] + 119872V
120597119901
120597119910
+ 120574119872119901
120597V120597119910
= 120594119864 (16)
where 120594119909and 120594
119910represent the viscous terms and 120594
119864refers
to the combination of the viscous and conduction terms inthe axial and transverse directions and 119862 = 119862
1015840
1198621015840
119900 The
amplitudes of the left-running waves (weierp1) the right-running
waves (weierp4) and the entropy waves (weierp
2 weierp3) in terms of the
characteristic velocities 120582119894rsquos are
weierp1= 1205821(
1
120574
120597119901
120597119909
minus 119872120588119862
120597119906
120597119909
) 1205821=
119906
119872
minus 119862 (17)
weierp2= 1205822(1198622120597120588
120597119909
minus
1
120574
120597119901
120597119909
) 1205822= 119872119906 (18)
weierp3= 1205823
120597V120597119909
1205823= 119872119906 (19)
weierp4= 1205824(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
) 1205824=
119906
119872
+ 119862 (20)
Two approaches are used to infer the values of theincoming amplitude-variations from the boundary condi-tions One is the local one-dimensional inviscid relations(LODI) technique and the second is the Navier-Stokescharacteristics boundary conditions (NSCBC) For the latterone the complete description of the numerical boundaryconditions at all boundaries is discussed below
24 Exit Boundary Conditions In this case a pressure nodeis assumed at the exit boundary The NSCBC relation (16)suggests that the amplitude of the reflected wave should be
weierp1= 2120594119864minus weierp4 (21)
and the remaining conditions are
weierp2= 119872119906(119862
2120597120588
120597119909
minus
1
120574
120597119901
120597119909
) weierp3= 119872119906
120597V120597119909
weierp4= (
119906
119872
+ 119888)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
(22)
Mathematical Problems in Engineering 7
The nondimensional equations can be written as follows
120597120588
120597119905
+ 119872119906
120597120588
120597119909
minus
119872119906
1205741198622
120597119901
120597119909
+ 119872V120597120588
120597119910
+
1198602
119872
(120574 minus 1) 119877119890119875119903
1
1198622
1205972
119879
1205971199102
= 0
120597119906
120597119905
+
1
120588119862
[(119906 +
119862
119872
)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)] + 119872V120597119906
120597119910
minus
1198602
(120574 minus 1) 119877119890119875119903
1
120588119862
1205972
119879
1205971199102
minus
1198721198602
119877119890
1
120588
1205972
119906
1205971199102
= 0
120597V120597119905
+ 119872(119906
120597V120597119909
+ V120597V120597119910
) = 0
119889119901
119889119905
= 0
(23)
Here again the nondimensional variables (120588 119906 and V)can be computed using one-sided finite-difference approxi-mations
25 Head End Boundary Condition The velocity componentnormal to the boundary (119906(0 119910 119905)) is zero since the head endis impermeable In contrast the transverse velocity compo-nent cannot be specified because the axial viscous terms inthe transversemomentum equation have been suppressed Asthe normal velocity drops to zero at 119909 = 0 the amplitudesweierp2and weierp
3 in (18) and (19) are zero and the characteristic
velocities are 1205821
= minus119862 and 1205824
= +119862 The reflected wave weierp4
enters the domain while the wave weierp1propagates out through
the boundary The NSCBC relation (19) indicates that theamplitude of the incoming wave weierp
4should be
weierp4= weierp1 (24)
and the remaining conditions are
weierp1= minus119888(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
weierp2= 0 weierp
3= 0
(25)
The equations can be written in nondimensional form asfollows
120597120588
120597119905
minus 119872120588
120597119906
120597119909
+
1
120574119862
120597119901
120597119909
+ 119872
120597120588V120597119910
= 0
120597119906
120597119905
= 0
120597V120597119905
+ 119872V120597V120597119910
= 0
120597119901
120597119905
minus 119888
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
+ 120574119872119901
120597V120597119910
minus
1205741198602
119872
(120574 minus 1) 119875119903119877119890
1205972
119879
1205971199102
= 0
(26)
More details about the NSCBC techniques are found in [2326]
26 CodeValidations Thecomputational codes are designedconstructed and validated for steady and unsteady flow as
shown in Figures 2 and 3 For the latter one the constructedcodes run for many wave cycles to detect the whole unsteadyrotational flow dynamic across the chamber
In Figure 2 comparison between the current numericaland asymptotic results dealing with the steady state flowsolutions is carried out The steady state flow is generatedby adding constant side wall mass injection (119881
119903119908= minus1)
from the injection surface of the prescribed chamber Similarexperiment in [18 27] was carried out and compared withthe current mathematical models in Figure 2The asymptoticanalysis solution represents the solid line while the dottedline represents the converged steady state numerical solutionfor viscous flow with wall Reynolds number 119877
119890119908= 5 lowast
103 The nonfilled points refer to the experimental measured
data The experimental results are recorded for flow in achannel with fully transpired wall Three different poroussurfaces using three different sizes of honeycomb cells areused [7] In this figure a comparison between the analyticaland the numerical solutions at the midlength of the chamberis illustratedThis comparison shows a reasonable agreementbetween the three approaches The comparison between thetheoretical and the computational approaches shows littlebit deviation This deviation is logically expected since theused honey combs have three different sizes The smallersize of honey combs the smaller deviations between the twoapproaches
The second set of the validations show the solution for theunsteady flow computations at nonresonance forcedmode Inthis set of the results the computational results are validatedby comparing with the asymptotic analysis These resultsrepresent the asymptotic solution when the sidewall injectionvelocity is represented by V
119903119908(119909 1 119905) = minus[1 + 120576120593
119903119908(1 minus
cos120596119905)] 120593119903119908
= cos(120573119899119909) In the current work and the work
by Hegab [7] the disturbance forcing frequency is taken tobe 120596 = 1 (nonresonance) with the amplitude of 120576 = 04
and the eigenvalues (120573119899
= 1198991205872 119899 = 1 3 5 7 9 etc)The temperature time history (119879) at themedlength location ispresented in Figure 3 for 119877
119890= 3 lowast 10
5119872 = 002 120575 = 20 and119899 = 10 For the nonresonance case the solution shows stableconditions but it is not purely harmonic This interestingbehavior shows the ability of the constructed code with thegood treatment to the boundary conditions to detect thenon-harmonic trend as detected by the analytical approachA comparison of the perturbed temperature (119879) (27) withthe numerical results in Figure 3 shows good agreementbetween the two approaches especially for 119905 lt 20 while thecomputational solution shows little less deviation than theanalytical results for 119905 gt 20 due to the nonlinearity effects
119879119900(119909 119905) =
infin
sum
119899=0
2120576120596 (120574 minus 1) int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
(27)
Generally the coexistence of the forced and the naturalmodes of the system is detected in the adapted computationaland analytical approaches as seen in the earlier investigationby Hegab [7] In contrast the computational techniques in
8 Mathematical Problems in Engineering
[10] show only pure harmonic oscillations The source ofthese deficiencies may be arising from the treatment of theboundary conditions especially at the head and aft ends ofthe simulated chamber
3 Results and Discussions
The impact of the injected wave number from the sidewallalong with the frequencies of the forced mode on thegenerated complex acoustical fields and their interactionwiththe fluid are examinedThe results have been categorized intothe following two sets
31 Nonresonance Solution In the current study a widerange of forcing frequencies away from or very close tothe first fundamental modes of the prescribed chamber isconsidered [7] a wide range of forcing frequencies away fromor very close to the first fundamental modes of the prescribedchamber is considered In Figure 4 the irrotational androtational fields across the whole chamber are presentedwhen 119905 = 42 for 120596 = 1 119877
119890= 3lowast10
5119872 = 002 120576 = 04 119899 = 13 and 5 and 120582
119899= 1205872 The latter one represents the Eigen
values of the chamberThese interesting graphs show that therotational fields are generated at the injection surface As timeincreases the generated unsteady vorticity is transferred to fillmost of the combustion chamber Moreover the constructedcodes are able to run for enough time and count about sevenvorticity wave cycles as 119905 = 42 It is also seen that theamplitude of the rotational field attainsmaximumvalues nearthe wall and found to be order of ten for the monotonicvariation (119899 = 1) and nonresonance frequency (120596 = 1)One notes that these magnitudes meet the scaling of theasymptotic solutions in [7 9 11 14] Moreover for 119899 = 1monotonic distribution of unsteady vorticity is noticed whilethe variation goes in nonmonotonic axial distribution for 119899 gt
1 For the nonresonance solution the longitudinal variationof the vorticity at the wall 120597|Ω
119908(119909 119905)|120597119909 gt 0 for 120582
119899=
1205872 (119899 = 1) is revealing that the magnitude of rotationalflow fields on the injection surface increases as axial distanceincreases
The vorticity contours adjacent to the injection surface forthe same parameters as in Figure 4 for several eigenvalues(a) 120582119899
= 1205872 (119899 = 1) (b) 31205872 (119899 = 3) and (c) 51205872
(119899 = 5) are present in Figure 5 It is clearly seen that adjacentto the combustion surface the generated unsteady vorticitymay impact the surface at downstream locations for 119899 = 1at two locations for 119899 = 3 and many locations for 119899 = 5The rotational flow fluctuates from negative to positive andvice versa at several successive times and may cause scouringon the combustion surface and in turn may increase theshearing stresses adjacent to the combustion surface Thisimportant conclusion may lead to increase of the burningrates over the designed ones especially for low wave numbersince the amplitude of the unsteady vorticity decreases as thewave number increases
Fortunately recent experimental work by Meteb [28] in2009 was directed to study the acoustic instability in hybridrocket system using different fuel grains and ignition systems
The experimental results verified qualitatively the currentanalytical and computational results for the generated com-plex wave patterns and the effect of the combustion surfaceon the acoustic fluid dynamics interactionmechanism In thisexperiment the pure polyethylene (PE) fuel grain and PEwithadditives of aluminum (Al) powder with different percentageare used to predict the hybrid rocket motor regression ratesThe flame geometry and structure for the burning of thepure PE and PE with additives of aluminum (Al) powderat 25 75 and 15 of the total weight of the wholesamples are illustrated in Figure 6The effect of theAl powderon the flame length geometry and the rate of reaction forthe burning of each sample is clearly seen In addition thepressure time histories at a point before the combustionchamber in the way of oxidizer are recorded and illustratedin Figure 7 for the pure PE and in Figures 8 and 9 for PEwith additives of aluminum (Al) powder at 75 and 15of the total weight respectively It is clearly seen that theamplitude of the pressure waves decreases as the percentageof Al fuel grain increases in the sample This interestingbehavior is similar to what we have seen when we increasethe wave number of the combustion surface As we knowmost of the Al fuel grains burn away from the combustionsurface consisting ofmany fragmentsThe escaping processesof Al fuel grains from the combustion surface generate manyof cavities that make the combustion surface more roughthan the smooth one in case of pure PE The experimentaland computational results are qualitatively in agreement Forthe first time the analytical and computational results forthe current cold model and the acoustic instabilities foundby Meteb [28] state an important and essential conclusionabout the effect of the combustion surface morphology onthe acoustic instability inside the rocket chamber and theaccompanying rotational flow Instantaneous total velocityvector map for the parameters in Figure 4 at (a) 119905 = 0 ldquosteadystate solutionrdquo and (b) 119905 = 16 is presented in Figures 10(a)and 10(b) The injection surface is located at 119910 = 1 on theupper part of the plot where the flow is completely verticalwhile the lower part of the plot represents the center lineat 119910 = 0 where the flow is completely axial The closedend is located at 119909 = 0 on the left hand side of the plotwhile at 119909 = 1 it represents the exit plane on the right handside One may notice the smooth and undisturbed fluid flowthrough the entire map for the steady state solution at 119905 = 0After initiating the side wall disturbances the coexisting ofacoustics and the generated rotational flow is clearly seenin the upper half part of the chamber and illustrates howthe interaction between the travelling acoustic waves and theinjected fluid affects the fluid flow in the chamber In additionat 119905 = 32 in Figure 11 there is an instantaneous reverseflow found at a location near the combustion surface Thecomplex acoustic fluid dynamics interaction structure mayincrease the shear stresses adjacent to the combustion surfaceand through the whole cavity These results may providethe rocket designers with conceptual tools and data to beconsidered in case of redesigning strategies
Mathematical Problems in Engineering 9
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
ldquoSteady state solutionrdquo t = 00
(a)
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 16Nonresonance (
(b)
Figure 10 Instantaneous total velocity vector map for the parameters in Figure 4 (a) 119905 = 0 ldquosteady state solutionrdquo and (b) 119905 = 16
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 32
Reverse flow region
Nonresonance (
Figure 11 Instantaneous total velocity vector map at 119905 = 32 for theparameters in Figure 4
32 Near and at Resonance Solution A comparison of theasymptotic analysis perturbed pressure time history withthe computational one at and near resonance frequenciesis presented in Figure 12 For 119905 lt 10 the results showgood agreement between the two approaches even with theasymptotic beat condition (120596 = 1205872 minus 00018) which isvery close to the resonance frequency While for 119905 gt 10
the asymptotic solution shows linear growth with time basedon the linear theory for the derived wave equation where120596 = 120573
119899lowast = 1205872 as derived by Hegab [7] In contrast
the computational solution shows beats at the resonancecondition due to the nonlinearity and the asymptotic beatsolutions have smaller amplitude than the resonance oneThe beats appear due to the interaction between the derivingfrequency mode 120596 = 1205872minus00018 and the first eigenfunctionmode (1205872) It is clearly seen that the amplitude of thevorticity at the resonance frequency is found to be ten timesthan the amplitudes at the nonresonance one in Figure 4
P
03
02
01
0
minus01
minus02
minus030 5 10 15 20 25 30 35 40
t
ComputationalAnalyticalAnalytical (beats)
x = 05 y = 00 120596 = 1205872 n = 1
Figure 12 Comparison between the computational and the asymp-totic solutions for the perturbed pressure time history with 120596 = 1205872
(resonance) 120576 = 04 119872 = 002 and 119877119890= 3 lowast 10
5 at the half-lengthof the chamber
Moreover it is found that the fluctuation of the transversetemperature gradient at the injection surface is found tobe about ten times than the fluctuation of the temperaturedisturbances This concludes that since the heat transferalong the sidewall is directly proportional to the transversetemperature gradient there is an unexpectedly large heat fluxwhich exists both at the wall and within the entire chambergiven the small size of the temperature distribution Theasymptotic analysis by Staab et al [11] is used to prove thisimportant conclusion
Finally the asymptotic and computational methods areemployed to study the dynamic process in an acoustically
10 Mathematical Problems in Engineering
simulated solid propellant rocket chamber with transientsidewall mass addition at different wave numbers
4 Conclusion Remarks
The current study emphasizes the effect of the combustionsurfacemorphology of the solid rocket systemon the acousticinstability and the accompanying vorticity and temperaturedynamics generation The rotational and irrotational fieldsgenerated in the prescribed solid rocket motor chamber withMach number are chosen to be O(10minus1) while the charac-teristic Reynolds number is O(105-106) at nonresonance andnear and at resonance frequencies have been described TheNSCBC techniques that applied in [7 29] are used to treat thenumerical boundary conditions
The numerical solution shows that the convergence tosteady state solution is achieved first and then additionaltransient side wall mass addition is initiated Moreover theresult shows that the forced mode of the nonresonance andresonance frequencies generates longitudinal acoustics waveswhich interact again with its sources at the injection surfaceto generate the rotational flow fields The generated unsteadyvorticity is transferred from the injection surface toward thecenterline to fill the entire chamber as time increases aftermore than seven wave cycles The solution shows nonpureharmonic oscillations as suggested from the theory in [11]Vorticity 3D graphs and contours illustrate that spatially themaximum amplitudes are found to be near the injectionsurface Moreover the largest shear stresses are found to be atthe injection surface near the exit end while the shear stressesimpact the injection surface at many locations as the wavenumber increasesThe longitudinal variation of the rotationalflow fields is found to be nonmonotonic for 119899 gt 1
Here for the first time the analytical and computationalresults for the current coldmodel and the recent experimentalwork by Meteb [28] revealed that the combustion surfacemorphology has a great effect on the acoustic instabilityinside the rocket chamber and the accompanying rotationalflow The computational approach reveals that there areunexpectedly high heat transfer and generated rotational flowfields found to be adjacent to the injection surface in bothnonresonance and resonance cases The amplitude of thelatter one is found to be approximately ten times than thatwith the nonresonance solution One may anticipate thatthese large oscillatory gradients will persist to the surface ofa burning solid propellant in a real rocket chamber
More traditional computational studies of turbulence inchannel flows with wall injection are reviewed by Chaouat[30] Mean axial velocity profiles are found to be laminaruntil about halfway down the channel and then show signs oftransition Chaouatrsquos results do not show the acoustic-relatedeffects discussed in the present work As a result it is difficultto know from the turbulent theory results if the source ofthe downstream vorticity is classical transition to turbulenceor partly the result of the kind of mechanism discussed inthis paper In our recent studies in 2011 and 2012 [31 32]the assessment of turbulence modeling for gas flow in thechamber and in the convergent-divergent rocket nozzle for
the steady state flow is studied The detailed information andrealities in the current research about the transient coexistingof the irrotational (acoustics) rotational (generated vorticityfrom the interaction between the injected flow and thetraveling acoustic waves) and our turbulent techniques thatwe have adapted in our studies in 2011 and 2012 for the steadyflow field in SRM chambernozzle geometry will be used toaccount all the complex flow fields in our future study
Finally results of the asymptotic and computationalapproaches may provide the rocket designer with codes andinformation making them able to interpret some reasonsbehind the combustion instability inside the combustionchamber of real SRM based on the composition of the solidpropellants and in turn the morphology of the combustionsurface
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The first author would like to thank Professor D R KassoyUniversity of Colorado at Boulder USA for their continuoushelp
References
[1] A Hegab T L Jackson J Buckmaster and D S Stewart ldquoTheburning of periodic sandwich propellantsrdquo in Proceedings of the36th AIAAASMESAEASEE Joint Propulsion Conference andExhibit Huntsville Ala USA July 2000
[2] A Hegab T L Jackson J Buckmaster and D S StewartldquoNonsteady burning of periodic sandwich propellants withcomplete coupling between the solid and gas phasesrdquo Journalof Combustion and Flame vol 125 no 1-2 pp 1055ndash1070 2001
[3] A M Hegab ldquoModeling of sandwich heterogeneous propellantwith two steps chemical kinetics and fluid mechanical effectsrdquoEngineering Research Journal vol 32 no 2 pp 119ndash130 2009
[4] J Buckmaster T L Jackson A Hegab S Kochevets andM Ulrich ldquoRandomly packed heterogeneous propellants andthe flames they supportrdquo in Proceedings of the 39th AerospaceSciences Meeting and Exhibit Reno NV USA January 2001
[5] S Kochevets J Buckmaster T L Jackson and A HegabldquoRandom packs and their use in modeling heterogeneous solidpropellant combustionrdquo Journal of Propulsion and Power vol 17no 4 pp 883ndash891 2001
[6] AMHegab S AHasanienH EMostafa andAMMaradenldquoExperimental and numerical investigations for the combustionof selected composite solid propellantrdquo Engineering ResearchJournal vol 35 no 3 2012
[7] A M Hegab ldquoVorticity generation and acoustic resonance ofsimulated solid rocket motor chamber with high wave numberwall injectionrdquo Computers and Fluids vol 38 no 6 pp 1258ndash1269 2009
[8] A M Hegab and D R Kassoy ldquoInternal flow temperatureand vorticity dynamics due to transient mass additionrdquo AIAAJournal vol 44 no 4 pp 812ndash826 2006
Mathematical Problems in Engineering 11
[9] K Kirkkopru D R Kassoy and Q Zhao ldquoUnsteady vorticitygeneration and evolution in a model of a solid rocket motorrdquoJournal of Propulsion and Power vol 12 no 4 pp 646ndash6541996
[10] K Kirkkopru D R Kassoy Q Zhao and P L Staab ldquoAcousti-cally generated unsteady vorticity field in a long narrow cylinderwith sidewall injectionrdquo Journal of Engineering Mathematicsvol 42 no 1 pp 65ndash90 2002
[11] P L StaabQ ZhaoD R Kassoy andKKirkkopru ldquoCoexistingacoustic-rotational flow in a cylinder with axisymmetric side-wall mass additionrdquo Physics of Fluids vol 11 no 10 pp 2935ndash2951 1999
[12] M Rempe P L Staab and D R Kassoy ldquoThermal response foran internal flow in a cylinderwith time dependent sidewallmassadditionrdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit AIAA 2000-0996 Reno Nev USA January 2000
[13] Q Zhao and D R Kassoy ldquoThe generation and evolutionof unsteady vorticity in a solid rocket engine chamberrdquo inProceedings of the 32th Aerospace Sciences Meeting AIAA paper94-0779 Reno Nev USA 1994
[14] Q Zhao P L Staab D R Kassoy and K Kirkkopru ldquoAcous-tically generated vorticity in an internal flowrdquo Journal of FluidMechanics vol 413 pp 247ndash285 2000
[15] G A Flandro ldquoEffects of vorticity on rocket combustionstabilityrdquo Journal of Propulsion and Power vol 11 no 4 pp 607ndash625 1995
[16] J Majdalani andW K VanMoorhem ldquoMultiple-scales solutionto the acoustic boundary layer in solid rocket motorsrdquo Journalof Propulsion and Power vol 13 no 2 pp 186ndash193 1997
[17] J Majdalani andW van Moorhem ldquoImproved time-dependentflowfield solution for solid rocket motorsrdquo AIAA Journal vol36 no 2 pp 241ndash248 1998
[18] Z Deng R J Adrian and C D Tomkinsm ldquoStructure ofturbulence in channel flow with a fully transpired wallrdquo inProceedings of the 39th Aerospace Science Meeting AIAA Paper2001-1019 Reno Nev USA 2001
[19] F Vuillot andG Avalon ldquoAcoustic boundary layers in solid pro-pellant rocketmotors usingNavier-Stokes equationsrdquo Journal ofPropulsion and Power vol 7 no 2 pp 231ndash239 1991
[20] T M Smith R L Roach and G A Flandro ldquoNumerical studyof the unsteady flow in a simulated solid rocket motorrdquo inProceedings of the 31th Aerospace Science Meeting AIAA Paper93-0112 Reno Nev USA 1993
[21] C F Tseng and I S Tseng ldquoInteraction between acoustic wavesand premixed flames in porous chamberrdquo in Proceedings of the32th Aerospace Science Meeting Reno Nev USA 1994 AIAAPaper 94 3328
[22] T S Roh and V Yang ldquoTransient combustion responses ofsolid propellants to acoustic disturbances in rocket motorsrdquo inProceedings of the 33rd Aerospace Sciences Meeting and ExhibitAIAA Paper 95-0602 Reno Nev USA 1995
[23] A M Hegab A study of acoustic phenomena in solid rocketengines [PhD thesis] Mechanical Power Engineering Depart-ment Menoufia University Menufia Egypt University of Col-orado Boulder Colo USA 1998
[24] R W MacCormack ldquoA numerical method for solving theequations of compressible viscous flowrdquo American Institute ofAeronautics and Astronautics Journal vol 20 no 9 pp 1275ndash1281 1982
[25] D Gottlieb and E Turkel ldquoDissipative two-four methods fortime-dependent problemsrdquo Mathematics of Computation vol30 no 136 pp 703ndash723 1976
[26] T J Poinsot and S K Lele ldquoBoundary conditions for directsimulations of compressible viscous flowsrdquo Journal of Compu-tational Physics vol 101 no 1 pp 104ndash129 1992
[27] Z Deng R J Adrian and C D Tomkins ldquoSensitivity ofturbulence in transpired channel to injection velocity small-scale nonuniformityrdquo AIAA Journal vol 40 no 11 pp 2241ndash2246 2002
[28] K J Meteb Prediction of hybrid rocket motor regression rate[MS thesis] Military Technical College Cairo Egypt 2009
[29] J W Anders V Magi and J Abraham ldquoLarge-eddy simulationin the near-field of a transient multi-component gas jet withdensity gradientsrdquoComputers amp Fluids vol 36 no 10 pp 1609ndash1620 2007
[30] B Chaouat ldquoNumerical simulations of channel flows with fluidinjection using Reynolds stressmodelrdquo AIAAPaper 2000-09922000
[31] W A El-Askary A Balabel S M El-Behery and A HegabldquoComputations of a compressible turbulent flow in a rocketmotor-chamber configuration with symmetric and asymmetricinjectionrdquoComputerModeling in EngineeringampSciences vol 82no 1 pp 29ndash54 2011
[32] A Balabel A M Hegab M Nasr and S M El-Behery ldquoAssess-ment of turbulence modeling for gas flow in two-dimensionalconvergent-divergent rocket nozzlerdquo Applied MathematicalModelling vol 35 no 7 pp 3408ndash3422 2011
2 Mathematical Problems in Engineering
00
y998400
H998400
Clos
ed en
dCombustion surface
Symmetry conditions
u998400= 0
998400= 0
(u998400
998400 P
998400 T
998400 120588
998400)
P998400= P
998400o ldquoPressure noderdquo
L x998400
Solid propellant
rb(x t)
rb(x t) cong 998400|wall(x t) = minus998400rws(x) +
998400o|wall(x
998400 t
998400)
998400= 0
120597(120575998400)
120597y998400= 0 120575
998400= (u
998400 T P
998400 120588
998400)
u998400= 0 T = T
998400o
Figure 1 The physical domain and the boundary conditions for the simulated solid rocket motor chamber
1
1
09
09
08
08
07
07
06
06
05
05
04
04
03
03
02
02
01
0100
y
Asymptotic analysisComputationExperimental Rew = 1030
Rew = 1302
Rew = 1565
u(05 y)u(05 0)
Figure 2 Normalized axial velocity profiles at the midlength of thechamber for the steady state flow
Numerical approaches for the presence of vorticity in aninjected internal flow are presented in [19 20] by solvingthe Navier-Stokes equation Moreover the studies in [9 10]presented detailed comparison of asymptotic and numericalapproaches without any consideration to the thermal effectIntensive numerical procedures for the effect of exothermicreactions very close to the sidewall have been carried out in[21 22]
Finally the root mean square value for the axial distri-bution for the coexistence of rotational and irrotational flowfield is calculated in [8 23]
In the present study asymptotic and computationalresults are obtained to consider the unsteady variation ofthe vorticity and temperature dynamic in the prescribedchamber Results are given for steady state solution generatedfrom spatially uniform injection velocity The convergentsolution is oscillated with similar amplitude unsteady massadditions with wide ranges of injection wave numbers Themagnitude of the superimposed injection velocity is chosen
0 5 10 15 20 25 30 35 40Time
02
015
01
005
0
minus005
minus01
minus015
minus02
ComputationalAnalytical
T
x = 05 y = 00
Figure 3 Comparison between the computational and the asymp-totic solutions for the perturbed temperature time history with 120596 =
1 (nonresonance) 120576 = 04 119872 = 002 and 119877119890= 3 lowast 10
5 at the half-length of the chamber
to be large enough to make sure that the fluid is alwaysinjected from the side wall and reduce the probability ofsucking outside the chamber
2 The Physical and Mathematical Formulation
The present numerical study is performed to study the effectof the complicated surface morphology of the combustionsurface on the generated vorticity in a chamber of length1198711015840 and half-height 119867
1015840 The aspect ratio in this study 120575 =
1198711015840
1198671015840 is taken to be twenty Pressure node is assigned at
the chamber exit while closed end is considered at the headend Air is injected from the side wall with a characteristicvelocity 119881
1015840
119910119900
to simulate the combustion products from real
Mathematical Problems in Engineering 3
002
0406
081
0
05
10
5
10
15
20
25
xy
120596
(a)
002
0406
081
0
05
1
0
1
2
3
4
xy
minus1
minus2
120596
(b)
002
0406
08 1
0
05
1
xy
0
1
2
3
4
minus1
minus2
120596
(c)
Figure 4 Vorticity topography for 120596 = 1 (nonresonance) and 120576 = 04119872 = 002 and 119877119890= 3 lowast 10
5 at 119905 = 42 at several axial wave numbers (a)119899 = 1 (b) 119899 = 3 and (c) 119899 = 5
burning of propellant This modeling generates an axial flowcharacterized by 119880
1015840
119911119900
= 1205751198811015840
119910119900
as in Figure 1The steady and the unsteady injection velocity are denoted
by V1015840119903119908119904
(119909) V1015840119900|wall(119909
1015840
1199051015840
) while the exit pressure and thetemperature of the fluid injected from the side wall aredenoted by 119875
1015840
119900and 119879
1015840
119900
The parabolized Navier Stokes equations for unsteadytwo dimensional compressible fluid flow are solved numeri-cally in the prescribed chamber andmay bewritten as follows
120597119876
120597119905
+
120597119864
120597119909
+
120597119865
120597119910
= 0 (1)
where
119876 = [120588 120588119906 120588V 119864119879]119879
119864 = [1198721205881199061198721205881199062
+
1
1205741198722119901119872120588119906V119872 119864
119879+ (120574 minus 1) 119901 119906]
119879
119865 = [119872120588V119872120588119906V minus
1205752
119872
119877119890
119906119910119872120588V2 +
1205752
120574119872
119901
119872 119864119879+ (120574 minus 1) 119901 V minus
1205741205752
119872
119877119890119875119903
119879119910]
119879
(2)
119864119879is the total energy (120588[119879+120574(120574minus1)119872
2
(1199062
+(V120575)2]) and theequation of state for a perfect gas is
119901 = 120588119879 (3)
In this study the specific heats the viscosity and conductivitycoefficients are considered to be constant since the change
4 Mathematical Problems in Engineering
0 02 04 06 0809
095
Injection surface
x
y
20
0
minus20
(a)
0 02 04 06 08x
09
095y
10
0
minus10
(b)
0 02 04 06 08x
09
095y
10
0
minus10
(c)
Figure 5 The vorticity contours adjacent to the injection surfacefor the same parameters as in Figure 4 for several eigenvalues (a)120582119899= 1205872 (119899 = 1) (b) 31205872 (119899 = 3) and (c) 51205872 (119899 = 5)
in the temperature in the cold model is relatively smallNondimensional variables are given by
119901 =
1199011015840
1199011015840
119900
120588 =
1205881015840
1205881015840
119900
119879 =
1198791015840
1198791015840
119900
119906 =
1199061015840
1198801015840
119911119900
V =
V1015840
(1198801015840
119911119900
120575)
119909 =
1199091015840
1198711015840
119910 =
1199101015840
1198671015840 119905 =
1199051015840
1199051015840
119886
(4)
The axial flow Mach number Prandtl number flow andacoustic Reynolds numbers are written as
119872 =
1198801015840
119911119900
1198621015840
119900
119875119903=
1205831015840
1199001198621015840
119901119900
1198961015840
119900
119877119890=
1205881015840
1199001198801015840
119911119900
1198711015840
1205831015840
119900
119877119890119860
=
119877119890
119872
(5)
The characteristic Mach number is chosen to be 119874(10minus1)while the characteristic Reynolds number is119874(105-106)Theseorders of magnitudes approximate the characteristics of thefluid flow in similar real rocket engineThe nondimensional-ized time 119905
1015840
119886is defined as the axial acoustic time 119905
1015840
119886= 1198711015840
1198621015840
119900
21 The Initial and Boundary Conditions For 119905 le 0 time-independent side wall transverse velocity with 119881|wall =
minus119881119903119908119904
(119909) is used to generate steady flow solution while for119905 gt 0 119881|wall = 119881
119903119908119904(119909 119905) = minus119881
119903119908119904(119909) minus 120576120593wall(119909)(1 minus cos120596119905)
as the unsteady longitudinal variation for the injectiontransverse at resonance or away from the resonance caseThe function 120593wall represents the longitudinal variation ofthe velocity at the injection surface Moreover the specifiedconditions at the four boundaries can be written as follows
119909 = 0 119906 = 0 V = 0
119909 = 1 119901 = 1 ldquopressure noderdquo
119910 = 0 V = 0
120597119906
120597119910
=
120597119879
120597119910
=
120597119901
120597119910
=
120597120588
120597119910
= 0
119910 = 1 119906 = 0 119879 = 1
V =
minusV119903119908119904
(119909) 119905 le 0
minusV119903119908119904
(119909) minus 120576120593119903119908
(1 minus cos120596119905) 119905 gt 0
(6)
and the initial condition is
119905 = 0 119906 = V = 0 119901 = 119879 = 1 (7)
The initial conditions for the unsteady flow computations arethe convergence solutions for the steady state conditions
22 The Analytical Approach
221 Steady and Unsteady Flow Hegab [7] has concludedthat the total axial velocity component includes two mainpartsThefirst part represents the steady state component andthe other part represents unsteady state component and canbe written as follows
(119906 V 119875 120588 119879) = (119906119904 V119904 119875119904 120588119904 119879119904) + ( V 120588
119879) (8)
Using the expended variables used inHegab [7] and insertingthem in (1) the side wall steady vorticity generation (Ω
119900119904|wall)
and the accompanying heat transfer (120597119879119900119904120597119910|wall) are derived
and written as follows
Ωos1003816100381610038161003816wall = minus
1
120574V119903119908119904
120597
120597119909
(
1205741205872
4
int
1
119909
V119903119908119904
(119909) int
119909
0
V119903119908119904
(120591) 119889120591 119889119909)
+
120597V119903119908119904
(119909)
120597119909
120597119879os120597119910
10038161003816100381610038161003816100381610038161003816wall
= 0
(9)
Equations (9) show that nonzero steady vorticity and zeroheat transfer are found with the use of no-slip and constantinjection boundary conditions
The sequence of the numerical solution consists of twosteps The first one is the initial steady state computationwhich arises from constant injection with isothermal con-dition The second one is the unsteady flow computationbased on initial conditions as computed from the first stepThe perturbation theory introduced by Staab et al [11] and
Mathematical Problems in Engineering 5
(a) (b)
(c) (d)
Figure 6 The flame geometry for the burning of hybrid rocket systems using (a) pure PE fuel grain (b) 975 PE +25 Al fuel grain (c)925 PE + 75 Al fuel grain and (d) 85 PE + 15 Al fuel grain [18]
200
201
202
203
204
205
600 625 650 675 700 725 750 775 800Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
Recording rate 60 points frequency about 20 cps and about 003 bar pure PEamplitude
Figure 7The recorded pressure time history for a burning of hybridmotor using pure polyethylene (PE) fuel grain
applied by Hegab [7] leads to the following weakly rotationalequations
119905 = 0 119900= 0
120597119900
120597119905
= 0
119909 = 0
120597119900
120597119909
= 0 119909 = 1 119900= 0
119910 = 0 V0= 0
119910 = 1 V119900= minus120576120593
119903119908(1 minus cos120596119905) 119905 gt 0
(10)
200
201
202
203
204
205
500 525 550 575 600 625 650 675 700Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
about 002 bar 75 AlRecording rate 60 points frequency about 20 cps and amplitude
Figure 8The recorded pressure time history for a burning of hybridmotor using 75 aluminum (Al) additive to PE fuel grain
205
204
203
202
201
200500 525 550 575 600 625 650 675 700
Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
Recording rate 60 points frequency about 20 cps about 0018 bar 15 Aland amplitude
Figure 9The recorded pressure time history for a burning of hybridmotor using 15 aluminum (Al) additive to PE fuel grain
6 Mathematical Problems in Engineering
The final nonresonance solution for the flow in the simulatedchamber in Figure 1 is
119900(119909 119905) =
infin
sum
119899=0
2120576120574120596int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
119900(119909 119905) =
infin
sum
119899=0
2120576120574120596int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
(cos120573119899119905 minus 1) minus
120573119899
120596
(cos120596119905 minus 1)
times sin120573119899119909
V119900(119909 119910 119905) = minus119910 120593
119903119908(119909) (1 minus cos120596119905)
(11)
Equations (11) show the time history of the perturbed valuesvariables (
119900 119900 V119900) at any axial locations
Following the same policies the resonance solutions arederived and formulated as
119900(119909 119905) =
120574119862119899lowast
120573119899lowast
sin120573119899lowast119905 minus 120573
119899lowast119905 cos120573
119899lowast119905 cos120573
119899lowast119909
119900(119909 119905) = 119862
119899lowast minus
2
120573119899lowast
(cos120573119899lowast119905 minus 1) minus 119905 sin120573
119899lowast119905
times cos120573119899lowast119909
(12)
where 119862119899lowast = 2 int
1
0
120593119903119908
(119909) cos120573119899lowast119909
23 The Computational Approach In the present studyhigher order accuracy difference equations are employedto solve the parabolized Navier-Stokes equations in theinterior of the prescribed chamber and at the boundariesThe numerical multisteps scheme in [24] is used to solve the2D unsteady and compressible Navier-Stokes equations forthe points located near the boundaries while the two-fourexplicit predictor-corrector scheme in [25] is employed tosolve the adapted equations at the interior points The lattermethod is a fourth-order accuracy variant of the fully explicitMacCormack scheme Uniform grids are used The accuracycriterion employed byMacCormack [24] is used to determinethe size of the time step
Here the NSCBC technique applied by [7] is presented indetail The full derivation for the parabolized Navier-Stokesat the upper and lower boundaries and at the head and exitends is formulated to specify the nonphysical conditions atthese boundaries Moreover this method is applied to reducethe generated reflected waves arising from the numericaltechniques
Based on the idea of characteristics for Euler equationsthe first derivatives normal to the boundaries appearing inthe convective terms of the Navier-Stokes equations may beexpressed in terms of the amplitude of incoming outcoming
and entropy waves weierp119894rsquos The fluid dynamics equations can
be written in nondimensional wave modes form as follows
120597120588
120597119905
+
1
1198622[weierp2+
1
2
(weierp4+ weierp1)] + 119872
120597120588V120597119910
= 0 (13)
120597119906
120597119905
+
1
2120588119862
[(weierp4minus weierp1)] + 119872V
120597119906
120597119910
=
120594119909
120588
(14)
120597V120597119905
+ weierp3+ 119872V
120597V120597119910
+
1198602
120574119872
1
120588
120597119901
120597119910
=
120594119910
120588
(15)
120597119901
120597119905
+
1
2
[(weierp4+ weierp1)] + 119872V
120597119901
120597119910
+ 120574119872119901
120597V120597119910
= 120594119864 (16)
where 120594119909and 120594
119910represent the viscous terms and 120594
119864refers
to the combination of the viscous and conduction terms inthe axial and transverse directions and 119862 = 119862
1015840
1198621015840
119900 The
amplitudes of the left-running waves (weierp1) the right-running
waves (weierp4) and the entropy waves (weierp
2 weierp3) in terms of the
characteristic velocities 120582119894rsquos are
weierp1= 1205821(
1
120574
120597119901
120597119909
minus 119872120588119862
120597119906
120597119909
) 1205821=
119906
119872
minus 119862 (17)
weierp2= 1205822(1198622120597120588
120597119909
minus
1
120574
120597119901
120597119909
) 1205822= 119872119906 (18)
weierp3= 1205823
120597V120597119909
1205823= 119872119906 (19)
weierp4= 1205824(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
) 1205824=
119906
119872
+ 119862 (20)
Two approaches are used to infer the values of theincoming amplitude-variations from the boundary condi-tions One is the local one-dimensional inviscid relations(LODI) technique and the second is the Navier-Stokescharacteristics boundary conditions (NSCBC) For the latterone the complete description of the numerical boundaryconditions at all boundaries is discussed below
24 Exit Boundary Conditions In this case a pressure nodeis assumed at the exit boundary The NSCBC relation (16)suggests that the amplitude of the reflected wave should be
weierp1= 2120594119864minus weierp4 (21)
and the remaining conditions are
weierp2= 119872119906(119862
2120597120588
120597119909
minus
1
120574
120597119901
120597119909
) weierp3= 119872119906
120597V120597119909
weierp4= (
119906
119872
+ 119888)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
(22)
Mathematical Problems in Engineering 7
The nondimensional equations can be written as follows
120597120588
120597119905
+ 119872119906
120597120588
120597119909
minus
119872119906
1205741198622
120597119901
120597119909
+ 119872V120597120588
120597119910
+
1198602
119872
(120574 minus 1) 119877119890119875119903
1
1198622
1205972
119879
1205971199102
= 0
120597119906
120597119905
+
1
120588119862
[(119906 +
119862
119872
)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)] + 119872V120597119906
120597119910
minus
1198602
(120574 minus 1) 119877119890119875119903
1
120588119862
1205972
119879
1205971199102
minus
1198721198602
119877119890
1
120588
1205972
119906
1205971199102
= 0
120597V120597119905
+ 119872(119906
120597V120597119909
+ V120597V120597119910
) = 0
119889119901
119889119905
= 0
(23)
Here again the nondimensional variables (120588 119906 and V)can be computed using one-sided finite-difference approxi-mations
25 Head End Boundary Condition The velocity componentnormal to the boundary (119906(0 119910 119905)) is zero since the head endis impermeable In contrast the transverse velocity compo-nent cannot be specified because the axial viscous terms inthe transversemomentum equation have been suppressed Asthe normal velocity drops to zero at 119909 = 0 the amplitudesweierp2and weierp
3 in (18) and (19) are zero and the characteristic
velocities are 1205821
= minus119862 and 1205824
= +119862 The reflected wave weierp4
enters the domain while the wave weierp1propagates out through
the boundary The NSCBC relation (19) indicates that theamplitude of the incoming wave weierp
4should be
weierp4= weierp1 (24)
and the remaining conditions are
weierp1= minus119888(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
weierp2= 0 weierp
3= 0
(25)
The equations can be written in nondimensional form asfollows
120597120588
120597119905
minus 119872120588
120597119906
120597119909
+
1
120574119862
120597119901
120597119909
+ 119872
120597120588V120597119910
= 0
120597119906
120597119905
= 0
120597V120597119905
+ 119872V120597V120597119910
= 0
120597119901
120597119905
minus 119888
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
+ 120574119872119901
120597V120597119910
minus
1205741198602
119872
(120574 minus 1) 119875119903119877119890
1205972
119879
1205971199102
= 0
(26)
More details about the NSCBC techniques are found in [2326]
26 CodeValidations Thecomputational codes are designedconstructed and validated for steady and unsteady flow as
shown in Figures 2 and 3 For the latter one the constructedcodes run for many wave cycles to detect the whole unsteadyrotational flow dynamic across the chamber
In Figure 2 comparison between the current numericaland asymptotic results dealing with the steady state flowsolutions is carried out The steady state flow is generatedby adding constant side wall mass injection (119881
119903119908= minus1)
from the injection surface of the prescribed chamber Similarexperiment in [18 27] was carried out and compared withthe current mathematical models in Figure 2The asymptoticanalysis solution represents the solid line while the dottedline represents the converged steady state numerical solutionfor viscous flow with wall Reynolds number 119877
119890119908= 5 lowast
103 The nonfilled points refer to the experimental measured
data The experimental results are recorded for flow in achannel with fully transpired wall Three different poroussurfaces using three different sizes of honeycomb cells areused [7] In this figure a comparison between the analyticaland the numerical solutions at the midlength of the chamberis illustratedThis comparison shows a reasonable agreementbetween the three approaches The comparison between thetheoretical and the computational approaches shows littlebit deviation This deviation is logically expected since theused honey combs have three different sizes The smallersize of honey combs the smaller deviations between the twoapproaches
The second set of the validations show the solution for theunsteady flow computations at nonresonance forcedmode Inthis set of the results the computational results are validatedby comparing with the asymptotic analysis These resultsrepresent the asymptotic solution when the sidewall injectionvelocity is represented by V
119903119908(119909 1 119905) = minus[1 + 120576120593
119903119908(1 minus
cos120596119905)] 120593119903119908
= cos(120573119899119909) In the current work and the work
by Hegab [7] the disturbance forcing frequency is taken tobe 120596 = 1 (nonresonance) with the amplitude of 120576 = 04
and the eigenvalues (120573119899
= 1198991205872 119899 = 1 3 5 7 9 etc)The temperature time history (119879) at themedlength location ispresented in Figure 3 for 119877
119890= 3 lowast 10
5119872 = 002 120575 = 20 and119899 = 10 For the nonresonance case the solution shows stableconditions but it is not purely harmonic This interestingbehavior shows the ability of the constructed code with thegood treatment to the boundary conditions to detect thenon-harmonic trend as detected by the analytical approachA comparison of the perturbed temperature (119879) (27) withthe numerical results in Figure 3 shows good agreementbetween the two approaches especially for 119905 lt 20 while thecomputational solution shows little less deviation than theanalytical results for 119905 gt 20 due to the nonlinearity effects
119879119900(119909 119905) =
infin
sum
119899=0
2120576120596 (120574 minus 1) int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
(27)
Generally the coexistence of the forced and the naturalmodes of the system is detected in the adapted computationaland analytical approaches as seen in the earlier investigationby Hegab [7] In contrast the computational techniques in
8 Mathematical Problems in Engineering
[10] show only pure harmonic oscillations The source ofthese deficiencies may be arising from the treatment of theboundary conditions especially at the head and aft ends ofthe simulated chamber
3 Results and Discussions
The impact of the injected wave number from the sidewallalong with the frequencies of the forced mode on thegenerated complex acoustical fields and their interactionwiththe fluid are examinedThe results have been categorized intothe following two sets
31 Nonresonance Solution In the current study a widerange of forcing frequencies away from or very close tothe first fundamental modes of the prescribed chamber isconsidered [7] a wide range of forcing frequencies away fromor very close to the first fundamental modes of the prescribedchamber is considered In Figure 4 the irrotational androtational fields across the whole chamber are presentedwhen 119905 = 42 for 120596 = 1 119877
119890= 3lowast10
5119872 = 002 120576 = 04 119899 = 13 and 5 and 120582
119899= 1205872 The latter one represents the Eigen
values of the chamberThese interesting graphs show that therotational fields are generated at the injection surface As timeincreases the generated unsteady vorticity is transferred to fillmost of the combustion chamber Moreover the constructedcodes are able to run for enough time and count about sevenvorticity wave cycles as 119905 = 42 It is also seen that theamplitude of the rotational field attainsmaximumvalues nearthe wall and found to be order of ten for the monotonicvariation (119899 = 1) and nonresonance frequency (120596 = 1)One notes that these magnitudes meet the scaling of theasymptotic solutions in [7 9 11 14] Moreover for 119899 = 1monotonic distribution of unsteady vorticity is noticed whilethe variation goes in nonmonotonic axial distribution for 119899 gt
1 For the nonresonance solution the longitudinal variationof the vorticity at the wall 120597|Ω
119908(119909 119905)|120597119909 gt 0 for 120582
119899=
1205872 (119899 = 1) is revealing that the magnitude of rotationalflow fields on the injection surface increases as axial distanceincreases
The vorticity contours adjacent to the injection surface forthe same parameters as in Figure 4 for several eigenvalues(a) 120582119899
= 1205872 (119899 = 1) (b) 31205872 (119899 = 3) and (c) 51205872
(119899 = 5) are present in Figure 5 It is clearly seen that adjacentto the combustion surface the generated unsteady vorticitymay impact the surface at downstream locations for 119899 = 1at two locations for 119899 = 3 and many locations for 119899 = 5The rotational flow fluctuates from negative to positive andvice versa at several successive times and may cause scouringon the combustion surface and in turn may increase theshearing stresses adjacent to the combustion surface Thisimportant conclusion may lead to increase of the burningrates over the designed ones especially for low wave numbersince the amplitude of the unsteady vorticity decreases as thewave number increases
Fortunately recent experimental work by Meteb [28] in2009 was directed to study the acoustic instability in hybridrocket system using different fuel grains and ignition systems
The experimental results verified qualitatively the currentanalytical and computational results for the generated com-plex wave patterns and the effect of the combustion surfaceon the acoustic fluid dynamics interactionmechanism In thisexperiment the pure polyethylene (PE) fuel grain and PEwithadditives of aluminum (Al) powder with different percentageare used to predict the hybrid rocket motor regression ratesThe flame geometry and structure for the burning of thepure PE and PE with additives of aluminum (Al) powderat 25 75 and 15 of the total weight of the wholesamples are illustrated in Figure 6The effect of theAl powderon the flame length geometry and the rate of reaction forthe burning of each sample is clearly seen In addition thepressure time histories at a point before the combustionchamber in the way of oxidizer are recorded and illustratedin Figure 7 for the pure PE and in Figures 8 and 9 for PEwith additives of aluminum (Al) powder at 75 and 15of the total weight respectively It is clearly seen that theamplitude of the pressure waves decreases as the percentageof Al fuel grain increases in the sample This interestingbehavior is similar to what we have seen when we increasethe wave number of the combustion surface As we knowmost of the Al fuel grains burn away from the combustionsurface consisting ofmany fragmentsThe escaping processesof Al fuel grains from the combustion surface generate manyof cavities that make the combustion surface more roughthan the smooth one in case of pure PE The experimentaland computational results are qualitatively in agreement Forthe first time the analytical and computational results forthe current cold model and the acoustic instabilities foundby Meteb [28] state an important and essential conclusionabout the effect of the combustion surface morphology onthe acoustic instability inside the rocket chamber and theaccompanying rotational flow Instantaneous total velocityvector map for the parameters in Figure 4 at (a) 119905 = 0 ldquosteadystate solutionrdquo and (b) 119905 = 16 is presented in Figures 10(a)and 10(b) The injection surface is located at 119910 = 1 on theupper part of the plot where the flow is completely verticalwhile the lower part of the plot represents the center lineat 119910 = 0 where the flow is completely axial The closedend is located at 119909 = 0 on the left hand side of the plotwhile at 119909 = 1 it represents the exit plane on the right handside One may notice the smooth and undisturbed fluid flowthrough the entire map for the steady state solution at 119905 = 0After initiating the side wall disturbances the coexisting ofacoustics and the generated rotational flow is clearly seenin the upper half part of the chamber and illustrates howthe interaction between the travelling acoustic waves and theinjected fluid affects the fluid flow in the chamber In additionat 119905 = 32 in Figure 11 there is an instantaneous reverseflow found at a location near the combustion surface Thecomplex acoustic fluid dynamics interaction structure mayincrease the shear stresses adjacent to the combustion surfaceand through the whole cavity These results may providethe rocket designers with conceptual tools and data to beconsidered in case of redesigning strategies
Mathematical Problems in Engineering 9
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
ldquoSteady state solutionrdquo t = 00
(a)
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 16Nonresonance (
(b)
Figure 10 Instantaneous total velocity vector map for the parameters in Figure 4 (a) 119905 = 0 ldquosteady state solutionrdquo and (b) 119905 = 16
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 32
Reverse flow region
Nonresonance (
Figure 11 Instantaneous total velocity vector map at 119905 = 32 for theparameters in Figure 4
32 Near and at Resonance Solution A comparison of theasymptotic analysis perturbed pressure time history withthe computational one at and near resonance frequenciesis presented in Figure 12 For 119905 lt 10 the results showgood agreement between the two approaches even with theasymptotic beat condition (120596 = 1205872 minus 00018) which isvery close to the resonance frequency While for 119905 gt 10
the asymptotic solution shows linear growth with time basedon the linear theory for the derived wave equation where120596 = 120573
119899lowast = 1205872 as derived by Hegab [7] In contrast
the computational solution shows beats at the resonancecondition due to the nonlinearity and the asymptotic beatsolutions have smaller amplitude than the resonance oneThe beats appear due to the interaction between the derivingfrequency mode 120596 = 1205872minus00018 and the first eigenfunctionmode (1205872) It is clearly seen that the amplitude of thevorticity at the resonance frequency is found to be ten timesthan the amplitudes at the nonresonance one in Figure 4
P
03
02
01
0
minus01
minus02
minus030 5 10 15 20 25 30 35 40
t
ComputationalAnalyticalAnalytical (beats)
x = 05 y = 00 120596 = 1205872 n = 1
Figure 12 Comparison between the computational and the asymp-totic solutions for the perturbed pressure time history with 120596 = 1205872
(resonance) 120576 = 04 119872 = 002 and 119877119890= 3 lowast 10
5 at the half-lengthof the chamber
Moreover it is found that the fluctuation of the transversetemperature gradient at the injection surface is found tobe about ten times than the fluctuation of the temperaturedisturbances This concludes that since the heat transferalong the sidewall is directly proportional to the transversetemperature gradient there is an unexpectedly large heat fluxwhich exists both at the wall and within the entire chambergiven the small size of the temperature distribution Theasymptotic analysis by Staab et al [11] is used to prove thisimportant conclusion
Finally the asymptotic and computational methods areemployed to study the dynamic process in an acoustically
10 Mathematical Problems in Engineering
simulated solid propellant rocket chamber with transientsidewall mass addition at different wave numbers
4 Conclusion Remarks
The current study emphasizes the effect of the combustionsurfacemorphology of the solid rocket systemon the acousticinstability and the accompanying vorticity and temperaturedynamics generation The rotational and irrotational fieldsgenerated in the prescribed solid rocket motor chamber withMach number are chosen to be O(10minus1) while the charac-teristic Reynolds number is O(105-106) at nonresonance andnear and at resonance frequencies have been described TheNSCBC techniques that applied in [7 29] are used to treat thenumerical boundary conditions
The numerical solution shows that the convergence tosteady state solution is achieved first and then additionaltransient side wall mass addition is initiated Moreover theresult shows that the forced mode of the nonresonance andresonance frequencies generates longitudinal acoustics waveswhich interact again with its sources at the injection surfaceto generate the rotational flow fields The generated unsteadyvorticity is transferred from the injection surface toward thecenterline to fill the entire chamber as time increases aftermore than seven wave cycles The solution shows nonpureharmonic oscillations as suggested from the theory in [11]Vorticity 3D graphs and contours illustrate that spatially themaximum amplitudes are found to be near the injectionsurface Moreover the largest shear stresses are found to be atthe injection surface near the exit end while the shear stressesimpact the injection surface at many locations as the wavenumber increasesThe longitudinal variation of the rotationalflow fields is found to be nonmonotonic for 119899 gt 1
Here for the first time the analytical and computationalresults for the current coldmodel and the recent experimentalwork by Meteb [28] revealed that the combustion surfacemorphology has a great effect on the acoustic instabilityinside the rocket chamber and the accompanying rotationalflow The computational approach reveals that there areunexpectedly high heat transfer and generated rotational flowfields found to be adjacent to the injection surface in bothnonresonance and resonance cases The amplitude of thelatter one is found to be approximately ten times than thatwith the nonresonance solution One may anticipate thatthese large oscillatory gradients will persist to the surface ofa burning solid propellant in a real rocket chamber
More traditional computational studies of turbulence inchannel flows with wall injection are reviewed by Chaouat[30] Mean axial velocity profiles are found to be laminaruntil about halfway down the channel and then show signs oftransition Chaouatrsquos results do not show the acoustic-relatedeffects discussed in the present work As a result it is difficultto know from the turbulent theory results if the source ofthe downstream vorticity is classical transition to turbulenceor partly the result of the kind of mechanism discussed inthis paper In our recent studies in 2011 and 2012 [31 32]the assessment of turbulence modeling for gas flow in thechamber and in the convergent-divergent rocket nozzle for
the steady state flow is studied The detailed information andrealities in the current research about the transient coexistingof the irrotational (acoustics) rotational (generated vorticityfrom the interaction between the injected flow and thetraveling acoustic waves) and our turbulent techniques thatwe have adapted in our studies in 2011 and 2012 for the steadyflow field in SRM chambernozzle geometry will be used toaccount all the complex flow fields in our future study
Finally results of the asymptotic and computationalapproaches may provide the rocket designer with codes andinformation making them able to interpret some reasonsbehind the combustion instability inside the combustionchamber of real SRM based on the composition of the solidpropellants and in turn the morphology of the combustionsurface
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The first author would like to thank Professor D R KassoyUniversity of Colorado at Boulder USA for their continuoushelp
References
[1] A Hegab T L Jackson J Buckmaster and D S Stewart ldquoTheburning of periodic sandwich propellantsrdquo in Proceedings of the36th AIAAASMESAEASEE Joint Propulsion Conference andExhibit Huntsville Ala USA July 2000
[2] A Hegab T L Jackson J Buckmaster and D S StewartldquoNonsteady burning of periodic sandwich propellants withcomplete coupling between the solid and gas phasesrdquo Journalof Combustion and Flame vol 125 no 1-2 pp 1055ndash1070 2001
[3] A M Hegab ldquoModeling of sandwich heterogeneous propellantwith two steps chemical kinetics and fluid mechanical effectsrdquoEngineering Research Journal vol 32 no 2 pp 119ndash130 2009
[4] J Buckmaster T L Jackson A Hegab S Kochevets andM Ulrich ldquoRandomly packed heterogeneous propellants andthe flames they supportrdquo in Proceedings of the 39th AerospaceSciences Meeting and Exhibit Reno NV USA January 2001
[5] S Kochevets J Buckmaster T L Jackson and A HegabldquoRandom packs and their use in modeling heterogeneous solidpropellant combustionrdquo Journal of Propulsion and Power vol 17no 4 pp 883ndash891 2001
[6] AMHegab S AHasanienH EMostafa andAMMaradenldquoExperimental and numerical investigations for the combustionof selected composite solid propellantrdquo Engineering ResearchJournal vol 35 no 3 2012
[7] A M Hegab ldquoVorticity generation and acoustic resonance ofsimulated solid rocket motor chamber with high wave numberwall injectionrdquo Computers and Fluids vol 38 no 6 pp 1258ndash1269 2009
[8] A M Hegab and D R Kassoy ldquoInternal flow temperatureand vorticity dynamics due to transient mass additionrdquo AIAAJournal vol 44 no 4 pp 812ndash826 2006
Mathematical Problems in Engineering 11
[9] K Kirkkopru D R Kassoy and Q Zhao ldquoUnsteady vorticitygeneration and evolution in a model of a solid rocket motorrdquoJournal of Propulsion and Power vol 12 no 4 pp 646ndash6541996
[10] K Kirkkopru D R Kassoy Q Zhao and P L Staab ldquoAcousti-cally generated unsteady vorticity field in a long narrow cylinderwith sidewall injectionrdquo Journal of Engineering Mathematicsvol 42 no 1 pp 65ndash90 2002
[11] P L StaabQ ZhaoD R Kassoy andKKirkkopru ldquoCoexistingacoustic-rotational flow in a cylinder with axisymmetric side-wall mass additionrdquo Physics of Fluids vol 11 no 10 pp 2935ndash2951 1999
[12] M Rempe P L Staab and D R Kassoy ldquoThermal response foran internal flow in a cylinderwith time dependent sidewallmassadditionrdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit AIAA 2000-0996 Reno Nev USA January 2000
[13] Q Zhao and D R Kassoy ldquoThe generation and evolutionof unsteady vorticity in a solid rocket engine chamberrdquo inProceedings of the 32th Aerospace Sciences Meeting AIAA paper94-0779 Reno Nev USA 1994
[14] Q Zhao P L Staab D R Kassoy and K Kirkkopru ldquoAcous-tically generated vorticity in an internal flowrdquo Journal of FluidMechanics vol 413 pp 247ndash285 2000
[15] G A Flandro ldquoEffects of vorticity on rocket combustionstabilityrdquo Journal of Propulsion and Power vol 11 no 4 pp 607ndash625 1995
[16] J Majdalani andW K VanMoorhem ldquoMultiple-scales solutionto the acoustic boundary layer in solid rocket motorsrdquo Journalof Propulsion and Power vol 13 no 2 pp 186ndash193 1997
[17] J Majdalani andW van Moorhem ldquoImproved time-dependentflowfield solution for solid rocket motorsrdquo AIAA Journal vol36 no 2 pp 241ndash248 1998
[18] Z Deng R J Adrian and C D Tomkinsm ldquoStructure ofturbulence in channel flow with a fully transpired wallrdquo inProceedings of the 39th Aerospace Science Meeting AIAA Paper2001-1019 Reno Nev USA 2001
[19] F Vuillot andG Avalon ldquoAcoustic boundary layers in solid pro-pellant rocketmotors usingNavier-Stokes equationsrdquo Journal ofPropulsion and Power vol 7 no 2 pp 231ndash239 1991
[20] T M Smith R L Roach and G A Flandro ldquoNumerical studyof the unsteady flow in a simulated solid rocket motorrdquo inProceedings of the 31th Aerospace Science Meeting AIAA Paper93-0112 Reno Nev USA 1993
[21] C F Tseng and I S Tseng ldquoInteraction between acoustic wavesand premixed flames in porous chamberrdquo in Proceedings of the32th Aerospace Science Meeting Reno Nev USA 1994 AIAAPaper 94 3328
[22] T S Roh and V Yang ldquoTransient combustion responses ofsolid propellants to acoustic disturbances in rocket motorsrdquo inProceedings of the 33rd Aerospace Sciences Meeting and ExhibitAIAA Paper 95-0602 Reno Nev USA 1995
[23] A M Hegab A study of acoustic phenomena in solid rocketengines [PhD thesis] Mechanical Power Engineering Depart-ment Menoufia University Menufia Egypt University of Col-orado Boulder Colo USA 1998
[24] R W MacCormack ldquoA numerical method for solving theequations of compressible viscous flowrdquo American Institute ofAeronautics and Astronautics Journal vol 20 no 9 pp 1275ndash1281 1982
[25] D Gottlieb and E Turkel ldquoDissipative two-four methods fortime-dependent problemsrdquo Mathematics of Computation vol30 no 136 pp 703ndash723 1976
[26] T J Poinsot and S K Lele ldquoBoundary conditions for directsimulations of compressible viscous flowsrdquo Journal of Compu-tational Physics vol 101 no 1 pp 104ndash129 1992
[27] Z Deng R J Adrian and C D Tomkins ldquoSensitivity ofturbulence in transpired channel to injection velocity small-scale nonuniformityrdquo AIAA Journal vol 40 no 11 pp 2241ndash2246 2002
[28] K J Meteb Prediction of hybrid rocket motor regression rate[MS thesis] Military Technical College Cairo Egypt 2009
[29] J W Anders V Magi and J Abraham ldquoLarge-eddy simulationin the near-field of a transient multi-component gas jet withdensity gradientsrdquoComputers amp Fluids vol 36 no 10 pp 1609ndash1620 2007
[30] B Chaouat ldquoNumerical simulations of channel flows with fluidinjection using Reynolds stressmodelrdquo AIAAPaper 2000-09922000
[31] W A El-Askary A Balabel S M El-Behery and A HegabldquoComputations of a compressible turbulent flow in a rocketmotor-chamber configuration with symmetric and asymmetricinjectionrdquoComputerModeling in EngineeringampSciences vol 82no 1 pp 29ndash54 2011
[32] A Balabel A M Hegab M Nasr and S M El-Behery ldquoAssess-ment of turbulence modeling for gas flow in two-dimensionalconvergent-divergent rocket nozzlerdquo Applied MathematicalModelling vol 35 no 7 pp 3408ndash3422 2011
Mathematical Problems in Engineering 3
002
0406
081
0
05
10
5
10
15
20
25
xy
120596
(a)
002
0406
081
0
05
1
0
1
2
3
4
xy
minus1
minus2
120596
(b)
002
0406
08 1
0
05
1
xy
0
1
2
3
4
minus1
minus2
120596
(c)
Figure 4 Vorticity topography for 120596 = 1 (nonresonance) and 120576 = 04119872 = 002 and 119877119890= 3 lowast 10
5 at 119905 = 42 at several axial wave numbers (a)119899 = 1 (b) 119899 = 3 and (c) 119899 = 5
burning of propellant This modeling generates an axial flowcharacterized by 119880
1015840
119911119900
= 1205751198811015840
119910119900
as in Figure 1The steady and the unsteady injection velocity are denoted
by V1015840119903119908119904
(119909) V1015840119900|wall(119909
1015840
1199051015840
) while the exit pressure and thetemperature of the fluid injected from the side wall aredenoted by 119875
1015840
119900and 119879
1015840
119900
The parabolized Navier Stokes equations for unsteadytwo dimensional compressible fluid flow are solved numeri-cally in the prescribed chamber andmay bewritten as follows
120597119876
120597119905
+
120597119864
120597119909
+
120597119865
120597119910
= 0 (1)
where
119876 = [120588 120588119906 120588V 119864119879]119879
119864 = [1198721205881199061198721205881199062
+
1
1205741198722119901119872120588119906V119872 119864
119879+ (120574 minus 1) 119901 119906]
119879
119865 = [119872120588V119872120588119906V minus
1205752
119872
119877119890
119906119910119872120588V2 +
1205752
120574119872
119901
119872 119864119879+ (120574 minus 1) 119901 V minus
1205741205752
119872
119877119890119875119903
119879119910]
119879
(2)
119864119879is the total energy (120588[119879+120574(120574minus1)119872
2
(1199062
+(V120575)2]) and theequation of state for a perfect gas is
119901 = 120588119879 (3)
In this study the specific heats the viscosity and conductivitycoefficients are considered to be constant since the change
4 Mathematical Problems in Engineering
0 02 04 06 0809
095
Injection surface
x
y
20
0
minus20
(a)
0 02 04 06 08x
09
095y
10
0
minus10
(b)
0 02 04 06 08x
09
095y
10
0
minus10
(c)
Figure 5 The vorticity contours adjacent to the injection surfacefor the same parameters as in Figure 4 for several eigenvalues (a)120582119899= 1205872 (119899 = 1) (b) 31205872 (119899 = 3) and (c) 51205872 (119899 = 5)
in the temperature in the cold model is relatively smallNondimensional variables are given by
119901 =
1199011015840
1199011015840
119900
120588 =
1205881015840
1205881015840
119900
119879 =
1198791015840
1198791015840
119900
119906 =
1199061015840
1198801015840
119911119900
V =
V1015840
(1198801015840
119911119900
120575)
119909 =
1199091015840
1198711015840
119910 =
1199101015840
1198671015840 119905 =
1199051015840
1199051015840
119886
(4)
The axial flow Mach number Prandtl number flow andacoustic Reynolds numbers are written as
119872 =
1198801015840
119911119900
1198621015840
119900
119875119903=
1205831015840
1199001198621015840
119901119900
1198961015840
119900
119877119890=
1205881015840
1199001198801015840
119911119900
1198711015840
1205831015840
119900
119877119890119860
=
119877119890
119872
(5)
The characteristic Mach number is chosen to be 119874(10minus1)while the characteristic Reynolds number is119874(105-106)Theseorders of magnitudes approximate the characteristics of thefluid flow in similar real rocket engineThe nondimensional-ized time 119905
1015840
119886is defined as the axial acoustic time 119905
1015840
119886= 1198711015840
1198621015840
119900
21 The Initial and Boundary Conditions For 119905 le 0 time-independent side wall transverse velocity with 119881|wall =
minus119881119903119908119904
(119909) is used to generate steady flow solution while for119905 gt 0 119881|wall = 119881
119903119908119904(119909 119905) = minus119881
119903119908119904(119909) minus 120576120593wall(119909)(1 minus cos120596119905)
as the unsteady longitudinal variation for the injectiontransverse at resonance or away from the resonance caseThe function 120593wall represents the longitudinal variation ofthe velocity at the injection surface Moreover the specifiedconditions at the four boundaries can be written as follows
119909 = 0 119906 = 0 V = 0
119909 = 1 119901 = 1 ldquopressure noderdquo
119910 = 0 V = 0
120597119906
120597119910
=
120597119879
120597119910
=
120597119901
120597119910
=
120597120588
120597119910
= 0
119910 = 1 119906 = 0 119879 = 1
V =
minusV119903119908119904
(119909) 119905 le 0
minusV119903119908119904
(119909) minus 120576120593119903119908
(1 minus cos120596119905) 119905 gt 0
(6)
and the initial condition is
119905 = 0 119906 = V = 0 119901 = 119879 = 1 (7)
The initial conditions for the unsteady flow computations arethe convergence solutions for the steady state conditions
22 The Analytical Approach
221 Steady and Unsteady Flow Hegab [7] has concludedthat the total axial velocity component includes two mainpartsThefirst part represents the steady state component andthe other part represents unsteady state component and canbe written as follows
(119906 V 119875 120588 119879) = (119906119904 V119904 119875119904 120588119904 119879119904) + ( V 120588
119879) (8)
Using the expended variables used inHegab [7] and insertingthem in (1) the side wall steady vorticity generation (Ω
119900119904|wall)
and the accompanying heat transfer (120597119879119900119904120597119910|wall) are derived
and written as follows
Ωos1003816100381610038161003816wall = minus
1
120574V119903119908119904
120597
120597119909
(
1205741205872
4
int
1
119909
V119903119908119904
(119909) int
119909
0
V119903119908119904
(120591) 119889120591 119889119909)
+
120597V119903119908119904
(119909)
120597119909
120597119879os120597119910
10038161003816100381610038161003816100381610038161003816wall
= 0
(9)
Equations (9) show that nonzero steady vorticity and zeroheat transfer are found with the use of no-slip and constantinjection boundary conditions
The sequence of the numerical solution consists of twosteps The first one is the initial steady state computationwhich arises from constant injection with isothermal con-dition The second one is the unsteady flow computationbased on initial conditions as computed from the first stepThe perturbation theory introduced by Staab et al [11] and
Mathematical Problems in Engineering 5
(a) (b)
(c) (d)
Figure 6 The flame geometry for the burning of hybrid rocket systems using (a) pure PE fuel grain (b) 975 PE +25 Al fuel grain (c)925 PE + 75 Al fuel grain and (d) 85 PE + 15 Al fuel grain [18]
200
201
202
203
204
205
600 625 650 675 700 725 750 775 800Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
Recording rate 60 points frequency about 20 cps and about 003 bar pure PEamplitude
Figure 7The recorded pressure time history for a burning of hybridmotor using pure polyethylene (PE) fuel grain
applied by Hegab [7] leads to the following weakly rotationalequations
119905 = 0 119900= 0
120597119900
120597119905
= 0
119909 = 0
120597119900
120597119909
= 0 119909 = 1 119900= 0
119910 = 0 V0= 0
119910 = 1 V119900= minus120576120593
119903119908(1 minus cos120596119905) 119905 gt 0
(10)
200
201
202
203
204
205
500 525 550 575 600 625 650 675 700Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
about 002 bar 75 AlRecording rate 60 points frequency about 20 cps and amplitude
Figure 8The recorded pressure time history for a burning of hybridmotor using 75 aluminum (Al) additive to PE fuel grain
205
204
203
202
201
200500 525 550 575 600 625 650 675 700
Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
Recording rate 60 points frequency about 20 cps about 0018 bar 15 Aland amplitude
Figure 9The recorded pressure time history for a burning of hybridmotor using 15 aluminum (Al) additive to PE fuel grain
6 Mathematical Problems in Engineering
The final nonresonance solution for the flow in the simulatedchamber in Figure 1 is
119900(119909 119905) =
infin
sum
119899=0
2120576120574120596int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
119900(119909 119905) =
infin
sum
119899=0
2120576120574120596int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
(cos120573119899119905 minus 1) minus
120573119899
120596
(cos120596119905 minus 1)
times sin120573119899119909
V119900(119909 119910 119905) = minus119910 120593
119903119908(119909) (1 minus cos120596119905)
(11)
Equations (11) show the time history of the perturbed valuesvariables (
119900 119900 V119900) at any axial locations
Following the same policies the resonance solutions arederived and formulated as
119900(119909 119905) =
120574119862119899lowast
120573119899lowast
sin120573119899lowast119905 minus 120573
119899lowast119905 cos120573
119899lowast119905 cos120573
119899lowast119909
119900(119909 119905) = 119862
119899lowast minus
2
120573119899lowast
(cos120573119899lowast119905 minus 1) minus 119905 sin120573
119899lowast119905
times cos120573119899lowast119909
(12)
where 119862119899lowast = 2 int
1
0
120593119903119908
(119909) cos120573119899lowast119909
23 The Computational Approach In the present studyhigher order accuracy difference equations are employedto solve the parabolized Navier-Stokes equations in theinterior of the prescribed chamber and at the boundariesThe numerical multisteps scheme in [24] is used to solve the2D unsteady and compressible Navier-Stokes equations forthe points located near the boundaries while the two-fourexplicit predictor-corrector scheme in [25] is employed tosolve the adapted equations at the interior points The lattermethod is a fourth-order accuracy variant of the fully explicitMacCormack scheme Uniform grids are used The accuracycriterion employed byMacCormack [24] is used to determinethe size of the time step
Here the NSCBC technique applied by [7] is presented indetail The full derivation for the parabolized Navier-Stokesat the upper and lower boundaries and at the head and exitends is formulated to specify the nonphysical conditions atthese boundaries Moreover this method is applied to reducethe generated reflected waves arising from the numericaltechniques
Based on the idea of characteristics for Euler equationsthe first derivatives normal to the boundaries appearing inthe convective terms of the Navier-Stokes equations may beexpressed in terms of the amplitude of incoming outcoming
and entropy waves weierp119894rsquos The fluid dynamics equations can
be written in nondimensional wave modes form as follows
120597120588
120597119905
+
1
1198622[weierp2+
1
2
(weierp4+ weierp1)] + 119872
120597120588V120597119910
= 0 (13)
120597119906
120597119905
+
1
2120588119862
[(weierp4minus weierp1)] + 119872V
120597119906
120597119910
=
120594119909
120588
(14)
120597V120597119905
+ weierp3+ 119872V
120597V120597119910
+
1198602
120574119872
1
120588
120597119901
120597119910
=
120594119910
120588
(15)
120597119901
120597119905
+
1
2
[(weierp4+ weierp1)] + 119872V
120597119901
120597119910
+ 120574119872119901
120597V120597119910
= 120594119864 (16)
where 120594119909and 120594
119910represent the viscous terms and 120594
119864refers
to the combination of the viscous and conduction terms inthe axial and transverse directions and 119862 = 119862
1015840
1198621015840
119900 The
amplitudes of the left-running waves (weierp1) the right-running
waves (weierp4) and the entropy waves (weierp
2 weierp3) in terms of the
characteristic velocities 120582119894rsquos are
weierp1= 1205821(
1
120574
120597119901
120597119909
minus 119872120588119862
120597119906
120597119909
) 1205821=
119906
119872
minus 119862 (17)
weierp2= 1205822(1198622120597120588
120597119909
minus
1
120574
120597119901
120597119909
) 1205822= 119872119906 (18)
weierp3= 1205823
120597V120597119909
1205823= 119872119906 (19)
weierp4= 1205824(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
) 1205824=
119906
119872
+ 119862 (20)
Two approaches are used to infer the values of theincoming amplitude-variations from the boundary condi-tions One is the local one-dimensional inviscid relations(LODI) technique and the second is the Navier-Stokescharacteristics boundary conditions (NSCBC) For the latterone the complete description of the numerical boundaryconditions at all boundaries is discussed below
24 Exit Boundary Conditions In this case a pressure nodeis assumed at the exit boundary The NSCBC relation (16)suggests that the amplitude of the reflected wave should be
weierp1= 2120594119864minus weierp4 (21)
and the remaining conditions are
weierp2= 119872119906(119862
2120597120588
120597119909
minus
1
120574
120597119901
120597119909
) weierp3= 119872119906
120597V120597119909
weierp4= (
119906
119872
+ 119888)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
(22)
Mathematical Problems in Engineering 7
The nondimensional equations can be written as follows
120597120588
120597119905
+ 119872119906
120597120588
120597119909
minus
119872119906
1205741198622
120597119901
120597119909
+ 119872V120597120588
120597119910
+
1198602
119872
(120574 minus 1) 119877119890119875119903
1
1198622
1205972
119879
1205971199102
= 0
120597119906
120597119905
+
1
120588119862
[(119906 +
119862
119872
)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)] + 119872V120597119906
120597119910
minus
1198602
(120574 minus 1) 119877119890119875119903
1
120588119862
1205972
119879
1205971199102
minus
1198721198602
119877119890
1
120588
1205972
119906
1205971199102
= 0
120597V120597119905
+ 119872(119906
120597V120597119909
+ V120597V120597119910
) = 0
119889119901
119889119905
= 0
(23)
Here again the nondimensional variables (120588 119906 and V)can be computed using one-sided finite-difference approxi-mations
25 Head End Boundary Condition The velocity componentnormal to the boundary (119906(0 119910 119905)) is zero since the head endis impermeable In contrast the transverse velocity compo-nent cannot be specified because the axial viscous terms inthe transversemomentum equation have been suppressed Asthe normal velocity drops to zero at 119909 = 0 the amplitudesweierp2and weierp
3 in (18) and (19) are zero and the characteristic
velocities are 1205821
= minus119862 and 1205824
= +119862 The reflected wave weierp4
enters the domain while the wave weierp1propagates out through
the boundary The NSCBC relation (19) indicates that theamplitude of the incoming wave weierp
4should be
weierp4= weierp1 (24)
and the remaining conditions are
weierp1= minus119888(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
weierp2= 0 weierp
3= 0
(25)
The equations can be written in nondimensional form asfollows
120597120588
120597119905
minus 119872120588
120597119906
120597119909
+
1
120574119862
120597119901
120597119909
+ 119872
120597120588V120597119910
= 0
120597119906
120597119905
= 0
120597V120597119905
+ 119872V120597V120597119910
= 0
120597119901
120597119905
minus 119888
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
+ 120574119872119901
120597V120597119910
minus
1205741198602
119872
(120574 minus 1) 119875119903119877119890
1205972
119879
1205971199102
= 0
(26)
More details about the NSCBC techniques are found in [2326]
26 CodeValidations Thecomputational codes are designedconstructed and validated for steady and unsteady flow as
shown in Figures 2 and 3 For the latter one the constructedcodes run for many wave cycles to detect the whole unsteadyrotational flow dynamic across the chamber
In Figure 2 comparison between the current numericaland asymptotic results dealing with the steady state flowsolutions is carried out The steady state flow is generatedby adding constant side wall mass injection (119881
119903119908= minus1)
from the injection surface of the prescribed chamber Similarexperiment in [18 27] was carried out and compared withthe current mathematical models in Figure 2The asymptoticanalysis solution represents the solid line while the dottedline represents the converged steady state numerical solutionfor viscous flow with wall Reynolds number 119877
119890119908= 5 lowast
103 The nonfilled points refer to the experimental measured
data The experimental results are recorded for flow in achannel with fully transpired wall Three different poroussurfaces using three different sizes of honeycomb cells areused [7] In this figure a comparison between the analyticaland the numerical solutions at the midlength of the chamberis illustratedThis comparison shows a reasonable agreementbetween the three approaches The comparison between thetheoretical and the computational approaches shows littlebit deviation This deviation is logically expected since theused honey combs have three different sizes The smallersize of honey combs the smaller deviations between the twoapproaches
The second set of the validations show the solution for theunsteady flow computations at nonresonance forcedmode Inthis set of the results the computational results are validatedby comparing with the asymptotic analysis These resultsrepresent the asymptotic solution when the sidewall injectionvelocity is represented by V
119903119908(119909 1 119905) = minus[1 + 120576120593
119903119908(1 minus
cos120596119905)] 120593119903119908
= cos(120573119899119909) In the current work and the work
by Hegab [7] the disturbance forcing frequency is taken tobe 120596 = 1 (nonresonance) with the amplitude of 120576 = 04
and the eigenvalues (120573119899
= 1198991205872 119899 = 1 3 5 7 9 etc)The temperature time history (119879) at themedlength location ispresented in Figure 3 for 119877
119890= 3 lowast 10
5119872 = 002 120575 = 20 and119899 = 10 For the nonresonance case the solution shows stableconditions but it is not purely harmonic This interestingbehavior shows the ability of the constructed code with thegood treatment to the boundary conditions to detect thenon-harmonic trend as detected by the analytical approachA comparison of the perturbed temperature (119879) (27) withthe numerical results in Figure 3 shows good agreementbetween the two approaches especially for 119905 lt 20 while thecomputational solution shows little less deviation than theanalytical results for 119905 gt 20 due to the nonlinearity effects
119879119900(119909 119905) =
infin
sum
119899=0
2120576120596 (120574 minus 1) int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
(27)
Generally the coexistence of the forced and the naturalmodes of the system is detected in the adapted computationaland analytical approaches as seen in the earlier investigationby Hegab [7] In contrast the computational techniques in
8 Mathematical Problems in Engineering
[10] show only pure harmonic oscillations The source ofthese deficiencies may be arising from the treatment of theboundary conditions especially at the head and aft ends ofthe simulated chamber
3 Results and Discussions
The impact of the injected wave number from the sidewallalong with the frequencies of the forced mode on thegenerated complex acoustical fields and their interactionwiththe fluid are examinedThe results have been categorized intothe following two sets
31 Nonresonance Solution In the current study a widerange of forcing frequencies away from or very close tothe first fundamental modes of the prescribed chamber isconsidered [7] a wide range of forcing frequencies away fromor very close to the first fundamental modes of the prescribedchamber is considered In Figure 4 the irrotational androtational fields across the whole chamber are presentedwhen 119905 = 42 for 120596 = 1 119877
119890= 3lowast10
5119872 = 002 120576 = 04 119899 = 13 and 5 and 120582
119899= 1205872 The latter one represents the Eigen
values of the chamberThese interesting graphs show that therotational fields are generated at the injection surface As timeincreases the generated unsteady vorticity is transferred to fillmost of the combustion chamber Moreover the constructedcodes are able to run for enough time and count about sevenvorticity wave cycles as 119905 = 42 It is also seen that theamplitude of the rotational field attainsmaximumvalues nearthe wall and found to be order of ten for the monotonicvariation (119899 = 1) and nonresonance frequency (120596 = 1)One notes that these magnitudes meet the scaling of theasymptotic solutions in [7 9 11 14] Moreover for 119899 = 1monotonic distribution of unsteady vorticity is noticed whilethe variation goes in nonmonotonic axial distribution for 119899 gt
1 For the nonresonance solution the longitudinal variationof the vorticity at the wall 120597|Ω
119908(119909 119905)|120597119909 gt 0 for 120582
119899=
1205872 (119899 = 1) is revealing that the magnitude of rotationalflow fields on the injection surface increases as axial distanceincreases
The vorticity contours adjacent to the injection surface forthe same parameters as in Figure 4 for several eigenvalues(a) 120582119899
= 1205872 (119899 = 1) (b) 31205872 (119899 = 3) and (c) 51205872
(119899 = 5) are present in Figure 5 It is clearly seen that adjacentto the combustion surface the generated unsteady vorticitymay impact the surface at downstream locations for 119899 = 1at two locations for 119899 = 3 and many locations for 119899 = 5The rotational flow fluctuates from negative to positive andvice versa at several successive times and may cause scouringon the combustion surface and in turn may increase theshearing stresses adjacent to the combustion surface Thisimportant conclusion may lead to increase of the burningrates over the designed ones especially for low wave numbersince the amplitude of the unsteady vorticity decreases as thewave number increases
Fortunately recent experimental work by Meteb [28] in2009 was directed to study the acoustic instability in hybridrocket system using different fuel grains and ignition systems
The experimental results verified qualitatively the currentanalytical and computational results for the generated com-plex wave patterns and the effect of the combustion surfaceon the acoustic fluid dynamics interactionmechanism In thisexperiment the pure polyethylene (PE) fuel grain and PEwithadditives of aluminum (Al) powder with different percentageare used to predict the hybrid rocket motor regression ratesThe flame geometry and structure for the burning of thepure PE and PE with additives of aluminum (Al) powderat 25 75 and 15 of the total weight of the wholesamples are illustrated in Figure 6The effect of theAl powderon the flame length geometry and the rate of reaction forthe burning of each sample is clearly seen In addition thepressure time histories at a point before the combustionchamber in the way of oxidizer are recorded and illustratedin Figure 7 for the pure PE and in Figures 8 and 9 for PEwith additives of aluminum (Al) powder at 75 and 15of the total weight respectively It is clearly seen that theamplitude of the pressure waves decreases as the percentageof Al fuel grain increases in the sample This interestingbehavior is similar to what we have seen when we increasethe wave number of the combustion surface As we knowmost of the Al fuel grains burn away from the combustionsurface consisting ofmany fragmentsThe escaping processesof Al fuel grains from the combustion surface generate manyof cavities that make the combustion surface more roughthan the smooth one in case of pure PE The experimentaland computational results are qualitatively in agreement Forthe first time the analytical and computational results forthe current cold model and the acoustic instabilities foundby Meteb [28] state an important and essential conclusionabout the effect of the combustion surface morphology onthe acoustic instability inside the rocket chamber and theaccompanying rotational flow Instantaneous total velocityvector map for the parameters in Figure 4 at (a) 119905 = 0 ldquosteadystate solutionrdquo and (b) 119905 = 16 is presented in Figures 10(a)and 10(b) The injection surface is located at 119910 = 1 on theupper part of the plot where the flow is completely verticalwhile the lower part of the plot represents the center lineat 119910 = 0 where the flow is completely axial The closedend is located at 119909 = 0 on the left hand side of the plotwhile at 119909 = 1 it represents the exit plane on the right handside One may notice the smooth and undisturbed fluid flowthrough the entire map for the steady state solution at 119905 = 0After initiating the side wall disturbances the coexisting ofacoustics and the generated rotational flow is clearly seenin the upper half part of the chamber and illustrates howthe interaction between the travelling acoustic waves and theinjected fluid affects the fluid flow in the chamber In additionat 119905 = 32 in Figure 11 there is an instantaneous reverseflow found at a location near the combustion surface Thecomplex acoustic fluid dynamics interaction structure mayincrease the shear stresses adjacent to the combustion surfaceand through the whole cavity These results may providethe rocket designers with conceptual tools and data to beconsidered in case of redesigning strategies
Mathematical Problems in Engineering 9
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
ldquoSteady state solutionrdquo t = 00
(a)
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 16Nonresonance (
(b)
Figure 10 Instantaneous total velocity vector map for the parameters in Figure 4 (a) 119905 = 0 ldquosteady state solutionrdquo and (b) 119905 = 16
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 32
Reverse flow region
Nonresonance (
Figure 11 Instantaneous total velocity vector map at 119905 = 32 for theparameters in Figure 4
32 Near and at Resonance Solution A comparison of theasymptotic analysis perturbed pressure time history withthe computational one at and near resonance frequenciesis presented in Figure 12 For 119905 lt 10 the results showgood agreement between the two approaches even with theasymptotic beat condition (120596 = 1205872 minus 00018) which isvery close to the resonance frequency While for 119905 gt 10
the asymptotic solution shows linear growth with time basedon the linear theory for the derived wave equation where120596 = 120573
119899lowast = 1205872 as derived by Hegab [7] In contrast
the computational solution shows beats at the resonancecondition due to the nonlinearity and the asymptotic beatsolutions have smaller amplitude than the resonance oneThe beats appear due to the interaction between the derivingfrequency mode 120596 = 1205872minus00018 and the first eigenfunctionmode (1205872) It is clearly seen that the amplitude of thevorticity at the resonance frequency is found to be ten timesthan the amplitudes at the nonresonance one in Figure 4
P
03
02
01
0
minus01
minus02
minus030 5 10 15 20 25 30 35 40
t
ComputationalAnalyticalAnalytical (beats)
x = 05 y = 00 120596 = 1205872 n = 1
Figure 12 Comparison between the computational and the asymp-totic solutions for the perturbed pressure time history with 120596 = 1205872
(resonance) 120576 = 04 119872 = 002 and 119877119890= 3 lowast 10
5 at the half-lengthof the chamber
Moreover it is found that the fluctuation of the transversetemperature gradient at the injection surface is found tobe about ten times than the fluctuation of the temperaturedisturbances This concludes that since the heat transferalong the sidewall is directly proportional to the transversetemperature gradient there is an unexpectedly large heat fluxwhich exists both at the wall and within the entire chambergiven the small size of the temperature distribution Theasymptotic analysis by Staab et al [11] is used to prove thisimportant conclusion
Finally the asymptotic and computational methods areemployed to study the dynamic process in an acoustically
10 Mathematical Problems in Engineering
simulated solid propellant rocket chamber with transientsidewall mass addition at different wave numbers
4 Conclusion Remarks
The current study emphasizes the effect of the combustionsurfacemorphology of the solid rocket systemon the acousticinstability and the accompanying vorticity and temperaturedynamics generation The rotational and irrotational fieldsgenerated in the prescribed solid rocket motor chamber withMach number are chosen to be O(10minus1) while the charac-teristic Reynolds number is O(105-106) at nonresonance andnear and at resonance frequencies have been described TheNSCBC techniques that applied in [7 29] are used to treat thenumerical boundary conditions
The numerical solution shows that the convergence tosteady state solution is achieved first and then additionaltransient side wall mass addition is initiated Moreover theresult shows that the forced mode of the nonresonance andresonance frequencies generates longitudinal acoustics waveswhich interact again with its sources at the injection surfaceto generate the rotational flow fields The generated unsteadyvorticity is transferred from the injection surface toward thecenterline to fill the entire chamber as time increases aftermore than seven wave cycles The solution shows nonpureharmonic oscillations as suggested from the theory in [11]Vorticity 3D graphs and contours illustrate that spatially themaximum amplitudes are found to be near the injectionsurface Moreover the largest shear stresses are found to be atthe injection surface near the exit end while the shear stressesimpact the injection surface at many locations as the wavenumber increasesThe longitudinal variation of the rotationalflow fields is found to be nonmonotonic for 119899 gt 1
Here for the first time the analytical and computationalresults for the current coldmodel and the recent experimentalwork by Meteb [28] revealed that the combustion surfacemorphology has a great effect on the acoustic instabilityinside the rocket chamber and the accompanying rotationalflow The computational approach reveals that there areunexpectedly high heat transfer and generated rotational flowfields found to be adjacent to the injection surface in bothnonresonance and resonance cases The amplitude of thelatter one is found to be approximately ten times than thatwith the nonresonance solution One may anticipate thatthese large oscillatory gradients will persist to the surface ofa burning solid propellant in a real rocket chamber
More traditional computational studies of turbulence inchannel flows with wall injection are reviewed by Chaouat[30] Mean axial velocity profiles are found to be laminaruntil about halfway down the channel and then show signs oftransition Chaouatrsquos results do not show the acoustic-relatedeffects discussed in the present work As a result it is difficultto know from the turbulent theory results if the source ofthe downstream vorticity is classical transition to turbulenceor partly the result of the kind of mechanism discussed inthis paper In our recent studies in 2011 and 2012 [31 32]the assessment of turbulence modeling for gas flow in thechamber and in the convergent-divergent rocket nozzle for
the steady state flow is studied The detailed information andrealities in the current research about the transient coexistingof the irrotational (acoustics) rotational (generated vorticityfrom the interaction between the injected flow and thetraveling acoustic waves) and our turbulent techniques thatwe have adapted in our studies in 2011 and 2012 for the steadyflow field in SRM chambernozzle geometry will be used toaccount all the complex flow fields in our future study
Finally results of the asymptotic and computationalapproaches may provide the rocket designer with codes andinformation making them able to interpret some reasonsbehind the combustion instability inside the combustionchamber of real SRM based on the composition of the solidpropellants and in turn the morphology of the combustionsurface
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The first author would like to thank Professor D R KassoyUniversity of Colorado at Boulder USA for their continuoushelp
References
[1] A Hegab T L Jackson J Buckmaster and D S Stewart ldquoTheburning of periodic sandwich propellantsrdquo in Proceedings of the36th AIAAASMESAEASEE Joint Propulsion Conference andExhibit Huntsville Ala USA July 2000
[2] A Hegab T L Jackson J Buckmaster and D S StewartldquoNonsteady burning of periodic sandwich propellants withcomplete coupling between the solid and gas phasesrdquo Journalof Combustion and Flame vol 125 no 1-2 pp 1055ndash1070 2001
[3] A M Hegab ldquoModeling of sandwich heterogeneous propellantwith two steps chemical kinetics and fluid mechanical effectsrdquoEngineering Research Journal vol 32 no 2 pp 119ndash130 2009
[4] J Buckmaster T L Jackson A Hegab S Kochevets andM Ulrich ldquoRandomly packed heterogeneous propellants andthe flames they supportrdquo in Proceedings of the 39th AerospaceSciences Meeting and Exhibit Reno NV USA January 2001
[5] S Kochevets J Buckmaster T L Jackson and A HegabldquoRandom packs and their use in modeling heterogeneous solidpropellant combustionrdquo Journal of Propulsion and Power vol 17no 4 pp 883ndash891 2001
[6] AMHegab S AHasanienH EMostafa andAMMaradenldquoExperimental and numerical investigations for the combustionof selected composite solid propellantrdquo Engineering ResearchJournal vol 35 no 3 2012
[7] A M Hegab ldquoVorticity generation and acoustic resonance ofsimulated solid rocket motor chamber with high wave numberwall injectionrdquo Computers and Fluids vol 38 no 6 pp 1258ndash1269 2009
[8] A M Hegab and D R Kassoy ldquoInternal flow temperatureand vorticity dynamics due to transient mass additionrdquo AIAAJournal vol 44 no 4 pp 812ndash826 2006
Mathematical Problems in Engineering 11
[9] K Kirkkopru D R Kassoy and Q Zhao ldquoUnsteady vorticitygeneration and evolution in a model of a solid rocket motorrdquoJournal of Propulsion and Power vol 12 no 4 pp 646ndash6541996
[10] K Kirkkopru D R Kassoy Q Zhao and P L Staab ldquoAcousti-cally generated unsteady vorticity field in a long narrow cylinderwith sidewall injectionrdquo Journal of Engineering Mathematicsvol 42 no 1 pp 65ndash90 2002
[11] P L StaabQ ZhaoD R Kassoy andKKirkkopru ldquoCoexistingacoustic-rotational flow in a cylinder with axisymmetric side-wall mass additionrdquo Physics of Fluids vol 11 no 10 pp 2935ndash2951 1999
[12] M Rempe P L Staab and D R Kassoy ldquoThermal response foran internal flow in a cylinderwith time dependent sidewallmassadditionrdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit AIAA 2000-0996 Reno Nev USA January 2000
[13] Q Zhao and D R Kassoy ldquoThe generation and evolutionof unsteady vorticity in a solid rocket engine chamberrdquo inProceedings of the 32th Aerospace Sciences Meeting AIAA paper94-0779 Reno Nev USA 1994
[14] Q Zhao P L Staab D R Kassoy and K Kirkkopru ldquoAcous-tically generated vorticity in an internal flowrdquo Journal of FluidMechanics vol 413 pp 247ndash285 2000
[15] G A Flandro ldquoEffects of vorticity on rocket combustionstabilityrdquo Journal of Propulsion and Power vol 11 no 4 pp 607ndash625 1995
[16] J Majdalani andW K VanMoorhem ldquoMultiple-scales solutionto the acoustic boundary layer in solid rocket motorsrdquo Journalof Propulsion and Power vol 13 no 2 pp 186ndash193 1997
[17] J Majdalani andW van Moorhem ldquoImproved time-dependentflowfield solution for solid rocket motorsrdquo AIAA Journal vol36 no 2 pp 241ndash248 1998
[18] Z Deng R J Adrian and C D Tomkinsm ldquoStructure ofturbulence in channel flow with a fully transpired wallrdquo inProceedings of the 39th Aerospace Science Meeting AIAA Paper2001-1019 Reno Nev USA 2001
[19] F Vuillot andG Avalon ldquoAcoustic boundary layers in solid pro-pellant rocketmotors usingNavier-Stokes equationsrdquo Journal ofPropulsion and Power vol 7 no 2 pp 231ndash239 1991
[20] T M Smith R L Roach and G A Flandro ldquoNumerical studyof the unsteady flow in a simulated solid rocket motorrdquo inProceedings of the 31th Aerospace Science Meeting AIAA Paper93-0112 Reno Nev USA 1993
[21] C F Tseng and I S Tseng ldquoInteraction between acoustic wavesand premixed flames in porous chamberrdquo in Proceedings of the32th Aerospace Science Meeting Reno Nev USA 1994 AIAAPaper 94 3328
[22] T S Roh and V Yang ldquoTransient combustion responses ofsolid propellants to acoustic disturbances in rocket motorsrdquo inProceedings of the 33rd Aerospace Sciences Meeting and ExhibitAIAA Paper 95-0602 Reno Nev USA 1995
[23] A M Hegab A study of acoustic phenomena in solid rocketengines [PhD thesis] Mechanical Power Engineering Depart-ment Menoufia University Menufia Egypt University of Col-orado Boulder Colo USA 1998
[24] R W MacCormack ldquoA numerical method for solving theequations of compressible viscous flowrdquo American Institute ofAeronautics and Astronautics Journal vol 20 no 9 pp 1275ndash1281 1982
[25] D Gottlieb and E Turkel ldquoDissipative two-four methods fortime-dependent problemsrdquo Mathematics of Computation vol30 no 136 pp 703ndash723 1976
[26] T J Poinsot and S K Lele ldquoBoundary conditions for directsimulations of compressible viscous flowsrdquo Journal of Compu-tational Physics vol 101 no 1 pp 104ndash129 1992
[27] Z Deng R J Adrian and C D Tomkins ldquoSensitivity ofturbulence in transpired channel to injection velocity small-scale nonuniformityrdquo AIAA Journal vol 40 no 11 pp 2241ndash2246 2002
[28] K J Meteb Prediction of hybrid rocket motor regression rate[MS thesis] Military Technical College Cairo Egypt 2009
[29] J W Anders V Magi and J Abraham ldquoLarge-eddy simulationin the near-field of a transient multi-component gas jet withdensity gradientsrdquoComputers amp Fluids vol 36 no 10 pp 1609ndash1620 2007
[30] B Chaouat ldquoNumerical simulations of channel flows with fluidinjection using Reynolds stressmodelrdquo AIAAPaper 2000-09922000
[31] W A El-Askary A Balabel S M El-Behery and A HegabldquoComputations of a compressible turbulent flow in a rocketmotor-chamber configuration with symmetric and asymmetricinjectionrdquoComputerModeling in EngineeringampSciences vol 82no 1 pp 29ndash54 2011
[32] A Balabel A M Hegab M Nasr and S M El-Behery ldquoAssess-ment of turbulence modeling for gas flow in two-dimensionalconvergent-divergent rocket nozzlerdquo Applied MathematicalModelling vol 35 no 7 pp 3408ndash3422 2011
4 Mathematical Problems in Engineering
0 02 04 06 0809
095
Injection surface
x
y
20
0
minus20
(a)
0 02 04 06 08x
09
095y
10
0
minus10
(b)
0 02 04 06 08x
09
095y
10
0
minus10
(c)
Figure 5 The vorticity contours adjacent to the injection surfacefor the same parameters as in Figure 4 for several eigenvalues (a)120582119899= 1205872 (119899 = 1) (b) 31205872 (119899 = 3) and (c) 51205872 (119899 = 5)
in the temperature in the cold model is relatively smallNondimensional variables are given by
119901 =
1199011015840
1199011015840
119900
120588 =
1205881015840
1205881015840
119900
119879 =
1198791015840
1198791015840
119900
119906 =
1199061015840
1198801015840
119911119900
V =
V1015840
(1198801015840
119911119900
120575)
119909 =
1199091015840
1198711015840
119910 =
1199101015840
1198671015840 119905 =
1199051015840
1199051015840
119886
(4)
The axial flow Mach number Prandtl number flow andacoustic Reynolds numbers are written as
119872 =
1198801015840
119911119900
1198621015840
119900
119875119903=
1205831015840
1199001198621015840
119901119900
1198961015840
119900
119877119890=
1205881015840
1199001198801015840
119911119900
1198711015840
1205831015840
119900
119877119890119860
=
119877119890
119872
(5)
The characteristic Mach number is chosen to be 119874(10minus1)while the characteristic Reynolds number is119874(105-106)Theseorders of magnitudes approximate the characteristics of thefluid flow in similar real rocket engineThe nondimensional-ized time 119905
1015840
119886is defined as the axial acoustic time 119905
1015840
119886= 1198711015840
1198621015840
119900
21 The Initial and Boundary Conditions For 119905 le 0 time-independent side wall transverse velocity with 119881|wall =
minus119881119903119908119904
(119909) is used to generate steady flow solution while for119905 gt 0 119881|wall = 119881
119903119908119904(119909 119905) = minus119881
119903119908119904(119909) minus 120576120593wall(119909)(1 minus cos120596119905)
as the unsteady longitudinal variation for the injectiontransverse at resonance or away from the resonance caseThe function 120593wall represents the longitudinal variation ofthe velocity at the injection surface Moreover the specifiedconditions at the four boundaries can be written as follows
119909 = 0 119906 = 0 V = 0
119909 = 1 119901 = 1 ldquopressure noderdquo
119910 = 0 V = 0
120597119906
120597119910
=
120597119879
120597119910
=
120597119901
120597119910
=
120597120588
120597119910
= 0
119910 = 1 119906 = 0 119879 = 1
V =
minusV119903119908119904
(119909) 119905 le 0
minusV119903119908119904
(119909) minus 120576120593119903119908
(1 minus cos120596119905) 119905 gt 0
(6)
and the initial condition is
119905 = 0 119906 = V = 0 119901 = 119879 = 1 (7)
The initial conditions for the unsteady flow computations arethe convergence solutions for the steady state conditions
22 The Analytical Approach
221 Steady and Unsteady Flow Hegab [7] has concludedthat the total axial velocity component includes two mainpartsThefirst part represents the steady state component andthe other part represents unsteady state component and canbe written as follows
(119906 V 119875 120588 119879) = (119906119904 V119904 119875119904 120588119904 119879119904) + ( V 120588
119879) (8)
Using the expended variables used inHegab [7] and insertingthem in (1) the side wall steady vorticity generation (Ω
119900119904|wall)
and the accompanying heat transfer (120597119879119900119904120597119910|wall) are derived
and written as follows
Ωos1003816100381610038161003816wall = minus
1
120574V119903119908119904
120597
120597119909
(
1205741205872
4
int
1
119909
V119903119908119904
(119909) int
119909
0
V119903119908119904
(120591) 119889120591 119889119909)
+
120597V119903119908119904
(119909)
120597119909
120597119879os120597119910
10038161003816100381610038161003816100381610038161003816wall
= 0
(9)
Equations (9) show that nonzero steady vorticity and zeroheat transfer are found with the use of no-slip and constantinjection boundary conditions
The sequence of the numerical solution consists of twosteps The first one is the initial steady state computationwhich arises from constant injection with isothermal con-dition The second one is the unsteady flow computationbased on initial conditions as computed from the first stepThe perturbation theory introduced by Staab et al [11] and
Mathematical Problems in Engineering 5
(a) (b)
(c) (d)
Figure 6 The flame geometry for the burning of hybrid rocket systems using (a) pure PE fuel grain (b) 975 PE +25 Al fuel grain (c)925 PE + 75 Al fuel grain and (d) 85 PE + 15 Al fuel grain [18]
200
201
202
203
204
205
600 625 650 675 700 725 750 775 800Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
Recording rate 60 points frequency about 20 cps and about 003 bar pure PEamplitude
Figure 7The recorded pressure time history for a burning of hybridmotor using pure polyethylene (PE) fuel grain
applied by Hegab [7] leads to the following weakly rotationalequations
119905 = 0 119900= 0
120597119900
120597119905
= 0
119909 = 0
120597119900
120597119909
= 0 119909 = 1 119900= 0
119910 = 0 V0= 0
119910 = 1 V119900= minus120576120593
119903119908(1 minus cos120596119905) 119905 gt 0
(10)
200
201
202
203
204
205
500 525 550 575 600 625 650 675 700Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
about 002 bar 75 AlRecording rate 60 points frequency about 20 cps and amplitude
Figure 8The recorded pressure time history for a burning of hybridmotor using 75 aluminum (Al) additive to PE fuel grain
205
204
203
202
201
200500 525 550 575 600 625 650 675 700
Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
Recording rate 60 points frequency about 20 cps about 0018 bar 15 Aland amplitude
Figure 9The recorded pressure time history for a burning of hybridmotor using 15 aluminum (Al) additive to PE fuel grain
6 Mathematical Problems in Engineering
The final nonresonance solution for the flow in the simulatedchamber in Figure 1 is
119900(119909 119905) =
infin
sum
119899=0
2120576120574120596int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
119900(119909 119905) =
infin
sum
119899=0
2120576120574120596int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
(cos120573119899119905 minus 1) minus
120573119899
120596
(cos120596119905 minus 1)
times sin120573119899119909
V119900(119909 119910 119905) = minus119910 120593
119903119908(119909) (1 minus cos120596119905)
(11)
Equations (11) show the time history of the perturbed valuesvariables (
119900 119900 V119900) at any axial locations
Following the same policies the resonance solutions arederived and formulated as
119900(119909 119905) =
120574119862119899lowast
120573119899lowast
sin120573119899lowast119905 minus 120573
119899lowast119905 cos120573
119899lowast119905 cos120573
119899lowast119909
119900(119909 119905) = 119862
119899lowast minus
2
120573119899lowast
(cos120573119899lowast119905 minus 1) minus 119905 sin120573
119899lowast119905
times cos120573119899lowast119909
(12)
where 119862119899lowast = 2 int
1
0
120593119903119908
(119909) cos120573119899lowast119909
23 The Computational Approach In the present studyhigher order accuracy difference equations are employedto solve the parabolized Navier-Stokes equations in theinterior of the prescribed chamber and at the boundariesThe numerical multisteps scheme in [24] is used to solve the2D unsteady and compressible Navier-Stokes equations forthe points located near the boundaries while the two-fourexplicit predictor-corrector scheme in [25] is employed tosolve the adapted equations at the interior points The lattermethod is a fourth-order accuracy variant of the fully explicitMacCormack scheme Uniform grids are used The accuracycriterion employed byMacCormack [24] is used to determinethe size of the time step
Here the NSCBC technique applied by [7] is presented indetail The full derivation for the parabolized Navier-Stokesat the upper and lower boundaries and at the head and exitends is formulated to specify the nonphysical conditions atthese boundaries Moreover this method is applied to reducethe generated reflected waves arising from the numericaltechniques
Based on the idea of characteristics for Euler equationsthe first derivatives normal to the boundaries appearing inthe convective terms of the Navier-Stokes equations may beexpressed in terms of the amplitude of incoming outcoming
and entropy waves weierp119894rsquos The fluid dynamics equations can
be written in nondimensional wave modes form as follows
120597120588
120597119905
+
1
1198622[weierp2+
1
2
(weierp4+ weierp1)] + 119872
120597120588V120597119910
= 0 (13)
120597119906
120597119905
+
1
2120588119862
[(weierp4minus weierp1)] + 119872V
120597119906
120597119910
=
120594119909
120588
(14)
120597V120597119905
+ weierp3+ 119872V
120597V120597119910
+
1198602
120574119872
1
120588
120597119901
120597119910
=
120594119910
120588
(15)
120597119901
120597119905
+
1
2
[(weierp4+ weierp1)] + 119872V
120597119901
120597119910
+ 120574119872119901
120597V120597119910
= 120594119864 (16)
where 120594119909and 120594
119910represent the viscous terms and 120594
119864refers
to the combination of the viscous and conduction terms inthe axial and transverse directions and 119862 = 119862
1015840
1198621015840
119900 The
amplitudes of the left-running waves (weierp1) the right-running
waves (weierp4) and the entropy waves (weierp
2 weierp3) in terms of the
characteristic velocities 120582119894rsquos are
weierp1= 1205821(
1
120574
120597119901
120597119909
minus 119872120588119862
120597119906
120597119909
) 1205821=
119906
119872
minus 119862 (17)
weierp2= 1205822(1198622120597120588
120597119909
minus
1
120574
120597119901
120597119909
) 1205822= 119872119906 (18)
weierp3= 1205823
120597V120597119909
1205823= 119872119906 (19)
weierp4= 1205824(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
) 1205824=
119906
119872
+ 119862 (20)
Two approaches are used to infer the values of theincoming amplitude-variations from the boundary condi-tions One is the local one-dimensional inviscid relations(LODI) technique and the second is the Navier-Stokescharacteristics boundary conditions (NSCBC) For the latterone the complete description of the numerical boundaryconditions at all boundaries is discussed below
24 Exit Boundary Conditions In this case a pressure nodeis assumed at the exit boundary The NSCBC relation (16)suggests that the amplitude of the reflected wave should be
weierp1= 2120594119864minus weierp4 (21)
and the remaining conditions are
weierp2= 119872119906(119862
2120597120588
120597119909
minus
1
120574
120597119901
120597119909
) weierp3= 119872119906
120597V120597119909
weierp4= (
119906
119872
+ 119888)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
(22)
Mathematical Problems in Engineering 7
The nondimensional equations can be written as follows
120597120588
120597119905
+ 119872119906
120597120588
120597119909
minus
119872119906
1205741198622
120597119901
120597119909
+ 119872V120597120588
120597119910
+
1198602
119872
(120574 minus 1) 119877119890119875119903
1
1198622
1205972
119879
1205971199102
= 0
120597119906
120597119905
+
1
120588119862
[(119906 +
119862
119872
)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)] + 119872V120597119906
120597119910
minus
1198602
(120574 minus 1) 119877119890119875119903
1
120588119862
1205972
119879
1205971199102
minus
1198721198602
119877119890
1
120588
1205972
119906
1205971199102
= 0
120597V120597119905
+ 119872(119906
120597V120597119909
+ V120597V120597119910
) = 0
119889119901
119889119905
= 0
(23)
Here again the nondimensional variables (120588 119906 and V)can be computed using one-sided finite-difference approxi-mations
25 Head End Boundary Condition The velocity componentnormal to the boundary (119906(0 119910 119905)) is zero since the head endis impermeable In contrast the transverse velocity compo-nent cannot be specified because the axial viscous terms inthe transversemomentum equation have been suppressed Asthe normal velocity drops to zero at 119909 = 0 the amplitudesweierp2and weierp
3 in (18) and (19) are zero and the characteristic
velocities are 1205821
= minus119862 and 1205824
= +119862 The reflected wave weierp4
enters the domain while the wave weierp1propagates out through
the boundary The NSCBC relation (19) indicates that theamplitude of the incoming wave weierp
4should be
weierp4= weierp1 (24)
and the remaining conditions are
weierp1= minus119888(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
weierp2= 0 weierp
3= 0
(25)
The equations can be written in nondimensional form asfollows
120597120588
120597119905
minus 119872120588
120597119906
120597119909
+
1
120574119862
120597119901
120597119909
+ 119872
120597120588V120597119910
= 0
120597119906
120597119905
= 0
120597V120597119905
+ 119872V120597V120597119910
= 0
120597119901
120597119905
minus 119888
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
+ 120574119872119901
120597V120597119910
minus
1205741198602
119872
(120574 minus 1) 119875119903119877119890
1205972
119879
1205971199102
= 0
(26)
More details about the NSCBC techniques are found in [2326]
26 CodeValidations Thecomputational codes are designedconstructed and validated for steady and unsteady flow as
shown in Figures 2 and 3 For the latter one the constructedcodes run for many wave cycles to detect the whole unsteadyrotational flow dynamic across the chamber
In Figure 2 comparison between the current numericaland asymptotic results dealing with the steady state flowsolutions is carried out The steady state flow is generatedby adding constant side wall mass injection (119881
119903119908= minus1)
from the injection surface of the prescribed chamber Similarexperiment in [18 27] was carried out and compared withthe current mathematical models in Figure 2The asymptoticanalysis solution represents the solid line while the dottedline represents the converged steady state numerical solutionfor viscous flow with wall Reynolds number 119877
119890119908= 5 lowast
103 The nonfilled points refer to the experimental measured
data The experimental results are recorded for flow in achannel with fully transpired wall Three different poroussurfaces using three different sizes of honeycomb cells areused [7] In this figure a comparison between the analyticaland the numerical solutions at the midlength of the chamberis illustratedThis comparison shows a reasonable agreementbetween the three approaches The comparison between thetheoretical and the computational approaches shows littlebit deviation This deviation is logically expected since theused honey combs have three different sizes The smallersize of honey combs the smaller deviations between the twoapproaches
The second set of the validations show the solution for theunsteady flow computations at nonresonance forcedmode Inthis set of the results the computational results are validatedby comparing with the asymptotic analysis These resultsrepresent the asymptotic solution when the sidewall injectionvelocity is represented by V
119903119908(119909 1 119905) = minus[1 + 120576120593
119903119908(1 minus
cos120596119905)] 120593119903119908
= cos(120573119899119909) In the current work and the work
by Hegab [7] the disturbance forcing frequency is taken tobe 120596 = 1 (nonresonance) with the amplitude of 120576 = 04
and the eigenvalues (120573119899
= 1198991205872 119899 = 1 3 5 7 9 etc)The temperature time history (119879) at themedlength location ispresented in Figure 3 for 119877
119890= 3 lowast 10
5119872 = 002 120575 = 20 and119899 = 10 For the nonresonance case the solution shows stableconditions but it is not purely harmonic This interestingbehavior shows the ability of the constructed code with thegood treatment to the boundary conditions to detect thenon-harmonic trend as detected by the analytical approachA comparison of the perturbed temperature (119879) (27) withthe numerical results in Figure 3 shows good agreementbetween the two approaches especially for 119905 lt 20 while thecomputational solution shows little less deviation than theanalytical results for 119905 gt 20 due to the nonlinearity effects
119879119900(119909 119905) =
infin
sum
119899=0
2120576120596 (120574 minus 1) int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
(27)
Generally the coexistence of the forced and the naturalmodes of the system is detected in the adapted computationaland analytical approaches as seen in the earlier investigationby Hegab [7] In contrast the computational techniques in
8 Mathematical Problems in Engineering
[10] show only pure harmonic oscillations The source ofthese deficiencies may be arising from the treatment of theboundary conditions especially at the head and aft ends ofthe simulated chamber
3 Results and Discussions
The impact of the injected wave number from the sidewallalong with the frequencies of the forced mode on thegenerated complex acoustical fields and their interactionwiththe fluid are examinedThe results have been categorized intothe following two sets
31 Nonresonance Solution In the current study a widerange of forcing frequencies away from or very close tothe first fundamental modes of the prescribed chamber isconsidered [7] a wide range of forcing frequencies away fromor very close to the first fundamental modes of the prescribedchamber is considered In Figure 4 the irrotational androtational fields across the whole chamber are presentedwhen 119905 = 42 for 120596 = 1 119877
119890= 3lowast10
5119872 = 002 120576 = 04 119899 = 13 and 5 and 120582
119899= 1205872 The latter one represents the Eigen
values of the chamberThese interesting graphs show that therotational fields are generated at the injection surface As timeincreases the generated unsteady vorticity is transferred to fillmost of the combustion chamber Moreover the constructedcodes are able to run for enough time and count about sevenvorticity wave cycles as 119905 = 42 It is also seen that theamplitude of the rotational field attainsmaximumvalues nearthe wall and found to be order of ten for the monotonicvariation (119899 = 1) and nonresonance frequency (120596 = 1)One notes that these magnitudes meet the scaling of theasymptotic solutions in [7 9 11 14] Moreover for 119899 = 1monotonic distribution of unsteady vorticity is noticed whilethe variation goes in nonmonotonic axial distribution for 119899 gt
1 For the nonresonance solution the longitudinal variationof the vorticity at the wall 120597|Ω
119908(119909 119905)|120597119909 gt 0 for 120582
119899=
1205872 (119899 = 1) is revealing that the magnitude of rotationalflow fields on the injection surface increases as axial distanceincreases
The vorticity contours adjacent to the injection surface forthe same parameters as in Figure 4 for several eigenvalues(a) 120582119899
= 1205872 (119899 = 1) (b) 31205872 (119899 = 3) and (c) 51205872
(119899 = 5) are present in Figure 5 It is clearly seen that adjacentto the combustion surface the generated unsteady vorticitymay impact the surface at downstream locations for 119899 = 1at two locations for 119899 = 3 and many locations for 119899 = 5The rotational flow fluctuates from negative to positive andvice versa at several successive times and may cause scouringon the combustion surface and in turn may increase theshearing stresses adjacent to the combustion surface Thisimportant conclusion may lead to increase of the burningrates over the designed ones especially for low wave numbersince the amplitude of the unsteady vorticity decreases as thewave number increases
Fortunately recent experimental work by Meteb [28] in2009 was directed to study the acoustic instability in hybridrocket system using different fuel grains and ignition systems
The experimental results verified qualitatively the currentanalytical and computational results for the generated com-plex wave patterns and the effect of the combustion surfaceon the acoustic fluid dynamics interactionmechanism In thisexperiment the pure polyethylene (PE) fuel grain and PEwithadditives of aluminum (Al) powder with different percentageare used to predict the hybrid rocket motor regression ratesThe flame geometry and structure for the burning of thepure PE and PE with additives of aluminum (Al) powderat 25 75 and 15 of the total weight of the wholesamples are illustrated in Figure 6The effect of theAl powderon the flame length geometry and the rate of reaction forthe burning of each sample is clearly seen In addition thepressure time histories at a point before the combustionchamber in the way of oxidizer are recorded and illustratedin Figure 7 for the pure PE and in Figures 8 and 9 for PEwith additives of aluminum (Al) powder at 75 and 15of the total weight respectively It is clearly seen that theamplitude of the pressure waves decreases as the percentageof Al fuel grain increases in the sample This interestingbehavior is similar to what we have seen when we increasethe wave number of the combustion surface As we knowmost of the Al fuel grains burn away from the combustionsurface consisting ofmany fragmentsThe escaping processesof Al fuel grains from the combustion surface generate manyof cavities that make the combustion surface more roughthan the smooth one in case of pure PE The experimentaland computational results are qualitatively in agreement Forthe first time the analytical and computational results forthe current cold model and the acoustic instabilities foundby Meteb [28] state an important and essential conclusionabout the effect of the combustion surface morphology onthe acoustic instability inside the rocket chamber and theaccompanying rotational flow Instantaneous total velocityvector map for the parameters in Figure 4 at (a) 119905 = 0 ldquosteadystate solutionrdquo and (b) 119905 = 16 is presented in Figures 10(a)and 10(b) The injection surface is located at 119910 = 1 on theupper part of the plot where the flow is completely verticalwhile the lower part of the plot represents the center lineat 119910 = 0 where the flow is completely axial The closedend is located at 119909 = 0 on the left hand side of the plotwhile at 119909 = 1 it represents the exit plane on the right handside One may notice the smooth and undisturbed fluid flowthrough the entire map for the steady state solution at 119905 = 0After initiating the side wall disturbances the coexisting ofacoustics and the generated rotational flow is clearly seenin the upper half part of the chamber and illustrates howthe interaction between the travelling acoustic waves and theinjected fluid affects the fluid flow in the chamber In additionat 119905 = 32 in Figure 11 there is an instantaneous reverseflow found at a location near the combustion surface Thecomplex acoustic fluid dynamics interaction structure mayincrease the shear stresses adjacent to the combustion surfaceand through the whole cavity These results may providethe rocket designers with conceptual tools and data to beconsidered in case of redesigning strategies
Mathematical Problems in Engineering 9
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
ldquoSteady state solutionrdquo t = 00
(a)
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 16Nonresonance (
(b)
Figure 10 Instantaneous total velocity vector map for the parameters in Figure 4 (a) 119905 = 0 ldquosteady state solutionrdquo and (b) 119905 = 16
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 32
Reverse flow region
Nonresonance (
Figure 11 Instantaneous total velocity vector map at 119905 = 32 for theparameters in Figure 4
32 Near and at Resonance Solution A comparison of theasymptotic analysis perturbed pressure time history withthe computational one at and near resonance frequenciesis presented in Figure 12 For 119905 lt 10 the results showgood agreement between the two approaches even with theasymptotic beat condition (120596 = 1205872 minus 00018) which isvery close to the resonance frequency While for 119905 gt 10
the asymptotic solution shows linear growth with time basedon the linear theory for the derived wave equation where120596 = 120573
119899lowast = 1205872 as derived by Hegab [7] In contrast
the computational solution shows beats at the resonancecondition due to the nonlinearity and the asymptotic beatsolutions have smaller amplitude than the resonance oneThe beats appear due to the interaction between the derivingfrequency mode 120596 = 1205872minus00018 and the first eigenfunctionmode (1205872) It is clearly seen that the amplitude of thevorticity at the resonance frequency is found to be ten timesthan the amplitudes at the nonresonance one in Figure 4
P
03
02
01
0
minus01
minus02
minus030 5 10 15 20 25 30 35 40
t
ComputationalAnalyticalAnalytical (beats)
x = 05 y = 00 120596 = 1205872 n = 1
Figure 12 Comparison between the computational and the asymp-totic solutions for the perturbed pressure time history with 120596 = 1205872
(resonance) 120576 = 04 119872 = 002 and 119877119890= 3 lowast 10
5 at the half-lengthof the chamber
Moreover it is found that the fluctuation of the transversetemperature gradient at the injection surface is found tobe about ten times than the fluctuation of the temperaturedisturbances This concludes that since the heat transferalong the sidewall is directly proportional to the transversetemperature gradient there is an unexpectedly large heat fluxwhich exists both at the wall and within the entire chambergiven the small size of the temperature distribution Theasymptotic analysis by Staab et al [11] is used to prove thisimportant conclusion
Finally the asymptotic and computational methods areemployed to study the dynamic process in an acoustically
10 Mathematical Problems in Engineering
simulated solid propellant rocket chamber with transientsidewall mass addition at different wave numbers
4 Conclusion Remarks
The current study emphasizes the effect of the combustionsurfacemorphology of the solid rocket systemon the acousticinstability and the accompanying vorticity and temperaturedynamics generation The rotational and irrotational fieldsgenerated in the prescribed solid rocket motor chamber withMach number are chosen to be O(10minus1) while the charac-teristic Reynolds number is O(105-106) at nonresonance andnear and at resonance frequencies have been described TheNSCBC techniques that applied in [7 29] are used to treat thenumerical boundary conditions
The numerical solution shows that the convergence tosteady state solution is achieved first and then additionaltransient side wall mass addition is initiated Moreover theresult shows that the forced mode of the nonresonance andresonance frequencies generates longitudinal acoustics waveswhich interact again with its sources at the injection surfaceto generate the rotational flow fields The generated unsteadyvorticity is transferred from the injection surface toward thecenterline to fill the entire chamber as time increases aftermore than seven wave cycles The solution shows nonpureharmonic oscillations as suggested from the theory in [11]Vorticity 3D graphs and contours illustrate that spatially themaximum amplitudes are found to be near the injectionsurface Moreover the largest shear stresses are found to be atthe injection surface near the exit end while the shear stressesimpact the injection surface at many locations as the wavenumber increasesThe longitudinal variation of the rotationalflow fields is found to be nonmonotonic for 119899 gt 1
Here for the first time the analytical and computationalresults for the current coldmodel and the recent experimentalwork by Meteb [28] revealed that the combustion surfacemorphology has a great effect on the acoustic instabilityinside the rocket chamber and the accompanying rotationalflow The computational approach reveals that there areunexpectedly high heat transfer and generated rotational flowfields found to be adjacent to the injection surface in bothnonresonance and resonance cases The amplitude of thelatter one is found to be approximately ten times than thatwith the nonresonance solution One may anticipate thatthese large oscillatory gradients will persist to the surface ofa burning solid propellant in a real rocket chamber
More traditional computational studies of turbulence inchannel flows with wall injection are reviewed by Chaouat[30] Mean axial velocity profiles are found to be laminaruntil about halfway down the channel and then show signs oftransition Chaouatrsquos results do not show the acoustic-relatedeffects discussed in the present work As a result it is difficultto know from the turbulent theory results if the source ofthe downstream vorticity is classical transition to turbulenceor partly the result of the kind of mechanism discussed inthis paper In our recent studies in 2011 and 2012 [31 32]the assessment of turbulence modeling for gas flow in thechamber and in the convergent-divergent rocket nozzle for
the steady state flow is studied The detailed information andrealities in the current research about the transient coexistingof the irrotational (acoustics) rotational (generated vorticityfrom the interaction between the injected flow and thetraveling acoustic waves) and our turbulent techniques thatwe have adapted in our studies in 2011 and 2012 for the steadyflow field in SRM chambernozzle geometry will be used toaccount all the complex flow fields in our future study
Finally results of the asymptotic and computationalapproaches may provide the rocket designer with codes andinformation making them able to interpret some reasonsbehind the combustion instability inside the combustionchamber of real SRM based on the composition of the solidpropellants and in turn the morphology of the combustionsurface
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The first author would like to thank Professor D R KassoyUniversity of Colorado at Boulder USA for their continuoushelp
References
[1] A Hegab T L Jackson J Buckmaster and D S Stewart ldquoTheburning of periodic sandwich propellantsrdquo in Proceedings of the36th AIAAASMESAEASEE Joint Propulsion Conference andExhibit Huntsville Ala USA July 2000
[2] A Hegab T L Jackson J Buckmaster and D S StewartldquoNonsteady burning of periodic sandwich propellants withcomplete coupling between the solid and gas phasesrdquo Journalof Combustion and Flame vol 125 no 1-2 pp 1055ndash1070 2001
[3] A M Hegab ldquoModeling of sandwich heterogeneous propellantwith two steps chemical kinetics and fluid mechanical effectsrdquoEngineering Research Journal vol 32 no 2 pp 119ndash130 2009
[4] J Buckmaster T L Jackson A Hegab S Kochevets andM Ulrich ldquoRandomly packed heterogeneous propellants andthe flames they supportrdquo in Proceedings of the 39th AerospaceSciences Meeting and Exhibit Reno NV USA January 2001
[5] S Kochevets J Buckmaster T L Jackson and A HegabldquoRandom packs and their use in modeling heterogeneous solidpropellant combustionrdquo Journal of Propulsion and Power vol 17no 4 pp 883ndash891 2001
[6] AMHegab S AHasanienH EMostafa andAMMaradenldquoExperimental and numerical investigations for the combustionof selected composite solid propellantrdquo Engineering ResearchJournal vol 35 no 3 2012
[7] A M Hegab ldquoVorticity generation and acoustic resonance ofsimulated solid rocket motor chamber with high wave numberwall injectionrdquo Computers and Fluids vol 38 no 6 pp 1258ndash1269 2009
[8] A M Hegab and D R Kassoy ldquoInternal flow temperatureand vorticity dynamics due to transient mass additionrdquo AIAAJournal vol 44 no 4 pp 812ndash826 2006
Mathematical Problems in Engineering 11
[9] K Kirkkopru D R Kassoy and Q Zhao ldquoUnsteady vorticitygeneration and evolution in a model of a solid rocket motorrdquoJournal of Propulsion and Power vol 12 no 4 pp 646ndash6541996
[10] K Kirkkopru D R Kassoy Q Zhao and P L Staab ldquoAcousti-cally generated unsteady vorticity field in a long narrow cylinderwith sidewall injectionrdquo Journal of Engineering Mathematicsvol 42 no 1 pp 65ndash90 2002
[11] P L StaabQ ZhaoD R Kassoy andKKirkkopru ldquoCoexistingacoustic-rotational flow in a cylinder with axisymmetric side-wall mass additionrdquo Physics of Fluids vol 11 no 10 pp 2935ndash2951 1999
[12] M Rempe P L Staab and D R Kassoy ldquoThermal response foran internal flow in a cylinderwith time dependent sidewallmassadditionrdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit AIAA 2000-0996 Reno Nev USA January 2000
[13] Q Zhao and D R Kassoy ldquoThe generation and evolutionof unsteady vorticity in a solid rocket engine chamberrdquo inProceedings of the 32th Aerospace Sciences Meeting AIAA paper94-0779 Reno Nev USA 1994
[14] Q Zhao P L Staab D R Kassoy and K Kirkkopru ldquoAcous-tically generated vorticity in an internal flowrdquo Journal of FluidMechanics vol 413 pp 247ndash285 2000
[15] G A Flandro ldquoEffects of vorticity on rocket combustionstabilityrdquo Journal of Propulsion and Power vol 11 no 4 pp 607ndash625 1995
[16] J Majdalani andW K VanMoorhem ldquoMultiple-scales solutionto the acoustic boundary layer in solid rocket motorsrdquo Journalof Propulsion and Power vol 13 no 2 pp 186ndash193 1997
[17] J Majdalani andW van Moorhem ldquoImproved time-dependentflowfield solution for solid rocket motorsrdquo AIAA Journal vol36 no 2 pp 241ndash248 1998
[18] Z Deng R J Adrian and C D Tomkinsm ldquoStructure ofturbulence in channel flow with a fully transpired wallrdquo inProceedings of the 39th Aerospace Science Meeting AIAA Paper2001-1019 Reno Nev USA 2001
[19] F Vuillot andG Avalon ldquoAcoustic boundary layers in solid pro-pellant rocketmotors usingNavier-Stokes equationsrdquo Journal ofPropulsion and Power vol 7 no 2 pp 231ndash239 1991
[20] T M Smith R L Roach and G A Flandro ldquoNumerical studyof the unsteady flow in a simulated solid rocket motorrdquo inProceedings of the 31th Aerospace Science Meeting AIAA Paper93-0112 Reno Nev USA 1993
[21] C F Tseng and I S Tseng ldquoInteraction between acoustic wavesand premixed flames in porous chamberrdquo in Proceedings of the32th Aerospace Science Meeting Reno Nev USA 1994 AIAAPaper 94 3328
[22] T S Roh and V Yang ldquoTransient combustion responses ofsolid propellants to acoustic disturbances in rocket motorsrdquo inProceedings of the 33rd Aerospace Sciences Meeting and ExhibitAIAA Paper 95-0602 Reno Nev USA 1995
[23] A M Hegab A study of acoustic phenomena in solid rocketengines [PhD thesis] Mechanical Power Engineering Depart-ment Menoufia University Menufia Egypt University of Col-orado Boulder Colo USA 1998
[24] R W MacCormack ldquoA numerical method for solving theequations of compressible viscous flowrdquo American Institute ofAeronautics and Astronautics Journal vol 20 no 9 pp 1275ndash1281 1982
[25] D Gottlieb and E Turkel ldquoDissipative two-four methods fortime-dependent problemsrdquo Mathematics of Computation vol30 no 136 pp 703ndash723 1976
[26] T J Poinsot and S K Lele ldquoBoundary conditions for directsimulations of compressible viscous flowsrdquo Journal of Compu-tational Physics vol 101 no 1 pp 104ndash129 1992
[27] Z Deng R J Adrian and C D Tomkins ldquoSensitivity ofturbulence in transpired channel to injection velocity small-scale nonuniformityrdquo AIAA Journal vol 40 no 11 pp 2241ndash2246 2002
[28] K J Meteb Prediction of hybrid rocket motor regression rate[MS thesis] Military Technical College Cairo Egypt 2009
[29] J W Anders V Magi and J Abraham ldquoLarge-eddy simulationin the near-field of a transient multi-component gas jet withdensity gradientsrdquoComputers amp Fluids vol 36 no 10 pp 1609ndash1620 2007
[30] B Chaouat ldquoNumerical simulations of channel flows with fluidinjection using Reynolds stressmodelrdquo AIAAPaper 2000-09922000
[31] W A El-Askary A Balabel S M El-Behery and A HegabldquoComputations of a compressible turbulent flow in a rocketmotor-chamber configuration with symmetric and asymmetricinjectionrdquoComputerModeling in EngineeringampSciences vol 82no 1 pp 29ndash54 2011
[32] A Balabel A M Hegab M Nasr and S M El-Behery ldquoAssess-ment of turbulence modeling for gas flow in two-dimensionalconvergent-divergent rocket nozzlerdquo Applied MathematicalModelling vol 35 no 7 pp 3408ndash3422 2011
Mathematical Problems in Engineering 5
(a) (b)
(c) (d)
Figure 6 The flame geometry for the burning of hybrid rocket systems using (a) pure PE fuel grain (b) 975 PE +25 Al fuel grain (c)925 PE + 75 Al fuel grain and (d) 85 PE + 15 Al fuel grain [18]
200
201
202
203
204
205
600 625 650 675 700 725 750 775 800Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
Recording rate 60 points frequency about 20 cps and about 003 bar pure PEamplitude
Figure 7The recorded pressure time history for a burning of hybridmotor using pure polyethylene (PE) fuel grain
applied by Hegab [7] leads to the following weakly rotationalequations
119905 = 0 119900= 0
120597119900
120597119905
= 0
119909 = 0
120597119900
120597119909
= 0 119909 = 1 119900= 0
119910 = 0 V0= 0
119910 = 1 V119900= minus120576120593
119903119908(1 minus cos120596119905) 119905 gt 0
(10)
200
201
202
203
204
205
500 525 550 575 600 625 650 675 700Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
about 002 bar 75 AlRecording rate 60 points frequency about 20 cps and amplitude
Figure 8The recorded pressure time history for a burning of hybridmotor using 75 aluminum (Al) additive to PE fuel grain
205
204
203
202
201
200500 525 550 575 600 625 650 675 700
Operating time (s)
Prec
ombu
stion
pre
ssur
e (ba
r)
Recording rate 60 points frequency about 20 cps about 0018 bar 15 Aland amplitude
Figure 9The recorded pressure time history for a burning of hybridmotor using 15 aluminum (Al) additive to PE fuel grain
6 Mathematical Problems in Engineering
The final nonresonance solution for the flow in the simulatedchamber in Figure 1 is
119900(119909 119905) =
infin
sum
119899=0
2120576120574120596int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
119900(119909 119905) =
infin
sum
119899=0
2120576120574120596int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
(cos120573119899119905 minus 1) minus
120573119899
120596
(cos120596119905 minus 1)
times sin120573119899119909
V119900(119909 119910 119905) = minus119910 120593
119903119908(119909) (1 minus cos120596119905)
(11)
Equations (11) show the time history of the perturbed valuesvariables (
119900 119900 V119900) at any axial locations
Following the same policies the resonance solutions arederived and formulated as
119900(119909 119905) =
120574119862119899lowast
120573119899lowast
sin120573119899lowast119905 minus 120573
119899lowast119905 cos120573
119899lowast119905 cos120573
119899lowast119909
119900(119909 119905) = 119862
119899lowast minus
2
120573119899lowast
(cos120573119899lowast119905 minus 1) minus 119905 sin120573
119899lowast119905
times cos120573119899lowast119909
(12)
where 119862119899lowast = 2 int
1
0
120593119903119908
(119909) cos120573119899lowast119909
23 The Computational Approach In the present studyhigher order accuracy difference equations are employedto solve the parabolized Navier-Stokes equations in theinterior of the prescribed chamber and at the boundariesThe numerical multisteps scheme in [24] is used to solve the2D unsteady and compressible Navier-Stokes equations forthe points located near the boundaries while the two-fourexplicit predictor-corrector scheme in [25] is employed tosolve the adapted equations at the interior points The lattermethod is a fourth-order accuracy variant of the fully explicitMacCormack scheme Uniform grids are used The accuracycriterion employed byMacCormack [24] is used to determinethe size of the time step
Here the NSCBC technique applied by [7] is presented indetail The full derivation for the parabolized Navier-Stokesat the upper and lower boundaries and at the head and exitends is formulated to specify the nonphysical conditions atthese boundaries Moreover this method is applied to reducethe generated reflected waves arising from the numericaltechniques
Based on the idea of characteristics for Euler equationsthe first derivatives normal to the boundaries appearing inthe convective terms of the Navier-Stokes equations may beexpressed in terms of the amplitude of incoming outcoming
and entropy waves weierp119894rsquos The fluid dynamics equations can
be written in nondimensional wave modes form as follows
120597120588
120597119905
+
1
1198622[weierp2+
1
2
(weierp4+ weierp1)] + 119872
120597120588V120597119910
= 0 (13)
120597119906
120597119905
+
1
2120588119862
[(weierp4minus weierp1)] + 119872V
120597119906
120597119910
=
120594119909
120588
(14)
120597V120597119905
+ weierp3+ 119872V
120597V120597119910
+
1198602
120574119872
1
120588
120597119901
120597119910
=
120594119910
120588
(15)
120597119901
120597119905
+
1
2
[(weierp4+ weierp1)] + 119872V
120597119901
120597119910
+ 120574119872119901
120597V120597119910
= 120594119864 (16)
where 120594119909and 120594
119910represent the viscous terms and 120594
119864refers
to the combination of the viscous and conduction terms inthe axial and transverse directions and 119862 = 119862
1015840
1198621015840
119900 The
amplitudes of the left-running waves (weierp1) the right-running
waves (weierp4) and the entropy waves (weierp
2 weierp3) in terms of the
characteristic velocities 120582119894rsquos are
weierp1= 1205821(
1
120574
120597119901
120597119909
minus 119872120588119862
120597119906
120597119909
) 1205821=
119906
119872
minus 119862 (17)
weierp2= 1205822(1198622120597120588
120597119909
minus
1
120574
120597119901
120597119909
) 1205822= 119872119906 (18)
weierp3= 1205823
120597V120597119909
1205823= 119872119906 (19)
weierp4= 1205824(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
) 1205824=
119906
119872
+ 119862 (20)
Two approaches are used to infer the values of theincoming amplitude-variations from the boundary condi-tions One is the local one-dimensional inviscid relations(LODI) technique and the second is the Navier-Stokescharacteristics boundary conditions (NSCBC) For the latterone the complete description of the numerical boundaryconditions at all boundaries is discussed below
24 Exit Boundary Conditions In this case a pressure nodeis assumed at the exit boundary The NSCBC relation (16)suggests that the amplitude of the reflected wave should be
weierp1= 2120594119864minus weierp4 (21)
and the remaining conditions are
weierp2= 119872119906(119862
2120597120588
120597119909
minus
1
120574
120597119901
120597119909
) weierp3= 119872119906
120597V120597119909
weierp4= (
119906
119872
+ 119888)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
(22)
Mathematical Problems in Engineering 7
The nondimensional equations can be written as follows
120597120588
120597119905
+ 119872119906
120597120588
120597119909
minus
119872119906
1205741198622
120597119901
120597119909
+ 119872V120597120588
120597119910
+
1198602
119872
(120574 minus 1) 119877119890119875119903
1
1198622
1205972
119879
1205971199102
= 0
120597119906
120597119905
+
1
120588119862
[(119906 +
119862
119872
)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)] + 119872V120597119906
120597119910
minus
1198602
(120574 minus 1) 119877119890119875119903
1
120588119862
1205972
119879
1205971199102
minus
1198721198602
119877119890
1
120588
1205972
119906
1205971199102
= 0
120597V120597119905
+ 119872(119906
120597V120597119909
+ V120597V120597119910
) = 0
119889119901
119889119905
= 0
(23)
Here again the nondimensional variables (120588 119906 and V)can be computed using one-sided finite-difference approxi-mations
25 Head End Boundary Condition The velocity componentnormal to the boundary (119906(0 119910 119905)) is zero since the head endis impermeable In contrast the transverse velocity compo-nent cannot be specified because the axial viscous terms inthe transversemomentum equation have been suppressed Asthe normal velocity drops to zero at 119909 = 0 the amplitudesweierp2and weierp
3 in (18) and (19) are zero and the characteristic
velocities are 1205821
= minus119862 and 1205824
= +119862 The reflected wave weierp4
enters the domain while the wave weierp1propagates out through
the boundary The NSCBC relation (19) indicates that theamplitude of the incoming wave weierp
4should be
weierp4= weierp1 (24)
and the remaining conditions are
weierp1= minus119888(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
weierp2= 0 weierp
3= 0
(25)
The equations can be written in nondimensional form asfollows
120597120588
120597119905
minus 119872120588
120597119906
120597119909
+
1
120574119862
120597119901
120597119909
+ 119872
120597120588V120597119910
= 0
120597119906
120597119905
= 0
120597V120597119905
+ 119872V120597V120597119910
= 0
120597119901
120597119905
minus 119888
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
+ 120574119872119901
120597V120597119910
minus
1205741198602
119872
(120574 minus 1) 119875119903119877119890
1205972
119879
1205971199102
= 0
(26)
More details about the NSCBC techniques are found in [2326]
26 CodeValidations Thecomputational codes are designedconstructed and validated for steady and unsteady flow as
shown in Figures 2 and 3 For the latter one the constructedcodes run for many wave cycles to detect the whole unsteadyrotational flow dynamic across the chamber
In Figure 2 comparison between the current numericaland asymptotic results dealing with the steady state flowsolutions is carried out The steady state flow is generatedby adding constant side wall mass injection (119881
119903119908= minus1)
from the injection surface of the prescribed chamber Similarexperiment in [18 27] was carried out and compared withthe current mathematical models in Figure 2The asymptoticanalysis solution represents the solid line while the dottedline represents the converged steady state numerical solutionfor viscous flow with wall Reynolds number 119877
119890119908= 5 lowast
103 The nonfilled points refer to the experimental measured
data The experimental results are recorded for flow in achannel with fully transpired wall Three different poroussurfaces using three different sizes of honeycomb cells areused [7] In this figure a comparison between the analyticaland the numerical solutions at the midlength of the chamberis illustratedThis comparison shows a reasonable agreementbetween the three approaches The comparison between thetheoretical and the computational approaches shows littlebit deviation This deviation is logically expected since theused honey combs have three different sizes The smallersize of honey combs the smaller deviations between the twoapproaches
The second set of the validations show the solution for theunsteady flow computations at nonresonance forcedmode Inthis set of the results the computational results are validatedby comparing with the asymptotic analysis These resultsrepresent the asymptotic solution when the sidewall injectionvelocity is represented by V
119903119908(119909 1 119905) = minus[1 + 120576120593
119903119908(1 minus
cos120596119905)] 120593119903119908
= cos(120573119899119909) In the current work and the work
by Hegab [7] the disturbance forcing frequency is taken tobe 120596 = 1 (nonresonance) with the amplitude of 120576 = 04
and the eigenvalues (120573119899
= 1198991205872 119899 = 1 3 5 7 9 etc)The temperature time history (119879) at themedlength location ispresented in Figure 3 for 119877
119890= 3 lowast 10
5119872 = 002 120575 = 20 and119899 = 10 For the nonresonance case the solution shows stableconditions but it is not purely harmonic This interestingbehavior shows the ability of the constructed code with thegood treatment to the boundary conditions to detect thenon-harmonic trend as detected by the analytical approachA comparison of the perturbed temperature (119879) (27) withthe numerical results in Figure 3 shows good agreementbetween the two approaches especially for 119905 lt 20 while thecomputational solution shows little less deviation than theanalytical results for 119905 gt 20 due to the nonlinearity effects
119879119900(119909 119905) =
infin
sum
119899=0
2120576120596 (120574 minus 1) int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
(27)
Generally the coexistence of the forced and the naturalmodes of the system is detected in the adapted computationaland analytical approaches as seen in the earlier investigationby Hegab [7] In contrast the computational techniques in
8 Mathematical Problems in Engineering
[10] show only pure harmonic oscillations The source ofthese deficiencies may be arising from the treatment of theboundary conditions especially at the head and aft ends ofthe simulated chamber
3 Results and Discussions
The impact of the injected wave number from the sidewallalong with the frequencies of the forced mode on thegenerated complex acoustical fields and their interactionwiththe fluid are examinedThe results have been categorized intothe following two sets
31 Nonresonance Solution In the current study a widerange of forcing frequencies away from or very close tothe first fundamental modes of the prescribed chamber isconsidered [7] a wide range of forcing frequencies away fromor very close to the first fundamental modes of the prescribedchamber is considered In Figure 4 the irrotational androtational fields across the whole chamber are presentedwhen 119905 = 42 for 120596 = 1 119877
119890= 3lowast10
5119872 = 002 120576 = 04 119899 = 13 and 5 and 120582
119899= 1205872 The latter one represents the Eigen
values of the chamberThese interesting graphs show that therotational fields are generated at the injection surface As timeincreases the generated unsteady vorticity is transferred to fillmost of the combustion chamber Moreover the constructedcodes are able to run for enough time and count about sevenvorticity wave cycles as 119905 = 42 It is also seen that theamplitude of the rotational field attainsmaximumvalues nearthe wall and found to be order of ten for the monotonicvariation (119899 = 1) and nonresonance frequency (120596 = 1)One notes that these magnitudes meet the scaling of theasymptotic solutions in [7 9 11 14] Moreover for 119899 = 1monotonic distribution of unsteady vorticity is noticed whilethe variation goes in nonmonotonic axial distribution for 119899 gt
1 For the nonresonance solution the longitudinal variationof the vorticity at the wall 120597|Ω
119908(119909 119905)|120597119909 gt 0 for 120582
119899=
1205872 (119899 = 1) is revealing that the magnitude of rotationalflow fields on the injection surface increases as axial distanceincreases
The vorticity contours adjacent to the injection surface forthe same parameters as in Figure 4 for several eigenvalues(a) 120582119899
= 1205872 (119899 = 1) (b) 31205872 (119899 = 3) and (c) 51205872
(119899 = 5) are present in Figure 5 It is clearly seen that adjacentto the combustion surface the generated unsteady vorticitymay impact the surface at downstream locations for 119899 = 1at two locations for 119899 = 3 and many locations for 119899 = 5The rotational flow fluctuates from negative to positive andvice versa at several successive times and may cause scouringon the combustion surface and in turn may increase theshearing stresses adjacent to the combustion surface Thisimportant conclusion may lead to increase of the burningrates over the designed ones especially for low wave numbersince the amplitude of the unsteady vorticity decreases as thewave number increases
Fortunately recent experimental work by Meteb [28] in2009 was directed to study the acoustic instability in hybridrocket system using different fuel grains and ignition systems
The experimental results verified qualitatively the currentanalytical and computational results for the generated com-plex wave patterns and the effect of the combustion surfaceon the acoustic fluid dynamics interactionmechanism In thisexperiment the pure polyethylene (PE) fuel grain and PEwithadditives of aluminum (Al) powder with different percentageare used to predict the hybrid rocket motor regression ratesThe flame geometry and structure for the burning of thepure PE and PE with additives of aluminum (Al) powderat 25 75 and 15 of the total weight of the wholesamples are illustrated in Figure 6The effect of theAl powderon the flame length geometry and the rate of reaction forthe burning of each sample is clearly seen In addition thepressure time histories at a point before the combustionchamber in the way of oxidizer are recorded and illustratedin Figure 7 for the pure PE and in Figures 8 and 9 for PEwith additives of aluminum (Al) powder at 75 and 15of the total weight respectively It is clearly seen that theamplitude of the pressure waves decreases as the percentageof Al fuel grain increases in the sample This interestingbehavior is similar to what we have seen when we increasethe wave number of the combustion surface As we knowmost of the Al fuel grains burn away from the combustionsurface consisting ofmany fragmentsThe escaping processesof Al fuel grains from the combustion surface generate manyof cavities that make the combustion surface more roughthan the smooth one in case of pure PE The experimentaland computational results are qualitatively in agreement Forthe first time the analytical and computational results forthe current cold model and the acoustic instabilities foundby Meteb [28] state an important and essential conclusionabout the effect of the combustion surface morphology onthe acoustic instability inside the rocket chamber and theaccompanying rotational flow Instantaneous total velocityvector map for the parameters in Figure 4 at (a) 119905 = 0 ldquosteadystate solutionrdquo and (b) 119905 = 16 is presented in Figures 10(a)and 10(b) The injection surface is located at 119910 = 1 on theupper part of the plot where the flow is completely verticalwhile the lower part of the plot represents the center lineat 119910 = 0 where the flow is completely axial The closedend is located at 119909 = 0 on the left hand side of the plotwhile at 119909 = 1 it represents the exit plane on the right handside One may notice the smooth and undisturbed fluid flowthrough the entire map for the steady state solution at 119905 = 0After initiating the side wall disturbances the coexisting ofacoustics and the generated rotational flow is clearly seenin the upper half part of the chamber and illustrates howthe interaction between the travelling acoustic waves and theinjected fluid affects the fluid flow in the chamber In additionat 119905 = 32 in Figure 11 there is an instantaneous reverseflow found at a location near the combustion surface Thecomplex acoustic fluid dynamics interaction structure mayincrease the shear stresses adjacent to the combustion surfaceand through the whole cavity These results may providethe rocket designers with conceptual tools and data to beconsidered in case of redesigning strategies
Mathematical Problems in Engineering 9
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
ldquoSteady state solutionrdquo t = 00
(a)
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 16Nonresonance (
(b)
Figure 10 Instantaneous total velocity vector map for the parameters in Figure 4 (a) 119905 = 0 ldquosteady state solutionrdquo and (b) 119905 = 16
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 32
Reverse flow region
Nonresonance (
Figure 11 Instantaneous total velocity vector map at 119905 = 32 for theparameters in Figure 4
32 Near and at Resonance Solution A comparison of theasymptotic analysis perturbed pressure time history withthe computational one at and near resonance frequenciesis presented in Figure 12 For 119905 lt 10 the results showgood agreement between the two approaches even with theasymptotic beat condition (120596 = 1205872 minus 00018) which isvery close to the resonance frequency While for 119905 gt 10
the asymptotic solution shows linear growth with time basedon the linear theory for the derived wave equation where120596 = 120573
119899lowast = 1205872 as derived by Hegab [7] In contrast
the computational solution shows beats at the resonancecondition due to the nonlinearity and the asymptotic beatsolutions have smaller amplitude than the resonance oneThe beats appear due to the interaction between the derivingfrequency mode 120596 = 1205872minus00018 and the first eigenfunctionmode (1205872) It is clearly seen that the amplitude of thevorticity at the resonance frequency is found to be ten timesthan the amplitudes at the nonresonance one in Figure 4
P
03
02
01
0
minus01
minus02
minus030 5 10 15 20 25 30 35 40
t
ComputationalAnalyticalAnalytical (beats)
x = 05 y = 00 120596 = 1205872 n = 1
Figure 12 Comparison between the computational and the asymp-totic solutions for the perturbed pressure time history with 120596 = 1205872
(resonance) 120576 = 04 119872 = 002 and 119877119890= 3 lowast 10
5 at the half-lengthof the chamber
Moreover it is found that the fluctuation of the transversetemperature gradient at the injection surface is found tobe about ten times than the fluctuation of the temperaturedisturbances This concludes that since the heat transferalong the sidewall is directly proportional to the transversetemperature gradient there is an unexpectedly large heat fluxwhich exists both at the wall and within the entire chambergiven the small size of the temperature distribution Theasymptotic analysis by Staab et al [11] is used to prove thisimportant conclusion
Finally the asymptotic and computational methods areemployed to study the dynamic process in an acoustically
10 Mathematical Problems in Engineering
simulated solid propellant rocket chamber with transientsidewall mass addition at different wave numbers
4 Conclusion Remarks
The current study emphasizes the effect of the combustionsurfacemorphology of the solid rocket systemon the acousticinstability and the accompanying vorticity and temperaturedynamics generation The rotational and irrotational fieldsgenerated in the prescribed solid rocket motor chamber withMach number are chosen to be O(10minus1) while the charac-teristic Reynolds number is O(105-106) at nonresonance andnear and at resonance frequencies have been described TheNSCBC techniques that applied in [7 29] are used to treat thenumerical boundary conditions
The numerical solution shows that the convergence tosteady state solution is achieved first and then additionaltransient side wall mass addition is initiated Moreover theresult shows that the forced mode of the nonresonance andresonance frequencies generates longitudinal acoustics waveswhich interact again with its sources at the injection surfaceto generate the rotational flow fields The generated unsteadyvorticity is transferred from the injection surface toward thecenterline to fill the entire chamber as time increases aftermore than seven wave cycles The solution shows nonpureharmonic oscillations as suggested from the theory in [11]Vorticity 3D graphs and contours illustrate that spatially themaximum amplitudes are found to be near the injectionsurface Moreover the largest shear stresses are found to be atthe injection surface near the exit end while the shear stressesimpact the injection surface at many locations as the wavenumber increasesThe longitudinal variation of the rotationalflow fields is found to be nonmonotonic for 119899 gt 1
Here for the first time the analytical and computationalresults for the current coldmodel and the recent experimentalwork by Meteb [28] revealed that the combustion surfacemorphology has a great effect on the acoustic instabilityinside the rocket chamber and the accompanying rotationalflow The computational approach reveals that there areunexpectedly high heat transfer and generated rotational flowfields found to be adjacent to the injection surface in bothnonresonance and resonance cases The amplitude of thelatter one is found to be approximately ten times than thatwith the nonresonance solution One may anticipate thatthese large oscillatory gradients will persist to the surface ofa burning solid propellant in a real rocket chamber
More traditional computational studies of turbulence inchannel flows with wall injection are reviewed by Chaouat[30] Mean axial velocity profiles are found to be laminaruntil about halfway down the channel and then show signs oftransition Chaouatrsquos results do not show the acoustic-relatedeffects discussed in the present work As a result it is difficultto know from the turbulent theory results if the source ofthe downstream vorticity is classical transition to turbulenceor partly the result of the kind of mechanism discussed inthis paper In our recent studies in 2011 and 2012 [31 32]the assessment of turbulence modeling for gas flow in thechamber and in the convergent-divergent rocket nozzle for
the steady state flow is studied The detailed information andrealities in the current research about the transient coexistingof the irrotational (acoustics) rotational (generated vorticityfrom the interaction between the injected flow and thetraveling acoustic waves) and our turbulent techniques thatwe have adapted in our studies in 2011 and 2012 for the steadyflow field in SRM chambernozzle geometry will be used toaccount all the complex flow fields in our future study
Finally results of the asymptotic and computationalapproaches may provide the rocket designer with codes andinformation making them able to interpret some reasonsbehind the combustion instability inside the combustionchamber of real SRM based on the composition of the solidpropellants and in turn the morphology of the combustionsurface
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The first author would like to thank Professor D R KassoyUniversity of Colorado at Boulder USA for their continuoushelp
References
[1] A Hegab T L Jackson J Buckmaster and D S Stewart ldquoTheburning of periodic sandwich propellantsrdquo in Proceedings of the36th AIAAASMESAEASEE Joint Propulsion Conference andExhibit Huntsville Ala USA July 2000
[2] A Hegab T L Jackson J Buckmaster and D S StewartldquoNonsteady burning of periodic sandwich propellants withcomplete coupling between the solid and gas phasesrdquo Journalof Combustion and Flame vol 125 no 1-2 pp 1055ndash1070 2001
[3] A M Hegab ldquoModeling of sandwich heterogeneous propellantwith two steps chemical kinetics and fluid mechanical effectsrdquoEngineering Research Journal vol 32 no 2 pp 119ndash130 2009
[4] J Buckmaster T L Jackson A Hegab S Kochevets andM Ulrich ldquoRandomly packed heterogeneous propellants andthe flames they supportrdquo in Proceedings of the 39th AerospaceSciences Meeting and Exhibit Reno NV USA January 2001
[5] S Kochevets J Buckmaster T L Jackson and A HegabldquoRandom packs and their use in modeling heterogeneous solidpropellant combustionrdquo Journal of Propulsion and Power vol 17no 4 pp 883ndash891 2001
[6] AMHegab S AHasanienH EMostafa andAMMaradenldquoExperimental and numerical investigations for the combustionof selected composite solid propellantrdquo Engineering ResearchJournal vol 35 no 3 2012
[7] A M Hegab ldquoVorticity generation and acoustic resonance ofsimulated solid rocket motor chamber with high wave numberwall injectionrdquo Computers and Fluids vol 38 no 6 pp 1258ndash1269 2009
[8] A M Hegab and D R Kassoy ldquoInternal flow temperatureand vorticity dynamics due to transient mass additionrdquo AIAAJournal vol 44 no 4 pp 812ndash826 2006
Mathematical Problems in Engineering 11
[9] K Kirkkopru D R Kassoy and Q Zhao ldquoUnsteady vorticitygeneration and evolution in a model of a solid rocket motorrdquoJournal of Propulsion and Power vol 12 no 4 pp 646ndash6541996
[10] K Kirkkopru D R Kassoy Q Zhao and P L Staab ldquoAcousti-cally generated unsteady vorticity field in a long narrow cylinderwith sidewall injectionrdquo Journal of Engineering Mathematicsvol 42 no 1 pp 65ndash90 2002
[11] P L StaabQ ZhaoD R Kassoy andKKirkkopru ldquoCoexistingacoustic-rotational flow in a cylinder with axisymmetric side-wall mass additionrdquo Physics of Fluids vol 11 no 10 pp 2935ndash2951 1999
[12] M Rempe P L Staab and D R Kassoy ldquoThermal response foran internal flow in a cylinderwith time dependent sidewallmassadditionrdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit AIAA 2000-0996 Reno Nev USA January 2000
[13] Q Zhao and D R Kassoy ldquoThe generation and evolutionof unsteady vorticity in a solid rocket engine chamberrdquo inProceedings of the 32th Aerospace Sciences Meeting AIAA paper94-0779 Reno Nev USA 1994
[14] Q Zhao P L Staab D R Kassoy and K Kirkkopru ldquoAcous-tically generated vorticity in an internal flowrdquo Journal of FluidMechanics vol 413 pp 247ndash285 2000
[15] G A Flandro ldquoEffects of vorticity on rocket combustionstabilityrdquo Journal of Propulsion and Power vol 11 no 4 pp 607ndash625 1995
[16] J Majdalani andW K VanMoorhem ldquoMultiple-scales solutionto the acoustic boundary layer in solid rocket motorsrdquo Journalof Propulsion and Power vol 13 no 2 pp 186ndash193 1997
[17] J Majdalani andW van Moorhem ldquoImproved time-dependentflowfield solution for solid rocket motorsrdquo AIAA Journal vol36 no 2 pp 241ndash248 1998
[18] Z Deng R J Adrian and C D Tomkinsm ldquoStructure ofturbulence in channel flow with a fully transpired wallrdquo inProceedings of the 39th Aerospace Science Meeting AIAA Paper2001-1019 Reno Nev USA 2001
[19] F Vuillot andG Avalon ldquoAcoustic boundary layers in solid pro-pellant rocketmotors usingNavier-Stokes equationsrdquo Journal ofPropulsion and Power vol 7 no 2 pp 231ndash239 1991
[20] T M Smith R L Roach and G A Flandro ldquoNumerical studyof the unsteady flow in a simulated solid rocket motorrdquo inProceedings of the 31th Aerospace Science Meeting AIAA Paper93-0112 Reno Nev USA 1993
[21] C F Tseng and I S Tseng ldquoInteraction between acoustic wavesand premixed flames in porous chamberrdquo in Proceedings of the32th Aerospace Science Meeting Reno Nev USA 1994 AIAAPaper 94 3328
[22] T S Roh and V Yang ldquoTransient combustion responses ofsolid propellants to acoustic disturbances in rocket motorsrdquo inProceedings of the 33rd Aerospace Sciences Meeting and ExhibitAIAA Paper 95-0602 Reno Nev USA 1995
[23] A M Hegab A study of acoustic phenomena in solid rocketengines [PhD thesis] Mechanical Power Engineering Depart-ment Menoufia University Menufia Egypt University of Col-orado Boulder Colo USA 1998
[24] R W MacCormack ldquoA numerical method for solving theequations of compressible viscous flowrdquo American Institute ofAeronautics and Astronautics Journal vol 20 no 9 pp 1275ndash1281 1982
[25] D Gottlieb and E Turkel ldquoDissipative two-four methods fortime-dependent problemsrdquo Mathematics of Computation vol30 no 136 pp 703ndash723 1976
[26] T J Poinsot and S K Lele ldquoBoundary conditions for directsimulations of compressible viscous flowsrdquo Journal of Compu-tational Physics vol 101 no 1 pp 104ndash129 1992
[27] Z Deng R J Adrian and C D Tomkins ldquoSensitivity ofturbulence in transpired channel to injection velocity small-scale nonuniformityrdquo AIAA Journal vol 40 no 11 pp 2241ndash2246 2002
[28] K J Meteb Prediction of hybrid rocket motor regression rate[MS thesis] Military Technical College Cairo Egypt 2009
[29] J W Anders V Magi and J Abraham ldquoLarge-eddy simulationin the near-field of a transient multi-component gas jet withdensity gradientsrdquoComputers amp Fluids vol 36 no 10 pp 1609ndash1620 2007
[30] B Chaouat ldquoNumerical simulations of channel flows with fluidinjection using Reynolds stressmodelrdquo AIAAPaper 2000-09922000
[31] W A El-Askary A Balabel S M El-Behery and A HegabldquoComputations of a compressible turbulent flow in a rocketmotor-chamber configuration with symmetric and asymmetricinjectionrdquoComputerModeling in EngineeringampSciences vol 82no 1 pp 29ndash54 2011
[32] A Balabel A M Hegab M Nasr and S M El-Behery ldquoAssess-ment of turbulence modeling for gas flow in two-dimensionalconvergent-divergent rocket nozzlerdquo Applied MathematicalModelling vol 35 no 7 pp 3408ndash3422 2011
6 Mathematical Problems in Engineering
The final nonresonance solution for the flow in the simulatedchamber in Figure 1 is
119900(119909 119905) =
infin
sum
119899=0
2120576120574120596int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
119900(119909 119905) =
infin
sum
119899=0
2120576120574120596int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
(cos120573119899119905 minus 1) minus
120573119899
120596
(cos120596119905 minus 1)
times sin120573119899119909
V119900(119909 119910 119905) = minus119910 120593
119903119908(119909) (1 minus cos120596119905)
(11)
Equations (11) show the time history of the perturbed valuesvariables (
119900 119900 V119900) at any axial locations
Following the same policies the resonance solutions arederived and formulated as
119900(119909 119905) =
120574119862119899lowast
120573119899lowast
sin120573119899lowast119905 minus 120573
119899lowast119905 cos120573
119899lowast119905 cos120573
119899lowast119909
119900(119909 119905) = 119862
119899lowast minus
2
120573119899lowast
(cos120573119899lowast119905 minus 1) minus 119905 sin120573
119899lowast119905
times cos120573119899lowast119909
(12)
where 119862119899lowast = 2 int
1
0
120593119903119908
(119909) cos120573119899lowast119909
23 The Computational Approach In the present studyhigher order accuracy difference equations are employedto solve the parabolized Navier-Stokes equations in theinterior of the prescribed chamber and at the boundariesThe numerical multisteps scheme in [24] is used to solve the2D unsteady and compressible Navier-Stokes equations forthe points located near the boundaries while the two-fourexplicit predictor-corrector scheme in [25] is employed tosolve the adapted equations at the interior points The lattermethod is a fourth-order accuracy variant of the fully explicitMacCormack scheme Uniform grids are used The accuracycriterion employed byMacCormack [24] is used to determinethe size of the time step
Here the NSCBC technique applied by [7] is presented indetail The full derivation for the parabolized Navier-Stokesat the upper and lower boundaries and at the head and exitends is formulated to specify the nonphysical conditions atthese boundaries Moreover this method is applied to reducethe generated reflected waves arising from the numericaltechniques
Based on the idea of characteristics for Euler equationsthe first derivatives normal to the boundaries appearing inthe convective terms of the Navier-Stokes equations may beexpressed in terms of the amplitude of incoming outcoming
and entropy waves weierp119894rsquos The fluid dynamics equations can
be written in nondimensional wave modes form as follows
120597120588
120597119905
+
1
1198622[weierp2+
1
2
(weierp4+ weierp1)] + 119872
120597120588V120597119910
= 0 (13)
120597119906
120597119905
+
1
2120588119862
[(weierp4minus weierp1)] + 119872V
120597119906
120597119910
=
120594119909
120588
(14)
120597V120597119905
+ weierp3+ 119872V
120597V120597119910
+
1198602
120574119872
1
120588
120597119901
120597119910
=
120594119910
120588
(15)
120597119901
120597119905
+
1
2
[(weierp4+ weierp1)] + 119872V
120597119901
120597119910
+ 120574119872119901
120597V120597119910
= 120594119864 (16)
where 120594119909and 120594
119910represent the viscous terms and 120594
119864refers
to the combination of the viscous and conduction terms inthe axial and transverse directions and 119862 = 119862
1015840
1198621015840
119900 The
amplitudes of the left-running waves (weierp1) the right-running
waves (weierp4) and the entropy waves (weierp
2 weierp3) in terms of the
characteristic velocities 120582119894rsquos are
weierp1= 1205821(
1
120574
120597119901
120597119909
minus 119872120588119862
120597119906
120597119909
) 1205821=
119906
119872
minus 119862 (17)
weierp2= 1205822(1198622120597120588
120597119909
minus
1
120574
120597119901
120597119909
) 1205822= 119872119906 (18)
weierp3= 1205823
120597V120597119909
1205823= 119872119906 (19)
weierp4= 1205824(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
) 1205824=
119906
119872
+ 119862 (20)
Two approaches are used to infer the values of theincoming amplitude-variations from the boundary condi-tions One is the local one-dimensional inviscid relations(LODI) technique and the second is the Navier-Stokescharacteristics boundary conditions (NSCBC) For the latterone the complete description of the numerical boundaryconditions at all boundaries is discussed below
24 Exit Boundary Conditions In this case a pressure nodeis assumed at the exit boundary The NSCBC relation (16)suggests that the amplitude of the reflected wave should be
weierp1= 2120594119864minus weierp4 (21)
and the remaining conditions are
weierp2= 119872119906(119862
2120597120588
120597119909
minus
1
120574
120597119901
120597119909
) weierp3= 119872119906
120597V120597119909
weierp4= (
119906
119872
+ 119888)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
(22)
Mathematical Problems in Engineering 7
The nondimensional equations can be written as follows
120597120588
120597119905
+ 119872119906
120597120588
120597119909
minus
119872119906
1205741198622
120597119901
120597119909
+ 119872V120597120588
120597119910
+
1198602
119872
(120574 minus 1) 119877119890119875119903
1
1198622
1205972
119879
1205971199102
= 0
120597119906
120597119905
+
1
120588119862
[(119906 +
119862
119872
)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)] + 119872V120597119906
120597119910
minus
1198602
(120574 minus 1) 119877119890119875119903
1
120588119862
1205972
119879
1205971199102
minus
1198721198602
119877119890
1
120588
1205972
119906
1205971199102
= 0
120597V120597119905
+ 119872(119906
120597V120597119909
+ V120597V120597119910
) = 0
119889119901
119889119905
= 0
(23)
Here again the nondimensional variables (120588 119906 and V)can be computed using one-sided finite-difference approxi-mations
25 Head End Boundary Condition The velocity componentnormal to the boundary (119906(0 119910 119905)) is zero since the head endis impermeable In contrast the transverse velocity compo-nent cannot be specified because the axial viscous terms inthe transversemomentum equation have been suppressed Asthe normal velocity drops to zero at 119909 = 0 the amplitudesweierp2and weierp
3 in (18) and (19) are zero and the characteristic
velocities are 1205821
= minus119862 and 1205824
= +119862 The reflected wave weierp4
enters the domain while the wave weierp1propagates out through
the boundary The NSCBC relation (19) indicates that theamplitude of the incoming wave weierp
4should be
weierp4= weierp1 (24)
and the remaining conditions are
weierp1= minus119888(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
weierp2= 0 weierp
3= 0
(25)
The equations can be written in nondimensional form asfollows
120597120588
120597119905
minus 119872120588
120597119906
120597119909
+
1
120574119862
120597119901
120597119909
+ 119872
120597120588V120597119910
= 0
120597119906
120597119905
= 0
120597V120597119905
+ 119872V120597V120597119910
= 0
120597119901
120597119905
minus 119888
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
+ 120574119872119901
120597V120597119910
minus
1205741198602
119872
(120574 minus 1) 119875119903119877119890
1205972
119879
1205971199102
= 0
(26)
More details about the NSCBC techniques are found in [2326]
26 CodeValidations Thecomputational codes are designedconstructed and validated for steady and unsteady flow as
shown in Figures 2 and 3 For the latter one the constructedcodes run for many wave cycles to detect the whole unsteadyrotational flow dynamic across the chamber
In Figure 2 comparison between the current numericaland asymptotic results dealing with the steady state flowsolutions is carried out The steady state flow is generatedby adding constant side wall mass injection (119881
119903119908= minus1)
from the injection surface of the prescribed chamber Similarexperiment in [18 27] was carried out and compared withthe current mathematical models in Figure 2The asymptoticanalysis solution represents the solid line while the dottedline represents the converged steady state numerical solutionfor viscous flow with wall Reynolds number 119877
119890119908= 5 lowast
103 The nonfilled points refer to the experimental measured
data The experimental results are recorded for flow in achannel with fully transpired wall Three different poroussurfaces using three different sizes of honeycomb cells areused [7] In this figure a comparison between the analyticaland the numerical solutions at the midlength of the chamberis illustratedThis comparison shows a reasonable agreementbetween the three approaches The comparison between thetheoretical and the computational approaches shows littlebit deviation This deviation is logically expected since theused honey combs have three different sizes The smallersize of honey combs the smaller deviations between the twoapproaches
The second set of the validations show the solution for theunsteady flow computations at nonresonance forcedmode Inthis set of the results the computational results are validatedby comparing with the asymptotic analysis These resultsrepresent the asymptotic solution when the sidewall injectionvelocity is represented by V
119903119908(119909 1 119905) = minus[1 + 120576120593
119903119908(1 minus
cos120596119905)] 120593119903119908
= cos(120573119899119909) In the current work and the work
by Hegab [7] the disturbance forcing frequency is taken tobe 120596 = 1 (nonresonance) with the amplitude of 120576 = 04
and the eigenvalues (120573119899
= 1198991205872 119899 = 1 3 5 7 9 etc)The temperature time history (119879) at themedlength location ispresented in Figure 3 for 119877
119890= 3 lowast 10
5119872 = 002 120575 = 20 and119899 = 10 For the nonresonance case the solution shows stableconditions but it is not purely harmonic This interestingbehavior shows the ability of the constructed code with thegood treatment to the boundary conditions to detect thenon-harmonic trend as detected by the analytical approachA comparison of the perturbed temperature (119879) (27) withthe numerical results in Figure 3 shows good agreementbetween the two approaches especially for 119905 lt 20 while thecomputational solution shows little less deviation than theanalytical results for 119905 gt 20 due to the nonlinearity effects
119879119900(119909 119905) =
infin
sum
119899=0
2120576120596 (120574 minus 1) int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
(27)
Generally the coexistence of the forced and the naturalmodes of the system is detected in the adapted computationaland analytical approaches as seen in the earlier investigationby Hegab [7] In contrast the computational techniques in
8 Mathematical Problems in Engineering
[10] show only pure harmonic oscillations The source ofthese deficiencies may be arising from the treatment of theboundary conditions especially at the head and aft ends ofthe simulated chamber
3 Results and Discussions
The impact of the injected wave number from the sidewallalong with the frequencies of the forced mode on thegenerated complex acoustical fields and their interactionwiththe fluid are examinedThe results have been categorized intothe following two sets
31 Nonresonance Solution In the current study a widerange of forcing frequencies away from or very close tothe first fundamental modes of the prescribed chamber isconsidered [7] a wide range of forcing frequencies away fromor very close to the first fundamental modes of the prescribedchamber is considered In Figure 4 the irrotational androtational fields across the whole chamber are presentedwhen 119905 = 42 for 120596 = 1 119877
119890= 3lowast10
5119872 = 002 120576 = 04 119899 = 13 and 5 and 120582
119899= 1205872 The latter one represents the Eigen
values of the chamberThese interesting graphs show that therotational fields are generated at the injection surface As timeincreases the generated unsteady vorticity is transferred to fillmost of the combustion chamber Moreover the constructedcodes are able to run for enough time and count about sevenvorticity wave cycles as 119905 = 42 It is also seen that theamplitude of the rotational field attainsmaximumvalues nearthe wall and found to be order of ten for the monotonicvariation (119899 = 1) and nonresonance frequency (120596 = 1)One notes that these magnitudes meet the scaling of theasymptotic solutions in [7 9 11 14] Moreover for 119899 = 1monotonic distribution of unsteady vorticity is noticed whilethe variation goes in nonmonotonic axial distribution for 119899 gt
1 For the nonresonance solution the longitudinal variationof the vorticity at the wall 120597|Ω
119908(119909 119905)|120597119909 gt 0 for 120582
119899=
1205872 (119899 = 1) is revealing that the magnitude of rotationalflow fields on the injection surface increases as axial distanceincreases
The vorticity contours adjacent to the injection surface forthe same parameters as in Figure 4 for several eigenvalues(a) 120582119899
= 1205872 (119899 = 1) (b) 31205872 (119899 = 3) and (c) 51205872
(119899 = 5) are present in Figure 5 It is clearly seen that adjacentto the combustion surface the generated unsteady vorticitymay impact the surface at downstream locations for 119899 = 1at two locations for 119899 = 3 and many locations for 119899 = 5The rotational flow fluctuates from negative to positive andvice versa at several successive times and may cause scouringon the combustion surface and in turn may increase theshearing stresses adjacent to the combustion surface Thisimportant conclusion may lead to increase of the burningrates over the designed ones especially for low wave numbersince the amplitude of the unsteady vorticity decreases as thewave number increases
Fortunately recent experimental work by Meteb [28] in2009 was directed to study the acoustic instability in hybridrocket system using different fuel grains and ignition systems
The experimental results verified qualitatively the currentanalytical and computational results for the generated com-plex wave patterns and the effect of the combustion surfaceon the acoustic fluid dynamics interactionmechanism In thisexperiment the pure polyethylene (PE) fuel grain and PEwithadditives of aluminum (Al) powder with different percentageare used to predict the hybrid rocket motor regression ratesThe flame geometry and structure for the burning of thepure PE and PE with additives of aluminum (Al) powderat 25 75 and 15 of the total weight of the wholesamples are illustrated in Figure 6The effect of theAl powderon the flame length geometry and the rate of reaction forthe burning of each sample is clearly seen In addition thepressure time histories at a point before the combustionchamber in the way of oxidizer are recorded and illustratedin Figure 7 for the pure PE and in Figures 8 and 9 for PEwith additives of aluminum (Al) powder at 75 and 15of the total weight respectively It is clearly seen that theamplitude of the pressure waves decreases as the percentageof Al fuel grain increases in the sample This interestingbehavior is similar to what we have seen when we increasethe wave number of the combustion surface As we knowmost of the Al fuel grains burn away from the combustionsurface consisting ofmany fragmentsThe escaping processesof Al fuel grains from the combustion surface generate manyof cavities that make the combustion surface more roughthan the smooth one in case of pure PE The experimentaland computational results are qualitatively in agreement Forthe first time the analytical and computational results forthe current cold model and the acoustic instabilities foundby Meteb [28] state an important and essential conclusionabout the effect of the combustion surface morphology onthe acoustic instability inside the rocket chamber and theaccompanying rotational flow Instantaneous total velocityvector map for the parameters in Figure 4 at (a) 119905 = 0 ldquosteadystate solutionrdquo and (b) 119905 = 16 is presented in Figures 10(a)and 10(b) The injection surface is located at 119910 = 1 on theupper part of the plot where the flow is completely verticalwhile the lower part of the plot represents the center lineat 119910 = 0 where the flow is completely axial The closedend is located at 119909 = 0 on the left hand side of the plotwhile at 119909 = 1 it represents the exit plane on the right handside One may notice the smooth and undisturbed fluid flowthrough the entire map for the steady state solution at 119905 = 0After initiating the side wall disturbances the coexisting ofacoustics and the generated rotational flow is clearly seenin the upper half part of the chamber and illustrates howthe interaction between the travelling acoustic waves and theinjected fluid affects the fluid flow in the chamber In additionat 119905 = 32 in Figure 11 there is an instantaneous reverseflow found at a location near the combustion surface Thecomplex acoustic fluid dynamics interaction structure mayincrease the shear stresses adjacent to the combustion surfaceand through the whole cavity These results may providethe rocket designers with conceptual tools and data to beconsidered in case of redesigning strategies
Mathematical Problems in Engineering 9
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
ldquoSteady state solutionrdquo t = 00
(a)
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 16Nonresonance (
(b)
Figure 10 Instantaneous total velocity vector map for the parameters in Figure 4 (a) 119905 = 0 ldquosteady state solutionrdquo and (b) 119905 = 16
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 32
Reverse flow region
Nonresonance (
Figure 11 Instantaneous total velocity vector map at 119905 = 32 for theparameters in Figure 4
32 Near and at Resonance Solution A comparison of theasymptotic analysis perturbed pressure time history withthe computational one at and near resonance frequenciesis presented in Figure 12 For 119905 lt 10 the results showgood agreement between the two approaches even with theasymptotic beat condition (120596 = 1205872 minus 00018) which isvery close to the resonance frequency While for 119905 gt 10
the asymptotic solution shows linear growth with time basedon the linear theory for the derived wave equation where120596 = 120573
119899lowast = 1205872 as derived by Hegab [7] In contrast
the computational solution shows beats at the resonancecondition due to the nonlinearity and the asymptotic beatsolutions have smaller amplitude than the resonance oneThe beats appear due to the interaction between the derivingfrequency mode 120596 = 1205872minus00018 and the first eigenfunctionmode (1205872) It is clearly seen that the amplitude of thevorticity at the resonance frequency is found to be ten timesthan the amplitudes at the nonresonance one in Figure 4
P
03
02
01
0
minus01
minus02
minus030 5 10 15 20 25 30 35 40
t
ComputationalAnalyticalAnalytical (beats)
x = 05 y = 00 120596 = 1205872 n = 1
Figure 12 Comparison between the computational and the asymp-totic solutions for the perturbed pressure time history with 120596 = 1205872
(resonance) 120576 = 04 119872 = 002 and 119877119890= 3 lowast 10
5 at the half-lengthof the chamber
Moreover it is found that the fluctuation of the transversetemperature gradient at the injection surface is found tobe about ten times than the fluctuation of the temperaturedisturbances This concludes that since the heat transferalong the sidewall is directly proportional to the transversetemperature gradient there is an unexpectedly large heat fluxwhich exists both at the wall and within the entire chambergiven the small size of the temperature distribution Theasymptotic analysis by Staab et al [11] is used to prove thisimportant conclusion
Finally the asymptotic and computational methods areemployed to study the dynamic process in an acoustically
10 Mathematical Problems in Engineering
simulated solid propellant rocket chamber with transientsidewall mass addition at different wave numbers
4 Conclusion Remarks
The current study emphasizes the effect of the combustionsurfacemorphology of the solid rocket systemon the acousticinstability and the accompanying vorticity and temperaturedynamics generation The rotational and irrotational fieldsgenerated in the prescribed solid rocket motor chamber withMach number are chosen to be O(10minus1) while the charac-teristic Reynolds number is O(105-106) at nonresonance andnear and at resonance frequencies have been described TheNSCBC techniques that applied in [7 29] are used to treat thenumerical boundary conditions
The numerical solution shows that the convergence tosteady state solution is achieved first and then additionaltransient side wall mass addition is initiated Moreover theresult shows that the forced mode of the nonresonance andresonance frequencies generates longitudinal acoustics waveswhich interact again with its sources at the injection surfaceto generate the rotational flow fields The generated unsteadyvorticity is transferred from the injection surface toward thecenterline to fill the entire chamber as time increases aftermore than seven wave cycles The solution shows nonpureharmonic oscillations as suggested from the theory in [11]Vorticity 3D graphs and contours illustrate that spatially themaximum amplitudes are found to be near the injectionsurface Moreover the largest shear stresses are found to be atthe injection surface near the exit end while the shear stressesimpact the injection surface at many locations as the wavenumber increasesThe longitudinal variation of the rotationalflow fields is found to be nonmonotonic for 119899 gt 1
Here for the first time the analytical and computationalresults for the current coldmodel and the recent experimentalwork by Meteb [28] revealed that the combustion surfacemorphology has a great effect on the acoustic instabilityinside the rocket chamber and the accompanying rotationalflow The computational approach reveals that there areunexpectedly high heat transfer and generated rotational flowfields found to be adjacent to the injection surface in bothnonresonance and resonance cases The amplitude of thelatter one is found to be approximately ten times than thatwith the nonresonance solution One may anticipate thatthese large oscillatory gradients will persist to the surface ofa burning solid propellant in a real rocket chamber
More traditional computational studies of turbulence inchannel flows with wall injection are reviewed by Chaouat[30] Mean axial velocity profiles are found to be laminaruntil about halfway down the channel and then show signs oftransition Chaouatrsquos results do not show the acoustic-relatedeffects discussed in the present work As a result it is difficultto know from the turbulent theory results if the source ofthe downstream vorticity is classical transition to turbulenceor partly the result of the kind of mechanism discussed inthis paper In our recent studies in 2011 and 2012 [31 32]the assessment of turbulence modeling for gas flow in thechamber and in the convergent-divergent rocket nozzle for
the steady state flow is studied The detailed information andrealities in the current research about the transient coexistingof the irrotational (acoustics) rotational (generated vorticityfrom the interaction between the injected flow and thetraveling acoustic waves) and our turbulent techniques thatwe have adapted in our studies in 2011 and 2012 for the steadyflow field in SRM chambernozzle geometry will be used toaccount all the complex flow fields in our future study
Finally results of the asymptotic and computationalapproaches may provide the rocket designer with codes andinformation making them able to interpret some reasonsbehind the combustion instability inside the combustionchamber of real SRM based on the composition of the solidpropellants and in turn the morphology of the combustionsurface
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The first author would like to thank Professor D R KassoyUniversity of Colorado at Boulder USA for their continuoushelp
References
[1] A Hegab T L Jackson J Buckmaster and D S Stewart ldquoTheburning of periodic sandwich propellantsrdquo in Proceedings of the36th AIAAASMESAEASEE Joint Propulsion Conference andExhibit Huntsville Ala USA July 2000
[2] A Hegab T L Jackson J Buckmaster and D S StewartldquoNonsteady burning of periodic sandwich propellants withcomplete coupling between the solid and gas phasesrdquo Journalof Combustion and Flame vol 125 no 1-2 pp 1055ndash1070 2001
[3] A M Hegab ldquoModeling of sandwich heterogeneous propellantwith two steps chemical kinetics and fluid mechanical effectsrdquoEngineering Research Journal vol 32 no 2 pp 119ndash130 2009
[4] J Buckmaster T L Jackson A Hegab S Kochevets andM Ulrich ldquoRandomly packed heterogeneous propellants andthe flames they supportrdquo in Proceedings of the 39th AerospaceSciences Meeting and Exhibit Reno NV USA January 2001
[5] S Kochevets J Buckmaster T L Jackson and A HegabldquoRandom packs and their use in modeling heterogeneous solidpropellant combustionrdquo Journal of Propulsion and Power vol 17no 4 pp 883ndash891 2001
[6] AMHegab S AHasanienH EMostafa andAMMaradenldquoExperimental and numerical investigations for the combustionof selected composite solid propellantrdquo Engineering ResearchJournal vol 35 no 3 2012
[7] A M Hegab ldquoVorticity generation and acoustic resonance ofsimulated solid rocket motor chamber with high wave numberwall injectionrdquo Computers and Fluids vol 38 no 6 pp 1258ndash1269 2009
[8] A M Hegab and D R Kassoy ldquoInternal flow temperatureand vorticity dynamics due to transient mass additionrdquo AIAAJournal vol 44 no 4 pp 812ndash826 2006
Mathematical Problems in Engineering 11
[9] K Kirkkopru D R Kassoy and Q Zhao ldquoUnsteady vorticitygeneration and evolution in a model of a solid rocket motorrdquoJournal of Propulsion and Power vol 12 no 4 pp 646ndash6541996
[10] K Kirkkopru D R Kassoy Q Zhao and P L Staab ldquoAcousti-cally generated unsteady vorticity field in a long narrow cylinderwith sidewall injectionrdquo Journal of Engineering Mathematicsvol 42 no 1 pp 65ndash90 2002
[11] P L StaabQ ZhaoD R Kassoy andKKirkkopru ldquoCoexistingacoustic-rotational flow in a cylinder with axisymmetric side-wall mass additionrdquo Physics of Fluids vol 11 no 10 pp 2935ndash2951 1999
[12] M Rempe P L Staab and D R Kassoy ldquoThermal response foran internal flow in a cylinderwith time dependent sidewallmassadditionrdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit AIAA 2000-0996 Reno Nev USA January 2000
[13] Q Zhao and D R Kassoy ldquoThe generation and evolutionof unsteady vorticity in a solid rocket engine chamberrdquo inProceedings of the 32th Aerospace Sciences Meeting AIAA paper94-0779 Reno Nev USA 1994
[14] Q Zhao P L Staab D R Kassoy and K Kirkkopru ldquoAcous-tically generated vorticity in an internal flowrdquo Journal of FluidMechanics vol 413 pp 247ndash285 2000
[15] G A Flandro ldquoEffects of vorticity on rocket combustionstabilityrdquo Journal of Propulsion and Power vol 11 no 4 pp 607ndash625 1995
[16] J Majdalani andW K VanMoorhem ldquoMultiple-scales solutionto the acoustic boundary layer in solid rocket motorsrdquo Journalof Propulsion and Power vol 13 no 2 pp 186ndash193 1997
[17] J Majdalani andW van Moorhem ldquoImproved time-dependentflowfield solution for solid rocket motorsrdquo AIAA Journal vol36 no 2 pp 241ndash248 1998
[18] Z Deng R J Adrian and C D Tomkinsm ldquoStructure ofturbulence in channel flow with a fully transpired wallrdquo inProceedings of the 39th Aerospace Science Meeting AIAA Paper2001-1019 Reno Nev USA 2001
[19] F Vuillot andG Avalon ldquoAcoustic boundary layers in solid pro-pellant rocketmotors usingNavier-Stokes equationsrdquo Journal ofPropulsion and Power vol 7 no 2 pp 231ndash239 1991
[20] T M Smith R L Roach and G A Flandro ldquoNumerical studyof the unsteady flow in a simulated solid rocket motorrdquo inProceedings of the 31th Aerospace Science Meeting AIAA Paper93-0112 Reno Nev USA 1993
[21] C F Tseng and I S Tseng ldquoInteraction between acoustic wavesand premixed flames in porous chamberrdquo in Proceedings of the32th Aerospace Science Meeting Reno Nev USA 1994 AIAAPaper 94 3328
[22] T S Roh and V Yang ldquoTransient combustion responses ofsolid propellants to acoustic disturbances in rocket motorsrdquo inProceedings of the 33rd Aerospace Sciences Meeting and ExhibitAIAA Paper 95-0602 Reno Nev USA 1995
[23] A M Hegab A study of acoustic phenomena in solid rocketengines [PhD thesis] Mechanical Power Engineering Depart-ment Menoufia University Menufia Egypt University of Col-orado Boulder Colo USA 1998
[24] R W MacCormack ldquoA numerical method for solving theequations of compressible viscous flowrdquo American Institute ofAeronautics and Astronautics Journal vol 20 no 9 pp 1275ndash1281 1982
[25] D Gottlieb and E Turkel ldquoDissipative two-four methods fortime-dependent problemsrdquo Mathematics of Computation vol30 no 136 pp 703ndash723 1976
[26] T J Poinsot and S K Lele ldquoBoundary conditions for directsimulations of compressible viscous flowsrdquo Journal of Compu-tational Physics vol 101 no 1 pp 104ndash129 1992
[27] Z Deng R J Adrian and C D Tomkins ldquoSensitivity ofturbulence in transpired channel to injection velocity small-scale nonuniformityrdquo AIAA Journal vol 40 no 11 pp 2241ndash2246 2002
[28] K J Meteb Prediction of hybrid rocket motor regression rate[MS thesis] Military Technical College Cairo Egypt 2009
[29] J W Anders V Magi and J Abraham ldquoLarge-eddy simulationin the near-field of a transient multi-component gas jet withdensity gradientsrdquoComputers amp Fluids vol 36 no 10 pp 1609ndash1620 2007
[30] B Chaouat ldquoNumerical simulations of channel flows with fluidinjection using Reynolds stressmodelrdquo AIAAPaper 2000-09922000
[31] W A El-Askary A Balabel S M El-Behery and A HegabldquoComputations of a compressible turbulent flow in a rocketmotor-chamber configuration with symmetric and asymmetricinjectionrdquoComputerModeling in EngineeringampSciences vol 82no 1 pp 29ndash54 2011
[32] A Balabel A M Hegab M Nasr and S M El-Behery ldquoAssess-ment of turbulence modeling for gas flow in two-dimensionalconvergent-divergent rocket nozzlerdquo Applied MathematicalModelling vol 35 no 7 pp 3408ndash3422 2011
Mathematical Problems in Engineering 7
The nondimensional equations can be written as follows
120597120588
120597119905
+ 119872119906
120597120588
120597119909
minus
119872119906
1205741198622
120597119901
120597119909
+ 119872V120597120588
120597119910
+
1198602
119872
(120574 minus 1) 119877119890119875119903
1
1198622
1205972
119879
1205971199102
= 0
120597119906
120597119905
+
1
120588119862
[(119906 +
119862
119872
)(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)] + 119872V120597119906
120597119910
minus
1198602
(120574 minus 1) 119877119890119875119903
1
120588119862
1205972
119879
1205971199102
minus
1198721198602
119877119890
1
120588
1205972
119906
1205971199102
= 0
120597V120597119905
+ 119872(119906
120597V120597119909
+ V120597V120597119910
) = 0
119889119901
119889119905
= 0
(23)
Here again the nondimensional variables (120588 119906 and V)can be computed using one-sided finite-difference approxi-mations
25 Head End Boundary Condition The velocity componentnormal to the boundary (119906(0 119910 119905)) is zero since the head endis impermeable In contrast the transverse velocity compo-nent cannot be specified because the axial viscous terms inthe transversemomentum equation have been suppressed Asthe normal velocity drops to zero at 119909 = 0 the amplitudesweierp2and weierp
3 in (18) and (19) are zero and the characteristic
velocities are 1205821
= minus119862 and 1205824
= +119862 The reflected wave weierp4
enters the domain while the wave weierp1propagates out through
the boundary The NSCBC relation (19) indicates that theamplitude of the incoming wave weierp
4should be
weierp4= weierp1 (24)
and the remaining conditions are
weierp1= minus119888(
1
120574
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
)
weierp2= 0 weierp
3= 0
(25)
The equations can be written in nondimensional form asfollows
120597120588
120597119905
minus 119872120588
120597119906
120597119909
+
1
120574119862
120597119901
120597119909
+ 119872
120597120588V120597119910
= 0
120597119906
120597119905
= 0
120597V120597119905
+ 119872V120597V120597119910
= 0
120597119901
120597119905
minus 119888
120597119901
120597119909
+ 119872120588119862
120597119906
120597119909
+ 120574119872119901
120597V120597119910
minus
1205741198602
119872
(120574 minus 1) 119875119903119877119890
1205972
119879
1205971199102
= 0
(26)
More details about the NSCBC techniques are found in [2326]
26 CodeValidations Thecomputational codes are designedconstructed and validated for steady and unsteady flow as
shown in Figures 2 and 3 For the latter one the constructedcodes run for many wave cycles to detect the whole unsteadyrotational flow dynamic across the chamber
In Figure 2 comparison between the current numericaland asymptotic results dealing with the steady state flowsolutions is carried out The steady state flow is generatedby adding constant side wall mass injection (119881
119903119908= minus1)
from the injection surface of the prescribed chamber Similarexperiment in [18 27] was carried out and compared withthe current mathematical models in Figure 2The asymptoticanalysis solution represents the solid line while the dottedline represents the converged steady state numerical solutionfor viscous flow with wall Reynolds number 119877
119890119908= 5 lowast
103 The nonfilled points refer to the experimental measured
data The experimental results are recorded for flow in achannel with fully transpired wall Three different poroussurfaces using three different sizes of honeycomb cells areused [7] In this figure a comparison between the analyticaland the numerical solutions at the midlength of the chamberis illustratedThis comparison shows a reasonable agreementbetween the three approaches The comparison between thetheoretical and the computational approaches shows littlebit deviation This deviation is logically expected since theused honey combs have three different sizes The smallersize of honey combs the smaller deviations between the twoapproaches
The second set of the validations show the solution for theunsteady flow computations at nonresonance forcedmode Inthis set of the results the computational results are validatedby comparing with the asymptotic analysis These resultsrepresent the asymptotic solution when the sidewall injectionvelocity is represented by V
119903119908(119909 1 119905) = minus[1 + 120576120593
119903119908(1 minus
cos120596119905)] 120593119903119908
= cos(120573119899119909) In the current work and the work
by Hegab [7] the disturbance forcing frequency is taken tobe 120596 = 1 (nonresonance) with the amplitude of 120576 = 04
and the eigenvalues (120573119899
= 1198991205872 119899 = 1 3 5 7 9 etc)The temperature time history (119879) at themedlength location ispresented in Figure 3 for 119877
119890= 3 lowast 10
5119872 = 002 120575 = 20 and119899 = 10 For the nonresonance case the solution shows stableconditions but it is not purely harmonic This interestingbehavior shows the ability of the constructed code with thegood treatment to the boundary conditions to detect thenon-harmonic trend as detected by the analytical approachA comparison of the perturbed temperature (119879) (27) withthe numerical results in Figure 3 shows good agreementbetween the two approaches especially for 119905 lt 20 while thecomputational solution shows little less deviation than theanalytical results for 119905 gt 20 due to the nonlinearity effects
119879119900(119909 119905) =
infin
sum
119899=0
2120576120596 (120574 minus 1) int
1
0
(120593119903119908
(119909) cos120573119899119909)
1205962minus 1205732
119899
times
120596
120573119899
sin120573119899119905 minus sin120596119905 cos120573
119899119909
(27)
Generally the coexistence of the forced and the naturalmodes of the system is detected in the adapted computationaland analytical approaches as seen in the earlier investigationby Hegab [7] In contrast the computational techniques in
8 Mathematical Problems in Engineering
[10] show only pure harmonic oscillations The source ofthese deficiencies may be arising from the treatment of theboundary conditions especially at the head and aft ends ofthe simulated chamber
3 Results and Discussions
The impact of the injected wave number from the sidewallalong with the frequencies of the forced mode on thegenerated complex acoustical fields and their interactionwiththe fluid are examinedThe results have been categorized intothe following two sets
31 Nonresonance Solution In the current study a widerange of forcing frequencies away from or very close tothe first fundamental modes of the prescribed chamber isconsidered [7] a wide range of forcing frequencies away fromor very close to the first fundamental modes of the prescribedchamber is considered In Figure 4 the irrotational androtational fields across the whole chamber are presentedwhen 119905 = 42 for 120596 = 1 119877
119890= 3lowast10
5119872 = 002 120576 = 04 119899 = 13 and 5 and 120582
119899= 1205872 The latter one represents the Eigen
values of the chamberThese interesting graphs show that therotational fields are generated at the injection surface As timeincreases the generated unsteady vorticity is transferred to fillmost of the combustion chamber Moreover the constructedcodes are able to run for enough time and count about sevenvorticity wave cycles as 119905 = 42 It is also seen that theamplitude of the rotational field attainsmaximumvalues nearthe wall and found to be order of ten for the monotonicvariation (119899 = 1) and nonresonance frequency (120596 = 1)One notes that these magnitudes meet the scaling of theasymptotic solutions in [7 9 11 14] Moreover for 119899 = 1monotonic distribution of unsteady vorticity is noticed whilethe variation goes in nonmonotonic axial distribution for 119899 gt
1 For the nonresonance solution the longitudinal variationof the vorticity at the wall 120597|Ω
119908(119909 119905)|120597119909 gt 0 for 120582
119899=
1205872 (119899 = 1) is revealing that the magnitude of rotationalflow fields on the injection surface increases as axial distanceincreases
The vorticity contours adjacent to the injection surface forthe same parameters as in Figure 4 for several eigenvalues(a) 120582119899
= 1205872 (119899 = 1) (b) 31205872 (119899 = 3) and (c) 51205872
(119899 = 5) are present in Figure 5 It is clearly seen that adjacentto the combustion surface the generated unsteady vorticitymay impact the surface at downstream locations for 119899 = 1at two locations for 119899 = 3 and many locations for 119899 = 5The rotational flow fluctuates from negative to positive andvice versa at several successive times and may cause scouringon the combustion surface and in turn may increase theshearing stresses adjacent to the combustion surface Thisimportant conclusion may lead to increase of the burningrates over the designed ones especially for low wave numbersince the amplitude of the unsteady vorticity decreases as thewave number increases
Fortunately recent experimental work by Meteb [28] in2009 was directed to study the acoustic instability in hybridrocket system using different fuel grains and ignition systems
The experimental results verified qualitatively the currentanalytical and computational results for the generated com-plex wave patterns and the effect of the combustion surfaceon the acoustic fluid dynamics interactionmechanism In thisexperiment the pure polyethylene (PE) fuel grain and PEwithadditives of aluminum (Al) powder with different percentageare used to predict the hybrid rocket motor regression ratesThe flame geometry and structure for the burning of thepure PE and PE with additives of aluminum (Al) powderat 25 75 and 15 of the total weight of the wholesamples are illustrated in Figure 6The effect of theAl powderon the flame length geometry and the rate of reaction forthe burning of each sample is clearly seen In addition thepressure time histories at a point before the combustionchamber in the way of oxidizer are recorded and illustratedin Figure 7 for the pure PE and in Figures 8 and 9 for PEwith additives of aluminum (Al) powder at 75 and 15of the total weight respectively It is clearly seen that theamplitude of the pressure waves decreases as the percentageof Al fuel grain increases in the sample This interestingbehavior is similar to what we have seen when we increasethe wave number of the combustion surface As we knowmost of the Al fuel grains burn away from the combustionsurface consisting ofmany fragmentsThe escaping processesof Al fuel grains from the combustion surface generate manyof cavities that make the combustion surface more roughthan the smooth one in case of pure PE The experimentaland computational results are qualitatively in agreement Forthe first time the analytical and computational results forthe current cold model and the acoustic instabilities foundby Meteb [28] state an important and essential conclusionabout the effect of the combustion surface morphology onthe acoustic instability inside the rocket chamber and theaccompanying rotational flow Instantaneous total velocityvector map for the parameters in Figure 4 at (a) 119905 = 0 ldquosteadystate solutionrdquo and (b) 119905 = 16 is presented in Figures 10(a)and 10(b) The injection surface is located at 119910 = 1 on theupper part of the plot where the flow is completely verticalwhile the lower part of the plot represents the center lineat 119910 = 0 where the flow is completely axial The closedend is located at 119909 = 0 on the left hand side of the plotwhile at 119909 = 1 it represents the exit plane on the right handside One may notice the smooth and undisturbed fluid flowthrough the entire map for the steady state solution at 119905 = 0After initiating the side wall disturbances the coexisting ofacoustics and the generated rotational flow is clearly seenin the upper half part of the chamber and illustrates howthe interaction between the travelling acoustic waves and theinjected fluid affects the fluid flow in the chamber In additionat 119905 = 32 in Figure 11 there is an instantaneous reverseflow found at a location near the combustion surface Thecomplex acoustic fluid dynamics interaction structure mayincrease the shear stresses adjacent to the combustion surfaceand through the whole cavity These results may providethe rocket designers with conceptual tools and data to beconsidered in case of redesigning strategies
Mathematical Problems in Engineering 9
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
ldquoSteady state solutionrdquo t = 00
(a)
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 16Nonresonance (
(b)
Figure 10 Instantaneous total velocity vector map for the parameters in Figure 4 (a) 119905 = 0 ldquosteady state solutionrdquo and (b) 119905 = 16
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 32
Reverse flow region
Nonresonance (
Figure 11 Instantaneous total velocity vector map at 119905 = 32 for theparameters in Figure 4
32 Near and at Resonance Solution A comparison of theasymptotic analysis perturbed pressure time history withthe computational one at and near resonance frequenciesis presented in Figure 12 For 119905 lt 10 the results showgood agreement between the two approaches even with theasymptotic beat condition (120596 = 1205872 minus 00018) which isvery close to the resonance frequency While for 119905 gt 10
the asymptotic solution shows linear growth with time basedon the linear theory for the derived wave equation where120596 = 120573
119899lowast = 1205872 as derived by Hegab [7] In contrast
the computational solution shows beats at the resonancecondition due to the nonlinearity and the asymptotic beatsolutions have smaller amplitude than the resonance oneThe beats appear due to the interaction between the derivingfrequency mode 120596 = 1205872minus00018 and the first eigenfunctionmode (1205872) It is clearly seen that the amplitude of thevorticity at the resonance frequency is found to be ten timesthan the amplitudes at the nonresonance one in Figure 4
P
03
02
01
0
minus01
minus02
minus030 5 10 15 20 25 30 35 40
t
ComputationalAnalyticalAnalytical (beats)
x = 05 y = 00 120596 = 1205872 n = 1
Figure 12 Comparison between the computational and the asymp-totic solutions for the perturbed pressure time history with 120596 = 1205872
(resonance) 120576 = 04 119872 = 002 and 119877119890= 3 lowast 10
5 at the half-lengthof the chamber
Moreover it is found that the fluctuation of the transversetemperature gradient at the injection surface is found tobe about ten times than the fluctuation of the temperaturedisturbances This concludes that since the heat transferalong the sidewall is directly proportional to the transversetemperature gradient there is an unexpectedly large heat fluxwhich exists both at the wall and within the entire chambergiven the small size of the temperature distribution Theasymptotic analysis by Staab et al [11] is used to prove thisimportant conclusion
Finally the asymptotic and computational methods areemployed to study the dynamic process in an acoustically
10 Mathematical Problems in Engineering
simulated solid propellant rocket chamber with transientsidewall mass addition at different wave numbers
4 Conclusion Remarks
The current study emphasizes the effect of the combustionsurfacemorphology of the solid rocket systemon the acousticinstability and the accompanying vorticity and temperaturedynamics generation The rotational and irrotational fieldsgenerated in the prescribed solid rocket motor chamber withMach number are chosen to be O(10minus1) while the charac-teristic Reynolds number is O(105-106) at nonresonance andnear and at resonance frequencies have been described TheNSCBC techniques that applied in [7 29] are used to treat thenumerical boundary conditions
The numerical solution shows that the convergence tosteady state solution is achieved first and then additionaltransient side wall mass addition is initiated Moreover theresult shows that the forced mode of the nonresonance andresonance frequencies generates longitudinal acoustics waveswhich interact again with its sources at the injection surfaceto generate the rotational flow fields The generated unsteadyvorticity is transferred from the injection surface toward thecenterline to fill the entire chamber as time increases aftermore than seven wave cycles The solution shows nonpureharmonic oscillations as suggested from the theory in [11]Vorticity 3D graphs and contours illustrate that spatially themaximum amplitudes are found to be near the injectionsurface Moreover the largest shear stresses are found to be atthe injection surface near the exit end while the shear stressesimpact the injection surface at many locations as the wavenumber increasesThe longitudinal variation of the rotationalflow fields is found to be nonmonotonic for 119899 gt 1
Here for the first time the analytical and computationalresults for the current coldmodel and the recent experimentalwork by Meteb [28] revealed that the combustion surfacemorphology has a great effect on the acoustic instabilityinside the rocket chamber and the accompanying rotationalflow The computational approach reveals that there areunexpectedly high heat transfer and generated rotational flowfields found to be adjacent to the injection surface in bothnonresonance and resonance cases The amplitude of thelatter one is found to be approximately ten times than thatwith the nonresonance solution One may anticipate thatthese large oscillatory gradients will persist to the surface ofa burning solid propellant in a real rocket chamber
More traditional computational studies of turbulence inchannel flows with wall injection are reviewed by Chaouat[30] Mean axial velocity profiles are found to be laminaruntil about halfway down the channel and then show signs oftransition Chaouatrsquos results do not show the acoustic-relatedeffects discussed in the present work As a result it is difficultto know from the turbulent theory results if the source ofthe downstream vorticity is classical transition to turbulenceor partly the result of the kind of mechanism discussed inthis paper In our recent studies in 2011 and 2012 [31 32]the assessment of turbulence modeling for gas flow in thechamber and in the convergent-divergent rocket nozzle for
the steady state flow is studied The detailed information andrealities in the current research about the transient coexistingof the irrotational (acoustics) rotational (generated vorticityfrom the interaction between the injected flow and thetraveling acoustic waves) and our turbulent techniques thatwe have adapted in our studies in 2011 and 2012 for the steadyflow field in SRM chambernozzle geometry will be used toaccount all the complex flow fields in our future study
Finally results of the asymptotic and computationalapproaches may provide the rocket designer with codes andinformation making them able to interpret some reasonsbehind the combustion instability inside the combustionchamber of real SRM based on the composition of the solidpropellants and in turn the morphology of the combustionsurface
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The first author would like to thank Professor D R KassoyUniversity of Colorado at Boulder USA for their continuoushelp
References
[1] A Hegab T L Jackson J Buckmaster and D S Stewart ldquoTheburning of periodic sandwich propellantsrdquo in Proceedings of the36th AIAAASMESAEASEE Joint Propulsion Conference andExhibit Huntsville Ala USA July 2000
[2] A Hegab T L Jackson J Buckmaster and D S StewartldquoNonsteady burning of periodic sandwich propellants withcomplete coupling between the solid and gas phasesrdquo Journalof Combustion and Flame vol 125 no 1-2 pp 1055ndash1070 2001
[3] A M Hegab ldquoModeling of sandwich heterogeneous propellantwith two steps chemical kinetics and fluid mechanical effectsrdquoEngineering Research Journal vol 32 no 2 pp 119ndash130 2009
[4] J Buckmaster T L Jackson A Hegab S Kochevets andM Ulrich ldquoRandomly packed heterogeneous propellants andthe flames they supportrdquo in Proceedings of the 39th AerospaceSciences Meeting and Exhibit Reno NV USA January 2001
[5] S Kochevets J Buckmaster T L Jackson and A HegabldquoRandom packs and their use in modeling heterogeneous solidpropellant combustionrdquo Journal of Propulsion and Power vol 17no 4 pp 883ndash891 2001
[6] AMHegab S AHasanienH EMostafa andAMMaradenldquoExperimental and numerical investigations for the combustionof selected composite solid propellantrdquo Engineering ResearchJournal vol 35 no 3 2012
[7] A M Hegab ldquoVorticity generation and acoustic resonance ofsimulated solid rocket motor chamber with high wave numberwall injectionrdquo Computers and Fluids vol 38 no 6 pp 1258ndash1269 2009
[8] A M Hegab and D R Kassoy ldquoInternal flow temperatureand vorticity dynamics due to transient mass additionrdquo AIAAJournal vol 44 no 4 pp 812ndash826 2006
Mathematical Problems in Engineering 11
[9] K Kirkkopru D R Kassoy and Q Zhao ldquoUnsteady vorticitygeneration and evolution in a model of a solid rocket motorrdquoJournal of Propulsion and Power vol 12 no 4 pp 646ndash6541996
[10] K Kirkkopru D R Kassoy Q Zhao and P L Staab ldquoAcousti-cally generated unsteady vorticity field in a long narrow cylinderwith sidewall injectionrdquo Journal of Engineering Mathematicsvol 42 no 1 pp 65ndash90 2002
[11] P L StaabQ ZhaoD R Kassoy andKKirkkopru ldquoCoexistingacoustic-rotational flow in a cylinder with axisymmetric side-wall mass additionrdquo Physics of Fluids vol 11 no 10 pp 2935ndash2951 1999
[12] M Rempe P L Staab and D R Kassoy ldquoThermal response foran internal flow in a cylinderwith time dependent sidewallmassadditionrdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit AIAA 2000-0996 Reno Nev USA January 2000
[13] Q Zhao and D R Kassoy ldquoThe generation and evolutionof unsteady vorticity in a solid rocket engine chamberrdquo inProceedings of the 32th Aerospace Sciences Meeting AIAA paper94-0779 Reno Nev USA 1994
[14] Q Zhao P L Staab D R Kassoy and K Kirkkopru ldquoAcous-tically generated vorticity in an internal flowrdquo Journal of FluidMechanics vol 413 pp 247ndash285 2000
[15] G A Flandro ldquoEffects of vorticity on rocket combustionstabilityrdquo Journal of Propulsion and Power vol 11 no 4 pp 607ndash625 1995
[16] J Majdalani andW K VanMoorhem ldquoMultiple-scales solutionto the acoustic boundary layer in solid rocket motorsrdquo Journalof Propulsion and Power vol 13 no 2 pp 186ndash193 1997
[17] J Majdalani andW van Moorhem ldquoImproved time-dependentflowfield solution for solid rocket motorsrdquo AIAA Journal vol36 no 2 pp 241ndash248 1998
[18] Z Deng R J Adrian and C D Tomkinsm ldquoStructure ofturbulence in channel flow with a fully transpired wallrdquo inProceedings of the 39th Aerospace Science Meeting AIAA Paper2001-1019 Reno Nev USA 2001
[19] F Vuillot andG Avalon ldquoAcoustic boundary layers in solid pro-pellant rocketmotors usingNavier-Stokes equationsrdquo Journal ofPropulsion and Power vol 7 no 2 pp 231ndash239 1991
[20] T M Smith R L Roach and G A Flandro ldquoNumerical studyof the unsteady flow in a simulated solid rocket motorrdquo inProceedings of the 31th Aerospace Science Meeting AIAA Paper93-0112 Reno Nev USA 1993
[21] C F Tseng and I S Tseng ldquoInteraction between acoustic wavesand premixed flames in porous chamberrdquo in Proceedings of the32th Aerospace Science Meeting Reno Nev USA 1994 AIAAPaper 94 3328
[22] T S Roh and V Yang ldquoTransient combustion responses ofsolid propellants to acoustic disturbances in rocket motorsrdquo inProceedings of the 33rd Aerospace Sciences Meeting and ExhibitAIAA Paper 95-0602 Reno Nev USA 1995
[23] A M Hegab A study of acoustic phenomena in solid rocketengines [PhD thesis] Mechanical Power Engineering Depart-ment Menoufia University Menufia Egypt University of Col-orado Boulder Colo USA 1998
[24] R W MacCormack ldquoA numerical method for solving theequations of compressible viscous flowrdquo American Institute ofAeronautics and Astronautics Journal vol 20 no 9 pp 1275ndash1281 1982
[25] D Gottlieb and E Turkel ldquoDissipative two-four methods fortime-dependent problemsrdquo Mathematics of Computation vol30 no 136 pp 703ndash723 1976
[26] T J Poinsot and S K Lele ldquoBoundary conditions for directsimulations of compressible viscous flowsrdquo Journal of Compu-tational Physics vol 101 no 1 pp 104ndash129 1992
[27] Z Deng R J Adrian and C D Tomkins ldquoSensitivity ofturbulence in transpired channel to injection velocity small-scale nonuniformityrdquo AIAA Journal vol 40 no 11 pp 2241ndash2246 2002
[28] K J Meteb Prediction of hybrid rocket motor regression rate[MS thesis] Military Technical College Cairo Egypt 2009
[29] J W Anders V Magi and J Abraham ldquoLarge-eddy simulationin the near-field of a transient multi-component gas jet withdensity gradientsrdquoComputers amp Fluids vol 36 no 10 pp 1609ndash1620 2007
[30] B Chaouat ldquoNumerical simulations of channel flows with fluidinjection using Reynolds stressmodelrdquo AIAAPaper 2000-09922000
[31] W A El-Askary A Balabel S M El-Behery and A HegabldquoComputations of a compressible turbulent flow in a rocketmotor-chamber configuration with symmetric and asymmetricinjectionrdquoComputerModeling in EngineeringampSciences vol 82no 1 pp 29ndash54 2011
[32] A Balabel A M Hegab M Nasr and S M El-Behery ldquoAssess-ment of turbulence modeling for gas flow in two-dimensionalconvergent-divergent rocket nozzlerdquo Applied MathematicalModelling vol 35 no 7 pp 3408ndash3422 2011
8 Mathematical Problems in Engineering
[10] show only pure harmonic oscillations The source ofthese deficiencies may be arising from the treatment of theboundary conditions especially at the head and aft ends ofthe simulated chamber
3 Results and Discussions
The impact of the injected wave number from the sidewallalong with the frequencies of the forced mode on thegenerated complex acoustical fields and their interactionwiththe fluid are examinedThe results have been categorized intothe following two sets
31 Nonresonance Solution In the current study a widerange of forcing frequencies away from or very close tothe first fundamental modes of the prescribed chamber isconsidered [7] a wide range of forcing frequencies away fromor very close to the first fundamental modes of the prescribedchamber is considered In Figure 4 the irrotational androtational fields across the whole chamber are presentedwhen 119905 = 42 for 120596 = 1 119877
119890= 3lowast10
5119872 = 002 120576 = 04 119899 = 13 and 5 and 120582
119899= 1205872 The latter one represents the Eigen
values of the chamberThese interesting graphs show that therotational fields are generated at the injection surface As timeincreases the generated unsteady vorticity is transferred to fillmost of the combustion chamber Moreover the constructedcodes are able to run for enough time and count about sevenvorticity wave cycles as 119905 = 42 It is also seen that theamplitude of the rotational field attainsmaximumvalues nearthe wall and found to be order of ten for the monotonicvariation (119899 = 1) and nonresonance frequency (120596 = 1)One notes that these magnitudes meet the scaling of theasymptotic solutions in [7 9 11 14] Moreover for 119899 = 1monotonic distribution of unsteady vorticity is noticed whilethe variation goes in nonmonotonic axial distribution for 119899 gt
1 For the nonresonance solution the longitudinal variationof the vorticity at the wall 120597|Ω
119908(119909 119905)|120597119909 gt 0 for 120582
119899=
1205872 (119899 = 1) is revealing that the magnitude of rotationalflow fields on the injection surface increases as axial distanceincreases
The vorticity contours adjacent to the injection surface forthe same parameters as in Figure 4 for several eigenvalues(a) 120582119899
= 1205872 (119899 = 1) (b) 31205872 (119899 = 3) and (c) 51205872
(119899 = 5) are present in Figure 5 It is clearly seen that adjacentto the combustion surface the generated unsteady vorticitymay impact the surface at downstream locations for 119899 = 1at two locations for 119899 = 3 and many locations for 119899 = 5The rotational flow fluctuates from negative to positive andvice versa at several successive times and may cause scouringon the combustion surface and in turn may increase theshearing stresses adjacent to the combustion surface Thisimportant conclusion may lead to increase of the burningrates over the designed ones especially for low wave numbersince the amplitude of the unsteady vorticity decreases as thewave number increases
Fortunately recent experimental work by Meteb [28] in2009 was directed to study the acoustic instability in hybridrocket system using different fuel grains and ignition systems
The experimental results verified qualitatively the currentanalytical and computational results for the generated com-plex wave patterns and the effect of the combustion surfaceon the acoustic fluid dynamics interactionmechanism In thisexperiment the pure polyethylene (PE) fuel grain and PEwithadditives of aluminum (Al) powder with different percentageare used to predict the hybrid rocket motor regression ratesThe flame geometry and structure for the burning of thepure PE and PE with additives of aluminum (Al) powderat 25 75 and 15 of the total weight of the wholesamples are illustrated in Figure 6The effect of theAl powderon the flame length geometry and the rate of reaction forthe burning of each sample is clearly seen In addition thepressure time histories at a point before the combustionchamber in the way of oxidizer are recorded and illustratedin Figure 7 for the pure PE and in Figures 8 and 9 for PEwith additives of aluminum (Al) powder at 75 and 15of the total weight respectively It is clearly seen that theamplitude of the pressure waves decreases as the percentageof Al fuel grain increases in the sample This interestingbehavior is similar to what we have seen when we increasethe wave number of the combustion surface As we knowmost of the Al fuel grains burn away from the combustionsurface consisting ofmany fragmentsThe escaping processesof Al fuel grains from the combustion surface generate manyof cavities that make the combustion surface more roughthan the smooth one in case of pure PE The experimentaland computational results are qualitatively in agreement Forthe first time the analytical and computational results forthe current cold model and the acoustic instabilities foundby Meteb [28] state an important and essential conclusionabout the effect of the combustion surface morphology onthe acoustic instability inside the rocket chamber and theaccompanying rotational flow Instantaneous total velocityvector map for the parameters in Figure 4 at (a) 119905 = 0 ldquosteadystate solutionrdquo and (b) 119905 = 16 is presented in Figures 10(a)and 10(b) The injection surface is located at 119910 = 1 on theupper part of the plot where the flow is completely verticalwhile the lower part of the plot represents the center lineat 119910 = 0 where the flow is completely axial The closedend is located at 119909 = 0 on the left hand side of the plotwhile at 119909 = 1 it represents the exit plane on the right handside One may notice the smooth and undisturbed fluid flowthrough the entire map for the steady state solution at 119905 = 0After initiating the side wall disturbances the coexisting ofacoustics and the generated rotational flow is clearly seenin the upper half part of the chamber and illustrates howthe interaction between the travelling acoustic waves and theinjected fluid affects the fluid flow in the chamber In additionat 119905 = 32 in Figure 11 there is an instantaneous reverseflow found at a location near the combustion surface Thecomplex acoustic fluid dynamics interaction structure mayincrease the shear stresses adjacent to the combustion surfaceand through the whole cavity These results may providethe rocket designers with conceptual tools and data to beconsidered in case of redesigning strategies
Mathematical Problems in Engineering 9
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
ldquoSteady state solutionrdquo t = 00
(a)
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 16Nonresonance (
(b)
Figure 10 Instantaneous total velocity vector map for the parameters in Figure 4 (a) 119905 = 0 ldquosteady state solutionrdquo and (b) 119905 = 16
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 32
Reverse flow region
Nonresonance (
Figure 11 Instantaneous total velocity vector map at 119905 = 32 for theparameters in Figure 4
32 Near and at Resonance Solution A comparison of theasymptotic analysis perturbed pressure time history withthe computational one at and near resonance frequenciesis presented in Figure 12 For 119905 lt 10 the results showgood agreement between the two approaches even with theasymptotic beat condition (120596 = 1205872 minus 00018) which isvery close to the resonance frequency While for 119905 gt 10
the asymptotic solution shows linear growth with time basedon the linear theory for the derived wave equation where120596 = 120573
119899lowast = 1205872 as derived by Hegab [7] In contrast
the computational solution shows beats at the resonancecondition due to the nonlinearity and the asymptotic beatsolutions have smaller amplitude than the resonance oneThe beats appear due to the interaction between the derivingfrequency mode 120596 = 1205872minus00018 and the first eigenfunctionmode (1205872) It is clearly seen that the amplitude of thevorticity at the resonance frequency is found to be ten timesthan the amplitudes at the nonresonance one in Figure 4
P
03
02
01
0
minus01
minus02
minus030 5 10 15 20 25 30 35 40
t
ComputationalAnalyticalAnalytical (beats)
x = 05 y = 00 120596 = 1205872 n = 1
Figure 12 Comparison between the computational and the asymp-totic solutions for the perturbed pressure time history with 120596 = 1205872
(resonance) 120576 = 04 119872 = 002 and 119877119890= 3 lowast 10
5 at the half-lengthof the chamber
Moreover it is found that the fluctuation of the transversetemperature gradient at the injection surface is found tobe about ten times than the fluctuation of the temperaturedisturbances This concludes that since the heat transferalong the sidewall is directly proportional to the transversetemperature gradient there is an unexpectedly large heat fluxwhich exists both at the wall and within the entire chambergiven the small size of the temperature distribution Theasymptotic analysis by Staab et al [11] is used to prove thisimportant conclusion
Finally the asymptotic and computational methods areemployed to study the dynamic process in an acoustically
10 Mathematical Problems in Engineering
simulated solid propellant rocket chamber with transientsidewall mass addition at different wave numbers
4 Conclusion Remarks
The current study emphasizes the effect of the combustionsurfacemorphology of the solid rocket systemon the acousticinstability and the accompanying vorticity and temperaturedynamics generation The rotational and irrotational fieldsgenerated in the prescribed solid rocket motor chamber withMach number are chosen to be O(10minus1) while the charac-teristic Reynolds number is O(105-106) at nonresonance andnear and at resonance frequencies have been described TheNSCBC techniques that applied in [7 29] are used to treat thenumerical boundary conditions
The numerical solution shows that the convergence tosteady state solution is achieved first and then additionaltransient side wall mass addition is initiated Moreover theresult shows that the forced mode of the nonresonance andresonance frequencies generates longitudinal acoustics waveswhich interact again with its sources at the injection surfaceto generate the rotational flow fields The generated unsteadyvorticity is transferred from the injection surface toward thecenterline to fill the entire chamber as time increases aftermore than seven wave cycles The solution shows nonpureharmonic oscillations as suggested from the theory in [11]Vorticity 3D graphs and contours illustrate that spatially themaximum amplitudes are found to be near the injectionsurface Moreover the largest shear stresses are found to be atthe injection surface near the exit end while the shear stressesimpact the injection surface at many locations as the wavenumber increasesThe longitudinal variation of the rotationalflow fields is found to be nonmonotonic for 119899 gt 1
Here for the first time the analytical and computationalresults for the current coldmodel and the recent experimentalwork by Meteb [28] revealed that the combustion surfacemorphology has a great effect on the acoustic instabilityinside the rocket chamber and the accompanying rotationalflow The computational approach reveals that there areunexpectedly high heat transfer and generated rotational flowfields found to be adjacent to the injection surface in bothnonresonance and resonance cases The amplitude of thelatter one is found to be approximately ten times than thatwith the nonresonance solution One may anticipate thatthese large oscillatory gradients will persist to the surface ofa burning solid propellant in a real rocket chamber
More traditional computational studies of turbulence inchannel flows with wall injection are reviewed by Chaouat[30] Mean axial velocity profiles are found to be laminaruntil about halfway down the channel and then show signs oftransition Chaouatrsquos results do not show the acoustic-relatedeffects discussed in the present work As a result it is difficultto know from the turbulent theory results if the source ofthe downstream vorticity is classical transition to turbulenceor partly the result of the kind of mechanism discussed inthis paper In our recent studies in 2011 and 2012 [31 32]the assessment of turbulence modeling for gas flow in thechamber and in the convergent-divergent rocket nozzle for
the steady state flow is studied The detailed information andrealities in the current research about the transient coexistingof the irrotational (acoustics) rotational (generated vorticityfrom the interaction between the injected flow and thetraveling acoustic waves) and our turbulent techniques thatwe have adapted in our studies in 2011 and 2012 for the steadyflow field in SRM chambernozzle geometry will be used toaccount all the complex flow fields in our future study
Finally results of the asymptotic and computationalapproaches may provide the rocket designer with codes andinformation making them able to interpret some reasonsbehind the combustion instability inside the combustionchamber of real SRM based on the composition of the solidpropellants and in turn the morphology of the combustionsurface
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The first author would like to thank Professor D R KassoyUniversity of Colorado at Boulder USA for their continuoushelp
References
[1] A Hegab T L Jackson J Buckmaster and D S Stewart ldquoTheburning of periodic sandwich propellantsrdquo in Proceedings of the36th AIAAASMESAEASEE Joint Propulsion Conference andExhibit Huntsville Ala USA July 2000
[2] A Hegab T L Jackson J Buckmaster and D S StewartldquoNonsteady burning of periodic sandwich propellants withcomplete coupling between the solid and gas phasesrdquo Journalof Combustion and Flame vol 125 no 1-2 pp 1055ndash1070 2001
[3] A M Hegab ldquoModeling of sandwich heterogeneous propellantwith two steps chemical kinetics and fluid mechanical effectsrdquoEngineering Research Journal vol 32 no 2 pp 119ndash130 2009
[4] J Buckmaster T L Jackson A Hegab S Kochevets andM Ulrich ldquoRandomly packed heterogeneous propellants andthe flames they supportrdquo in Proceedings of the 39th AerospaceSciences Meeting and Exhibit Reno NV USA January 2001
[5] S Kochevets J Buckmaster T L Jackson and A HegabldquoRandom packs and their use in modeling heterogeneous solidpropellant combustionrdquo Journal of Propulsion and Power vol 17no 4 pp 883ndash891 2001
[6] AMHegab S AHasanienH EMostafa andAMMaradenldquoExperimental and numerical investigations for the combustionof selected composite solid propellantrdquo Engineering ResearchJournal vol 35 no 3 2012
[7] A M Hegab ldquoVorticity generation and acoustic resonance ofsimulated solid rocket motor chamber with high wave numberwall injectionrdquo Computers and Fluids vol 38 no 6 pp 1258ndash1269 2009
[8] A M Hegab and D R Kassoy ldquoInternal flow temperatureand vorticity dynamics due to transient mass additionrdquo AIAAJournal vol 44 no 4 pp 812ndash826 2006
Mathematical Problems in Engineering 11
[9] K Kirkkopru D R Kassoy and Q Zhao ldquoUnsteady vorticitygeneration and evolution in a model of a solid rocket motorrdquoJournal of Propulsion and Power vol 12 no 4 pp 646ndash6541996
[10] K Kirkkopru D R Kassoy Q Zhao and P L Staab ldquoAcousti-cally generated unsteady vorticity field in a long narrow cylinderwith sidewall injectionrdquo Journal of Engineering Mathematicsvol 42 no 1 pp 65ndash90 2002
[11] P L StaabQ ZhaoD R Kassoy andKKirkkopru ldquoCoexistingacoustic-rotational flow in a cylinder with axisymmetric side-wall mass additionrdquo Physics of Fluids vol 11 no 10 pp 2935ndash2951 1999
[12] M Rempe P L Staab and D R Kassoy ldquoThermal response foran internal flow in a cylinderwith time dependent sidewallmassadditionrdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit AIAA 2000-0996 Reno Nev USA January 2000
[13] Q Zhao and D R Kassoy ldquoThe generation and evolutionof unsteady vorticity in a solid rocket engine chamberrdquo inProceedings of the 32th Aerospace Sciences Meeting AIAA paper94-0779 Reno Nev USA 1994
[14] Q Zhao P L Staab D R Kassoy and K Kirkkopru ldquoAcous-tically generated vorticity in an internal flowrdquo Journal of FluidMechanics vol 413 pp 247ndash285 2000
[15] G A Flandro ldquoEffects of vorticity on rocket combustionstabilityrdquo Journal of Propulsion and Power vol 11 no 4 pp 607ndash625 1995
[16] J Majdalani andW K VanMoorhem ldquoMultiple-scales solutionto the acoustic boundary layer in solid rocket motorsrdquo Journalof Propulsion and Power vol 13 no 2 pp 186ndash193 1997
[17] J Majdalani andW van Moorhem ldquoImproved time-dependentflowfield solution for solid rocket motorsrdquo AIAA Journal vol36 no 2 pp 241ndash248 1998
[18] Z Deng R J Adrian and C D Tomkinsm ldquoStructure ofturbulence in channel flow with a fully transpired wallrdquo inProceedings of the 39th Aerospace Science Meeting AIAA Paper2001-1019 Reno Nev USA 2001
[19] F Vuillot andG Avalon ldquoAcoustic boundary layers in solid pro-pellant rocketmotors usingNavier-Stokes equationsrdquo Journal ofPropulsion and Power vol 7 no 2 pp 231ndash239 1991
[20] T M Smith R L Roach and G A Flandro ldquoNumerical studyof the unsteady flow in a simulated solid rocket motorrdquo inProceedings of the 31th Aerospace Science Meeting AIAA Paper93-0112 Reno Nev USA 1993
[21] C F Tseng and I S Tseng ldquoInteraction between acoustic wavesand premixed flames in porous chamberrdquo in Proceedings of the32th Aerospace Science Meeting Reno Nev USA 1994 AIAAPaper 94 3328
[22] T S Roh and V Yang ldquoTransient combustion responses ofsolid propellants to acoustic disturbances in rocket motorsrdquo inProceedings of the 33rd Aerospace Sciences Meeting and ExhibitAIAA Paper 95-0602 Reno Nev USA 1995
[23] A M Hegab A study of acoustic phenomena in solid rocketengines [PhD thesis] Mechanical Power Engineering Depart-ment Menoufia University Menufia Egypt University of Col-orado Boulder Colo USA 1998
[24] R W MacCormack ldquoA numerical method for solving theequations of compressible viscous flowrdquo American Institute ofAeronautics and Astronautics Journal vol 20 no 9 pp 1275ndash1281 1982
[25] D Gottlieb and E Turkel ldquoDissipative two-four methods fortime-dependent problemsrdquo Mathematics of Computation vol30 no 136 pp 703ndash723 1976
[26] T J Poinsot and S K Lele ldquoBoundary conditions for directsimulations of compressible viscous flowsrdquo Journal of Compu-tational Physics vol 101 no 1 pp 104ndash129 1992
[27] Z Deng R J Adrian and C D Tomkins ldquoSensitivity ofturbulence in transpired channel to injection velocity small-scale nonuniformityrdquo AIAA Journal vol 40 no 11 pp 2241ndash2246 2002
[28] K J Meteb Prediction of hybrid rocket motor regression rate[MS thesis] Military Technical College Cairo Egypt 2009
[29] J W Anders V Magi and J Abraham ldquoLarge-eddy simulationin the near-field of a transient multi-component gas jet withdensity gradientsrdquoComputers amp Fluids vol 36 no 10 pp 1609ndash1620 2007
[30] B Chaouat ldquoNumerical simulations of channel flows with fluidinjection using Reynolds stressmodelrdquo AIAAPaper 2000-09922000
[31] W A El-Askary A Balabel S M El-Behery and A HegabldquoComputations of a compressible turbulent flow in a rocketmotor-chamber configuration with symmetric and asymmetricinjectionrdquoComputerModeling in EngineeringampSciences vol 82no 1 pp 29ndash54 2011
[32] A Balabel A M Hegab M Nasr and S M El-Behery ldquoAssess-ment of turbulence modeling for gas flow in two-dimensionalconvergent-divergent rocket nozzlerdquo Applied MathematicalModelling vol 35 no 7 pp 3408ndash3422 2011
Mathematical Problems in Engineering 9
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
ldquoSteady state solutionrdquo t = 00
(a)
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 16Nonresonance (
(b)
Figure 10 Instantaneous total velocity vector map for the parameters in Figure 4 (a) 119905 = 0 ldquosteady state solutionrdquo and (b) 119905 = 16
10
10
08
08
06
06
04
04
02
02
00
00
Ytit
le
X title
120596 = 10) n = 1 at t = 32
Reverse flow region
Nonresonance (
Figure 11 Instantaneous total velocity vector map at 119905 = 32 for theparameters in Figure 4
32 Near and at Resonance Solution A comparison of theasymptotic analysis perturbed pressure time history withthe computational one at and near resonance frequenciesis presented in Figure 12 For 119905 lt 10 the results showgood agreement between the two approaches even with theasymptotic beat condition (120596 = 1205872 minus 00018) which isvery close to the resonance frequency While for 119905 gt 10
the asymptotic solution shows linear growth with time basedon the linear theory for the derived wave equation where120596 = 120573
119899lowast = 1205872 as derived by Hegab [7] In contrast
the computational solution shows beats at the resonancecondition due to the nonlinearity and the asymptotic beatsolutions have smaller amplitude than the resonance oneThe beats appear due to the interaction between the derivingfrequency mode 120596 = 1205872minus00018 and the first eigenfunctionmode (1205872) It is clearly seen that the amplitude of thevorticity at the resonance frequency is found to be ten timesthan the amplitudes at the nonresonance one in Figure 4
P
03
02
01
0
minus01
minus02
minus030 5 10 15 20 25 30 35 40
t
ComputationalAnalyticalAnalytical (beats)
x = 05 y = 00 120596 = 1205872 n = 1
Figure 12 Comparison between the computational and the asymp-totic solutions for the perturbed pressure time history with 120596 = 1205872
(resonance) 120576 = 04 119872 = 002 and 119877119890= 3 lowast 10
5 at the half-lengthof the chamber
Moreover it is found that the fluctuation of the transversetemperature gradient at the injection surface is found tobe about ten times than the fluctuation of the temperaturedisturbances This concludes that since the heat transferalong the sidewall is directly proportional to the transversetemperature gradient there is an unexpectedly large heat fluxwhich exists both at the wall and within the entire chambergiven the small size of the temperature distribution Theasymptotic analysis by Staab et al [11] is used to prove thisimportant conclusion
Finally the asymptotic and computational methods areemployed to study the dynamic process in an acoustically
10 Mathematical Problems in Engineering
simulated solid propellant rocket chamber with transientsidewall mass addition at different wave numbers
4 Conclusion Remarks
The current study emphasizes the effect of the combustionsurfacemorphology of the solid rocket systemon the acousticinstability and the accompanying vorticity and temperaturedynamics generation The rotational and irrotational fieldsgenerated in the prescribed solid rocket motor chamber withMach number are chosen to be O(10minus1) while the charac-teristic Reynolds number is O(105-106) at nonresonance andnear and at resonance frequencies have been described TheNSCBC techniques that applied in [7 29] are used to treat thenumerical boundary conditions
The numerical solution shows that the convergence tosteady state solution is achieved first and then additionaltransient side wall mass addition is initiated Moreover theresult shows that the forced mode of the nonresonance andresonance frequencies generates longitudinal acoustics waveswhich interact again with its sources at the injection surfaceto generate the rotational flow fields The generated unsteadyvorticity is transferred from the injection surface toward thecenterline to fill the entire chamber as time increases aftermore than seven wave cycles The solution shows nonpureharmonic oscillations as suggested from the theory in [11]Vorticity 3D graphs and contours illustrate that spatially themaximum amplitudes are found to be near the injectionsurface Moreover the largest shear stresses are found to be atthe injection surface near the exit end while the shear stressesimpact the injection surface at many locations as the wavenumber increasesThe longitudinal variation of the rotationalflow fields is found to be nonmonotonic for 119899 gt 1
Here for the first time the analytical and computationalresults for the current coldmodel and the recent experimentalwork by Meteb [28] revealed that the combustion surfacemorphology has a great effect on the acoustic instabilityinside the rocket chamber and the accompanying rotationalflow The computational approach reveals that there areunexpectedly high heat transfer and generated rotational flowfields found to be adjacent to the injection surface in bothnonresonance and resonance cases The amplitude of thelatter one is found to be approximately ten times than thatwith the nonresonance solution One may anticipate thatthese large oscillatory gradients will persist to the surface ofa burning solid propellant in a real rocket chamber
More traditional computational studies of turbulence inchannel flows with wall injection are reviewed by Chaouat[30] Mean axial velocity profiles are found to be laminaruntil about halfway down the channel and then show signs oftransition Chaouatrsquos results do not show the acoustic-relatedeffects discussed in the present work As a result it is difficultto know from the turbulent theory results if the source ofthe downstream vorticity is classical transition to turbulenceor partly the result of the kind of mechanism discussed inthis paper In our recent studies in 2011 and 2012 [31 32]the assessment of turbulence modeling for gas flow in thechamber and in the convergent-divergent rocket nozzle for
the steady state flow is studied The detailed information andrealities in the current research about the transient coexistingof the irrotational (acoustics) rotational (generated vorticityfrom the interaction between the injected flow and thetraveling acoustic waves) and our turbulent techniques thatwe have adapted in our studies in 2011 and 2012 for the steadyflow field in SRM chambernozzle geometry will be used toaccount all the complex flow fields in our future study
Finally results of the asymptotic and computationalapproaches may provide the rocket designer with codes andinformation making them able to interpret some reasonsbehind the combustion instability inside the combustionchamber of real SRM based on the composition of the solidpropellants and in turn the morphology of the combustionsurface
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The first author would like to thank Professor D R KassoyUniversity of Colorado at Boulder USA for their continuoushelp
References
[1] A Hegab T L Jackson J Buckmaster and D S Stewart ldquoTheburning of periodic sandwich propellantsrdquo in Proceedings of the36th AIAAASMESAEASEE Joint Propulsion Conference andExhibit Huntsville Ala USA July 2000
[2] A Hegab T L Jackson J Buckmaster and D S StewartldquoNonsteady burning of periodic sandwich propellants withcomplete coupling between the solid and gas phasesrdquo Journalof Combustion and Flame vol 125 no 1-2 pp 1055ndash1070 2001
[3] A M Hegab ldquoModeling of sandwich heterogeneous propellantwith two steps chemical kinetics and fluid mechanical effectsrdquoEngineering Research Journal vol 32 no 2 pp 119ndash130 2009
[4] J Buckmaster T L Jackson A Hegab S Kochevets andM Ulrich ldquoRandomly packed heterogeneous propellants andthe flames they supportrdquo in Proceedings of the 39th AerospaceSciences Meeting and Exhibit Reno NV USA January 2001
[5] S Kochevets J Buckmaster T L Jackson and A HegabldquoRandom packs and their use in modeling heterogeneous solidpropellant combustionrdquo Journal of Propulsion and Power vol 17no 4 pp 883ndash891 2001
[6] AMHegab S AHasanienH EMostafa andAMMaradenldquoExperimental and numerical investigations for the combustionof selected composite solid propellantrdquo Engineering ResearchJournal vol 35 no 3 2012
[7] A M Hegab ldquoVorticity generation and acoustic resonance ofsimulated solid rocket motor chamber with high wave numberwall injectionrdquo Computers and Fluids vol 38 no 6 pp 1258ndash1269 2009
[8] A M Hegab and D R Kassoy ldquoInternal flow temperatureand vorticity dynamics due to transient mass additionrdquo AIAAJournal vol 44 no 4 pp 812ndash826 2006
Mathematical Problems in Engineering 11
[9] K Kirkkopru D R Kassoy and Q Zhao ldquoUnsteady vorticitygeneration and evolution in a model of a solid rocket motorrdquoJournal of Propulsion and Power vol 12 no 4 pp 646ndash6541996
[10] K Kirkkopru D R Kassoy Q Zhao and P L Staab ldquoAcousti-cally generated unsteady vorticity field in a long narrow cylinderwith sidewall injectionrdquo Journal of Engineering Mathematicsvol 42 no 1 pp 65ndash90 2002
[11] P L StaabQ ZhaoD R Kassoy andKKirkkopru ldquoCoexistingacoustic-rotational flow in a cylinder with axisymmetric side-wall mass additionrdquo Physics of Fluids vol 11 no 10 pp 2935ndash2951 1999
[12] M Rempe P L Staab and D R Kassoy ldquoThermal response foran internal flow in a cylinderwith time dependent sidewallmassadditionrdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit AIAA 2000-0996 Reno Nev USA January 2000
[13] Q Zhao and D R Kassoy ldquoThe generation and evolutionof unsteady vorticity in a solid rocket engine chamberrdquo inProceedings of the 32th Aerospace Sciences Meeting AIAA paper94-0779 Reno Nev USA 1994
[14] Q Zhao P L Staab D R Kassoy and K Kirkkopru ldquoAcous-tically generated vorticity in an internal flowrdquo Journal of FluidMechanics vol 413 pp 247ndash285 2000
[15] G A Flandro ldquoEffects of vorticity on rocket combustionstabilityrdquo Journal of Propulsion and Power vol 11 no 4 pp 607ndash625 1995
[16] J Majdalani andW K VanMoorhem ldquoMultiple-scales solutionto the acoustic boundary layer in solid rocket motorsrdquo Journalof Propulsion and Power vol 13 no 2 pp 186ndash193 1997
[17] J Majdalani andW van Moorhem ldquoImproved time-dependentflowfield solution for solid rocket motorsrdquo AIAA Journal vol36 no 2 pp 241ndash248 1998
[18] Z Deng R J Adrian and C D Tomkinsm ldquoStructure ofturbulence in channel flow with a fully transpired wallrdquo inProceedings of the 39th Aerospace Science Meeting AIAA Paper2001-1019 Reno Nev USA 2001
[19] F Vuillot andG Avalon ldquoAcoustic boundary layers in solid pro-pellant rocketmotors usingNavier-Stokes equationsrdquo Journal ofPropulsion and Power vol 7 no 2 pp 231ndash239 1991
[20] T M Smith R L Roach and G A Flandro ldquoNumerical studyof the unsteady flow in a simulated solid rocket motorrdquo inProceedings of the 31th Aerospace Science Meeting AIAA Paper93-0112 Reno Nev USA 1993
[21] C F Tseng and I S Tseng ldquoInteraction between acoustic wavesand premixed flames in porous chamberrdquo in Proceedings of the32th Aerospace Science Meeting Reno Nev USA 1994 AIAAPaper 94 3328
[22] T S Roh and V Yang ldquoTransient combustion responses ofsolid propellants to acoustic disturbances in rocket motorsrdquo inProceedings of the 33rd Aerospace Sciences Meeting and ExhibitAIAA Paper 95-0602 Reno Nev USA 1995
[23] A M Hegab A study of acoustic phenomena in solid rocketengines [PhD thesis] Mechanical Power Engineering Depart-ment Menoufia University Menufia Egypt University of Col-orado Boulder Colo USA 1998
[24] R W MacCormack ldquoA numerical method for solving theequations of compressible viscous flowrdquo American Institute ofAeronautics and Astronautics Journal vol 20 no 9 pp 1275ndash1281 1982
[25] D Gottlieb and E Turkel ldquoDissipative two-four methods fortime-dependent problemsrdquo Mathematics of Computation vol30 no 136 pp 703ndash723 1976
[26] T J Poinsot and S K Lele ldquoBoundary conditions for directsimulations of compressible viscous flowsrdquo Journal of Compu-tational Physics vol 101 no 1 pp 104ndash129 1992
[27] Z Deng R J Adrian and C D Tomkins ldquoSensitivity ofturbulence in transpired channel to injection velocity small-scale nonuniformityrdquo AIAA Journal vol 40 no 11 pp 2241ndash2246 2002
[28] K J Meteb Prediction of hybrid rocket motor regression rate[MS thesis] Military Technical College Cairo Egypt 2009
[29] J W Anders V Magi and J Abraham ldquoLarge-eddy simulationin the near-field of a transient multi-component gas jet withdensity gradientsrdquoComputers amp Fluids vol 36 no 10 pp 1609ndash1620 2007
[30] B Chaouat ldquoNumerical simulations of channel flows with fluidinjection using Reynolds stressmodelrdquo AIAAPaper 2000-09922000
[31] W A El-Askary A Balabel S M El-Behery and A HegabldquoComputations of a compressible turbulent flow in a rocketmotor-chamber configuration with symmetric and asymmetricinjectionrdquoComputerModeling in EngineeringampSciences vol 82no 1 pp 29ndash54 2011
[32] A Balabel A M Hegab M Nasr and S M El-Behery ldquoAssess-ment of turbulence modeling for gas flow in two-dimensionalconvergent-divergent rocket nozzlerdquo Applied MathematicalModelling vol 35 no 7 pp 3408ndash3422 2011
10 Mathematical Problems in Engineering
simulated solid propellant rocket chamber with transientsidewall mass addition at different wave numbers
4 Conclusion Remarks
The current study emphasizes the effect of the combustionsurfacemorphology of the solid rocket systemon the acousticinstability and the accompanying vorticity and temperaturedynamics generation The rotational and irrotational fieldsgenerated in the prescribed solid rocket motor chamber withMach number are chosen to be O(10minus1) while the charac-teristic Reynolds number is O(105-106) at nonresonance andnear and at resonance frequencies have been described TheNSCBC techniques that applied in [7 29] are used to treat thenumerical boundary conditions
The numerical solution shows that the convergence tosteady state solution is achieved first and then additionaltransient side wall mass addition is initiated Moreover theresult shows that the forced mode of the nonresonance andresonance frequencies generates longitudinal acoustics waveswhich interact again with its sources at the injection surfaceto generate the rotational flow fields The generated unsteadyvorticity is transferred from the injection surface toward thecenterline to fill the entire chamber as time increases aftermore than seven wave cycles The solution shows nonpureharmonic oscillations as suggested from the theory in [11]Vorticity 3D graphs and contours illustrate that spatially themaximum amplitudes are found to be near the injectionsurface Moreover the largest shear stresses are found to be atthe injection surface near the exit end while the shear stressesimpact the injection surface at many locations as the wavenumber increasesThe longitudinal variation of the rotationalflow fields is found to be nonmonotonic for 119899 gt 1
Here for the first time the analytical and computationalresults for the current coldmodel and the recent experimentalwork by Meteb [28] revealed that the combustion surfacemorphology has a great effect on the acoustic instabilityinside the rocket chamber and the accompanying rotationalflow The computational approach reveals that there areunexpectedly high heat transfer and generated rotational flowfields found to be adjacent to the injection surface in bothnonresonance and resonance cases The amplitude of thelatter one is found to be approximately ten times than thatwith the nonresonance solution One may anticipate thatthese large oscillatory gradients will persist to the surface ofa burning solid propellant in a real rocket chamber
More traditional computational studies of turbulence inchannel flows with wall injection are reviewed by Chaouat[30] Mean axial velocity profiles are found to be laminaruntil about halfway down the channel and then show signs oftransition Chaouatrsquos results do not show the acoustic-relatedeffects discussed in the present work As a result it is difficultto know from the turbulent theory results if the source ofthe downstream vorticity is classical transition to turbulenceor partly the result of the kind of mechanism discussed inthis paper In our recent studies in 2011 and 2012 [31 32]the assessment of turbulence modeling for gas flow in thechamber and in the convergent-divergent rocket nozzle for
the steady state flow is studied The detailed information andrealities in the current research about the transient coexistingof the irrotational (acoustics) rotational (generated vorticityfrom the interaction between the injected flow and thetraveling acoustic waves) and our turbulent techniques thatwe have adapted in our studies in 2011 and 2012 for the steadyflow field in SRM chambernozzle geometry will be used toaccount all the complex flow fields in our future study
Finally results of the asymptotic and computationalapproaches may provide the rocket designer with codes andinformation making them able to interpret some reasonsbehind the combustion instability inside the combustionchamber of real SRM based on the composition of the solidpropellants and in turn the morphology of the combustionsurface
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The first author would like to thank Professor D R KassoyUniversity of Colorado at Boulder USA for their continuoushelp
References
[1] A Hegab T L Jackson J Buckmaster and D S Stewart ldquoTheburning of periodic sandwich propellantsrdquo in Proceedings of the36th AIAAASMESAEASEE Joint Propulsion Conference andExhibit Huntsville Ala USA July 2000
[2] A Hegab T L Jackson J Buckmaster and D S StewartldquoNonsteady burning of periodic sandwich propellants withcomplete coupling between the solid and gas phasesrdquo Journalof Combustion and Flame vol 125 no 1-2 pp 1055ndash1070 2001
[3] A M Hegab ldquoModeling of sandwich heterogeneous propellantwith two steps chemical kinetics and fluid mechanical effectsrdquoEngineering Research Journal vol 32 no 2 pp 119ndash130 2009
[4] J Buckmaster T L Jackson A Hegab S Kochevets andM Ulrich ldquoRandomly packed heterogeneous propellants andthe flames they supportrdquo in Proceedings of the 39th AerospaceSciences Meeting and Exhibit Reno NV USA January 2001
[5] S Kochevets J Buckmaster T L Jackson and A HegabldquoRandom packs and their use in modeling heterogeneous solidpropellant combustionrdquo Journal of Propulsion and Power vol 17no 4 pp 883ndash891 2001
[6] AMHegab S AHasanienH EMostafa andAMMaradenldquoExperimental and numerical investigations for the combustionof selected composite solid propellantrdquo Engineering ResearchJournal vol 35 no 3 2012
[7] A M Hegab ldquoVorticity generation and acoustic resonance ofsimulated solid rocket motor chamber with high wave numberwall injectionrdquo Computers and Fluids vol 38 no 6 pp 1258ndash1269 2009
[8] A M Hegab and D R Kassoy ldquoInternal flow temperatureand vorticity dynamics due to transient mass additionrdquo AIAAJournal vol 44 no 4 pp 812ndash826 2006
Mathematical Problems in Engineering 11
[9] K Kirkkopru D R Kassoy and Q Zhao ldquoUnsteady vorticitygeneration and evolution in a model of a solid rocket motorrdquoJournal of Propulsion and Power vol 12 no 4 pp 646ndash6541996
[10] K Kirkkopru D R Kassoy Q Zhao and P L Staab ldquoAcousti-cally generated unsteady vorticity field in a long narrow cylinderwith sidewall injectionrdquo Journal of Engineering Mathematicsvol 42 no 1 pp 65ndash90 2002
[11] P L StaabQ ZhaoD R Kassoy andKKirkkopru ldquoCoexistingacoustic-rotational flow in a cylinder with axisymmetric side-wall mass additionrdquo Physics of Fluids vol 11 no 10 pp 2935ndash2951 1999
[12] M Rempe P L Staab and D R Kassoy ldquoThermal response foran internal flow in a cylinderwith time dependent sidewallmassadditionrdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit AIAA 2000-0996 Reno Nev USA January 2000
[13] Q Zhao and D R Kassoy ldquoThe generation and evolutionof unsteady vorticity in a solid rocket engine chamberrdquo inProceedings of the 32th Aerospace Sciences Meeting AIAA paper94-0779 Reno Nev USA 1994
[14] Q Zhao P L Staab D R Kassoy and K Kirkkopru ldquoAcous-tically generated vorticity in an internal flowrdquo Journal of FluidMechanics vol 413 pp 247ndash285 2000
[15] G A Flandro ldquoEffects of vorticity on rocket combustionstabilityrdquo Journal of Propulsion and Power vol 11 no 4 pp 607ndash625 1995
[16] J Majdalani andW K VanMoorhem ldquoMultiple-scales solutionto the acoustic boundary layer in solid rocket motorsrdquo Journalof Propulsion and Power vol 13 no 2 pp 186ndash193 1997
[17] J Majdalani andW van Moorhem ldquoImproved time-dependentflowfield solution for solid rocket motorsrdquo AIAA Journal vol36 no 2 pp 241ndash248 1998
[18] Z Deng R J Adrian and C D Tomkinsm ldquoStructure ofturbulence in channel flow with a fully transpired wallrdquo inProceedings of the 39th Aerospace Science Meeting AIAA Paper2001-1019 Reno Nev USA 2001
[19] F Vuillot andG Avalon ldquoAcoustic boundary layers in solid pro-pellant rocketmotors usingNavier-Stokes equationsrdquo Journal ofPropulsion and Power vol 7 no 2 pp 231ndash239 1991
[20] T M Smith R L Roach and G A Flandro ldquoNumerical studyof the unsteady flow in a simulated solid rocket motorrdquo inProceedings of the 31th Aerospace Science Meeting AIAA Paper93-0112 Reno Nev USA 1993
[21] C F Tseng and I S Tseng ldquoInteraction between acoustic wavesand premixed flames in porous chamberrdquo in Proceedings of the32th Aerospace Science Meeting Reno Nev USA 1994 AIAAPaper 94 3328
[22] T S Roh and V Yang ldquoTransient combustion responses ofsolid propellants to acoustic disturbances in rocket motorsrdquo inProceedings of the 33rd Aerospace Sciences Meeting and ExhibitAIAA Paper 95-0602 Reno Nev USA 1995
[23] A M Hegab A study of acoustic phenomena in solid rocketengines [PhD thesis] Mechanical Power Engineering Depart-ment Menoufia University Menufia Egypt University of Col-orado Boulder Colo USA 1998
[24] R W MacCormack ldquoA numerical method for solving theequations of compressible viscous flowrdquo American Institute ofAeronautics and Astronautics Journal vol 20 no 9 pp 1275ndash1281 1982
[25] D Gottlieb and E Turkel ldquoDissipative two-four methods fortime-dependent problemsrdquo Mathematics of Computation vol30 no 136 pp 703ndash723 1976
[26] T J Poinsot and S K Lele ldquoBoundary conditions for directsimulations of compressible viscous flowsrdquo Journal of Compu-tational Physics vol 101 no 1 pp 104ndash129 1992
[27] Z Deng R J Adrian and C D Tomkins ldquoSensitivity ofturbulence in transpired channel to injection velocity small-scale nonuniformityrdquo AIAA Journal vol 40 no 11 pp 2241ndash2246 2002
[28] K J Meteb Prediction of hybrid rocket motor regression rate[MS thesis] Military Technical College Cairo Egypt 2009
[29] J W Anders V Magi and J Abraham ldquoLarge-eddy simulationin the near-field of a transient multi-component gas jet withdensity gradientsrdquoComputers amp Fluids vol 36 no 10 pp 1609ndash1620 2007
[30] B Chaouat ldquoNumerical simulations of channel flows with fluidinjection using Reynolds stressmodelrdquo AIAAPaper 2000-09922000
[31] W A El-Askary A Balabel S M El-Behery and A HegabldquoComputations of a compressible turbulent flow in a rocketmotor-chamber configuration with symmetric and asymmetricinjectionrdquoComputerModeling in EngineeringampSciences vol 82no 1 pp 29ndash54 2011
[32] A Balabel A M Hegab M Nasr and S M El-Behery ldquoAssess-ment of turbulence modeling for gas flow in two-dimensionalconvergent-divergent rocket nozzlerdquo Applied MathematicalModelling vol 35 no 7 pp 3408ndash3422 2011
Mathematical Problems in Engineering 11
[9] K Kirkkopru D R Kassoy and Q Zhao ldquoUnsteady vorticitygeneration and evolution in a model of a solid rocket motorrdquoJournal of Propulsion and Power vol 12 no 4 pp 646ndash6541996
[10] K Kirkkopru D R Kassoy Q Zhao and P L Staab ldquoAcousti-cally generated unsteady vorticity field in a long narrow cylinderwith sidewall injectionrdquo Journal of Engineering Mathematicsvol 42 no 1 pp 65ndash90 2002
[11] P L StaabQ ZhaoD R Kassoy andKKirkkopru ldquoCoexistingacoustic-rotational flow in a cylinder with axisymmetric side-wall mass additionrdquo Physics of Fluids vol 11 no 10 pp 2935ndash2951 1999
[12] M Rempe P L Staab and D R Kassoy ldquoThermal response foran internal flow in a cylinderwith time dependent sidewallmassadditionrdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit AIAA 2000-0996 Reno Nev USA January 2000
[13] Q Zhao and D R Kassoy ldquoThe generation and evolutionof unsteady vorticity in a solid rocket engine chamberrdquo inProceedings of the 32th Aerospace Sciences Meeting AIAA paper94-0779 Reno Nev USA 1994
[14] Q Zhao P L Staab D R Kassoy and K Kirkkopru ldquoAcous-tically generated vorticity in an internal flowrdquo Journal of FluidMechanics vol 413 pp 247ndash285 2000
[15] G A Flandro ldquoEffects of vorticity on rocket combustionstabilityrdquo Journal of Propulsion and Power vol 11 no 4 pp 607ndash625 1995
[16] J Majdalani andW K VanMoorhem ldquoMultiple-scales solutionto the acoustic boundary layer in solid rocket motorsrdquo Journalof Propulsion and Power vol 13 no 2 pp 186ndash193 1997
[17] J Majdalani andW van Moorhem ldquoImproved time-dependentflowfield solution for solid rocket motorsrdquo AIAA Journal vol36 no 2 pp 241ndash248 1998
[18] Z Deng R J Adrian and C D Tomkinsm ldquoStructure ofturbulence in channel flow with a fully transpired wallrdquo inProceedings of the 39th Aerospace Science Meeting AIAA Paper2001-1019 Reno Nev USA 2001
[19] F Vuillot andG Avalon ldquoAcoustic boundary layers in solid pro-pellant rocketmotors usingNavier-Stokes equationsrdquo Journal ofPropulsion and Power vol 7 no 2 pp 231ndash239 1991
[20] T M Smith R L Roach and G A Flandro ldquoNumerical studyof the unsteady flow in a simulated solid rocket motorrdquo inProceedings of the 31th Aerospace Science Meeting AIAA Paper93-0112 Reno Nev USA 1993
[21] C F Tseng and I S Tseng ldquoInteraction between acoustic wavesand premixed flames in porous chamberrdquo in Proceedings of the32th Aerospace Science Meeting Reno Nev USA 1994 AIAAPaper 94 3328
[22] T S Roh and V Yang ldquoTransient combustion responses ofsolid propellants to acoustic disturbances in rocket motorsrdquo inProceedings of the 33rd Aerospace Sciences Meeting and ExhibitAIAA Paper 95-0602 Reno Nev USA 1995
[23] A M Hegab A study of acoustic phenomena in solid rocketengines [PhD thesis] Mechanical Power Engineering Depart-ment Menoufia University Menufia Egypt University of Col-orado Boulder Colo USA 1998
[24] R W MacCormack ldquoA numerical method for solving theequations of compressible viscous flowrdquo American Institute ofAeronautics and Astronautics Journal vol 20 no 9 pp 1275ndash1281 1982
[25] D Gottlieb and E Turkel ldquoDissipative two-four methods fortime-dependent problemsrdquo Mathematics of Computation vol30 no 136 pp 703ndash723 1976
[26] T J Poinsot and S K Lele ldquoBoundary conditions for directsimulations of compressible viscous flowsrdquo Journal of Compu-tational Physics vol 101 no 1 pp 104ndash129 1992
[27] Z Deng R J Adrian and C D Tomkins ldquoSensitivity ofturbulence in transpired channel to injection velocity small-scale nonuniformityrdquo AIAA Journal vol 40 no 11 pp 2241ndash2246 2002
[28] K J Meteb Prediction of hybrid rocket motor regression rate[MS thesis] Military Technical College Cairo Egypt 2009
[29] J W Anders V Magi and J Abraham ldquoLarge-eddy simulationin the near-field of a transient multi-component gas jet withdensity gradientsrdquoComputers amp Fluids vol 36 no 10 pp 1609ndash1620 2007
[30] B Chaouat ldquoNumerical simulations of channel flows with fluidinjection using Reynolds stressmodelrdquo AIAAPaper 2000-09922000
[31] W A El-Askary A Balabel S M El-Behery and A HegabldquoComputations of a compressible turbulent flow in a rocketmotor-chamber configuration with symmetric and asymmetricinjectionrdquoComputerModeling in EngineeringampSciences vol 82no 1 pp 29ndash54 2011
[32] A Balabel A M Hegab M Nasr and S M El-Behery ldquoAssess-ment of turbulence modeling for gas flow in two-dimensionalconvergent-divergent rocket nozzlerdquo Applied MathematicalModelling vol 35 no 7 pp 3408ndash3422 2011
