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AIAA JOURNALVol. 38, No. 7, July 2000

Computational Fluid Dynamics Algorithms for UnsteadyShock-Induced Combustion, Part 1: Validation

Jeong-Yeol Choi

Pusan National University, Pusan 609-735, Republic of Korea

and

In-Seuck Jeung and Youngbin Yoon

Seoul National University, Seoul 151-742, Republic of Korea

A computationalstudy is carried out to develop a fully implicit and time-accurate computationaluid dynamics

code for the analysis of reactive ow systems. Periodically oscillating shock-induced combustion around a blunt

body in a stoichiometric hydrogenair mixture is used as a validation problem of examining various numerical

considerations. Euler equations and species conservation equations are used as the governing equations for the

chemically reacting ow. Spatial discretization of the g overning equation is based on Roes approximateRiemann

solver with a MUSCL-type total variation diminishing scheme for higher-order spatial resolution. The second-

order-accurate timeintegrationmethodis based on a loweruppersymmetric GaussSeidelscheme, using a Newtonsubiteration method and StegerWarming ux Jacobian splitting. As a rst step of the validation procedure,

simulations of experimental results were carried out to conrm the reliability of the baseline method. In t he next

step, the general aspects of the baseline method were examined, including order of time integration, number of

subiterations, and use of approximate ux Jacobian splitting. Appropriateness of the grid system is also examined

by using a grid renement study.

Introduction

S HOCK-INDUCED combustion is the self-ignited combustionphenomena of a premixed gas induced by the ight of a hyper-velocity projectile in a combustible gas mixture. It has gained pub-lic interest as a promising combustion mechanism for hypersonicpropulsion devices such as oblique detonation wave engine1 andram accelerators.2 In additionto much experimental and theoreticalresearch in the 1960s and 1970s, a computational uid dynamicsapproach has been used as a major research tool of the phenomenain the 1980s and 1990s, with developments in numerical methodsand computational power.

The shock-induced combustion oweld is characterizedby thehypersonic range ow velocity and the nite rate exothermicchem-istry behind the bow shock. Thus, the coupling and the interactionbetween the bow shock and the reaction wave show the variousand distinctive ow features according to the chemical and uiddynamic conditions. Among the various features of shock-inducedcombustion, a periodically unstable regime would be the most in-

teresting due to its naturally oscillating characteristics. Figure 1is the shadowgraph of the unstable phenomena taken by Lehr3 inthe 1960s. After the experimentalobservationsin 1960s and 1970s,manyresearchersshowed theirinterestin the phenomena.The oscil-lating mechanism of shock-induced combustion was suggested byToongs4 in the 1970s and was proved numerically by Matsuo andFujiwara5 and Matsuo et al.6 in 1990s as a wave interactionmecha-nism between the shock wave, combustion wave, compression andexpansionwaves, and projectile surface.

For the analysis of the unsteady combustion phenomena, a sta-ble and time-accurate numerical approach is required to capturethe strong bow shock and weak pressure waves without numeri-

Presented as Paper 98-3217 at the AIAA/ASME/SAE/ASEE 34th JointPropulsion Conference, Cleveland, OH, 13 15 July 1998; received 14 Jan-uary 1999; revision received 13 December 1999; accepted for publication

29 December 1999. Copyright c 2000 by the authors. Published by theAmerican Institute of Aeronautics and Astronautics, Inc., with permission.

Assistant Professor, Department of Aerospace Engineering. Member

AIAA. Professor, Department of Aerospace Engineering. Senior Member

AIAA.

cal instability or distortion because the phenomena is the result ofwave interactions between the bow shock, combustion wave, andbody surface. In addition, a detailed combustion mechanism thatincludes a sufcient numberof chemical species and elementaryre-actionstepsshould be used to reproduceaccuratelythe experimental

resultsof shockstandoffdistance,inductiondistance,and oscillationfrequencybecause these characteristicsare determinedby niteratereaction mechanisms. The inclusion of more species results in theincrease of the variables proportionalto the number of species andthe increase of the matrix inversion time proportional to the cubicnumberof equations.Moreover,a hugegridsystem is alsoneededtoresolve the wave interactionsbetween bow shock and body surface.Therefore, enormously large memory capacity and computing timeseem to be inevitable for the analysis of the unsteady combustionphenomena. The reliable simulation results of the oscillating com-bustion phenomena began to appear in the 1990s with the help ofdevelopments in numerical methods and computational power.5 7

At present, the development of computing power makes the simu-lation possible on a desktop computer, and the numerical approachis used in the analysis of the unknown shock-induced combustionphenomena for which the experimental or analytic methods do notapply adequately.5,6

During the pastseveral years,many researchersperformedsimu-lations of experimentallyknown shock-inducedcombustionand re-producedthe experimental result with sufcient accuracy.5 9 How-ever, there are some discrepancies between their simulations andexperimental results because they used different numerical meth-ods and computationalconditions.They explainedthe discrepanciesby the inaccuracies of the chemistry mechanisms, numerical meth-ods, etc. However, the huge computational expense of the analysispreventedthe needed validationalor comparativestudiesbefore us-ing the computational methods in the analysisof unknown physicalproblems.

In this study, a comparativestudy of numerical methods is madeas a part of the process of developing an efcient computationaluid dynamics code capable of analyzing unsteady reacting ows.A time-accurate fully implicit algorithm and a third-orderspatiallyaccurate upwind scheme are presented as baseline methods. Val-idation of these baseline methods is made with the simulation ofexperimental results, and then the results from different numericalapproaches are compared with those of the baseline method.

1179

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1180 CHOI, JEUNG, AND YOON

Fig. 1 Experimental shadowgraph ofperiodically oscillating shock-induced

combustion.3

Governing Equation and Chemistry Mechanism

The oscillatory phenomena in shock-inducedcombustionare ba-sically due to the interactionbetween the shock and the combustionwave. The effect of transport properties is known to be negligible.4

Therefore, species conservationequations and inviscid Euler equa-tions are used as governingequationsof the reactingow aroundanaxisymmetric bluntbody.The conservationformof these governingequationsset involvingNnumber of speciesis written in curvilinearcoordinate form as follows:

@Q

@t+

@F

@n+

@G

@g+ H =W (1)

where

Q =1

J

26666664

q1...qN

qu

qv

e

37777775

, F =1

J

2666666664

q1U

.

..

qNU

quU +nxp

qvU + nyp

(e + p)U

3777777775

G =1

J

2666666664

q1 V...

qNV

qu V + gxp

qv V + gyp

(e + p)V

3777777775

, H =1

y J

266666664

q1v

.

..qNv

quv

qv 2

(e + p)v

377777775

W =1

J

26666664

w 1...

wN

0

0

0

37777775(2)

Here,u and v are the velocity components in Cartesian coordinatesx and y , andU andV are the contravariantvelocity components inthe generalized coordinates:

U =nx u +ny v , V =gx u + gy v (3)

Coordinate transformation metrics nx , ny , gx , and gy and metricJacobian J are obtained from coordinate transform relation as fol-lows:

J =1/(xnyg xgyn) (4)

nx =J yg, ny =J xg, gx =J yn , gy =J xn (5)

Total density qis expressed as a sum of the partial densityqkofeach species:

q =

NXk=1

qk (6)

Pressure p is evaluated from the ideal gas law for a mixture ofthermallyperfectgases, and totalenergy per unitvolume e isdenedas a sumof kineticenergyand internalenergy.The denitionof totalenergy is used for the implicit evaluation of temperature T by theNewtonRaphson iteration method:

p =

N

Xk =1

qk

MkRT (7)

e =q

2(u2 + v 2 ) +

NXk =1

qk

Z T

Cpk R

MkdT + h 0k

! (8)

Here,Mkis themolecularweightfor kth speciesandR is the univer-sal gas constant. The specic heats of each species are obtained asfunctions of temperature from NASA thermochemical polynomialdata that are valid up to 6000 K (Ref. 10):

Cpk/R =a1k +a 2kT + a3kT2+ a 4kT

3+ a5kT

4 (9)

For the elementary reaction steps generally expressed as

NXk =1

m0k,rXk$

NXk =1

m00k,rXk (10)

the mass production rate of each species in Eq.(2)is written as

w k =Mk

N rXr =1

(m00k,r m0k,r)

"kfr

NYk =1

qk

Mk

m0k,r

kbr

NYk =1

qk

Mk

m00k,r

#

(11)

where forward reaction rate constantkfr for therth reaction step isexpressed in Arrhenius form:

kfr =ArTBr expEr R T (12)

Thereactioncoefcients Ar, Br, andactivationenergyEr inEq. (12)

are taken from Jachimowskis detailed chemistry mechanisms forhydrogen air combustion11 (see Table 1). The original mechanismincludesthe oxidationprocess of nitrogen,but the reaction steps are

Table 1 Jachimowskis mechanisms for hydrogenair combustion

without nitrogen oxidation processa

Reaction Ar Br Er

Forward reaction rate coefcients

H2 + O2 $HO2 + H 1.00 1014 0.00 56,000

H + O2 $OH + O 2.60 1014 0.00 16,800

O + H2 $OH + H 1.80 1010 1.00 8,900

OH + H2 $H2 O+ H 2.20 1013 0.00 5,150OH + OH $H2O + O 6.30 10

12 0.00 1,090

H + OH + X$ H2O + X 2.20 1022 2.00 0

H + H + X$ H2 + X 6.40 1017 1.00 0

H + O + X$ OH + X 6.00 1016 0.60 0

H + O2 + X$ HO2 + X 2.10 1015 0.00 1,000

HO2 + H $OH + OH 1.40 1014 0.00 1,080

HO2 + H $H2 O+ O 1.00 1013 0.00 1,080

HO2 + O $O2 + OH 1.50 1013 0.00 950

HO2 + OH $H2 O + O2 8.00 1012 0.00 0

HO2 $H2 O2 + O2 2.00 1012 0.00 0

H + H2 O2 $H2 + HO2 1.40 1012 0.00 3,600

O + H2 O2 $OH + HO2 1.40 1013 0.00 6,400

OH + H2 O2 $H2O + HO2 6.10 1012 0.00 1,430

H2 O2+

X$ OH+

OH+

X 1.20 1017

0.00 45,500O + O + X$ O2 + X 6.00 1013 0.00 1,800

Third-body efciencies relative to N2H + OH + X$ H2O + X H2O = 6.0

H + H + X$ H2 + X H2 O = 6.0, H2 = 2.0H + O + X$ OH + X H2O = 6.0

H + O2 + X$ HO2 + X H2 O = 6.0, H2 = 2.0H2 O2 + X$ H + OH + X H2 O = 15.0

aUnitsof reaction constants are seconds,moles, cubiccentimeters,calories, andKelvin.

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CHOI, JEUNG, AND YOON 1181

neglected and nitrogen is assumed to be an inert species becausethe oxidation process does not have a signicant effect on the uiddynamics of shock-induced combustion. This combustion modelhas been successfullyused and validated in a number of supersonicreacting ow studies.5,7,8,11

Backward reaction rate constant kbr in Eq.(11)is evaluated fromthe forward reaction rate constantand equilibriumconstant:

Kreq =kfr/kbr (13)

The equilibrium constant is calculated from the Gibbs free-energyminimum condition as follows:

Kreq =

1atm

RT

PNk=1

(m00k,r

m0k,r

)

exp

" NXk =1

(m00k,r m0k,r)

S0

k

R

Hk

R T

#

(14)

Here, entropyS0k

and enthalpyHkare obtainedas functions of tem-perature from the specic heat data at standard state:

Hk

R T =

1

RT

(Z T

Cpkd T + H0f k

) (15)

S0k

R=

Z T

dHk

R T=

Z T

CpkdT

RT(16)

Baseline Numerical Method

Time-Accurate Implicit Integration with Newton Subiteration

A fully implicit second-order-accurate time-integration methodis used for the analysis of an unsteady supersonic reacting ow.A Newton subiteration method is used to reduce the error of tem-poral discretization and to ensure the second-order time accuracyand stability in the case of a large time step equivalent to Courant

FriedrichsLewy (CFL) number greater than one. By the takingof the second-orderap proximationof time derivative and applyingNewtons method to evaluateconservativevariablesat the next timestep, the governingequation(1) could be written in discretizedformas follows:

c0I

@R

@Q

n + 1, m

i,j

DQn + 1, m

i,j =RHS

n + 1, m

i,j

RHSn + 1, m

i, j =c0Q

n + 1, m

i,j c1Q

ni, j c2Q

n 1i,j

+ RQn + 1, mi,j (17)where the residual vector based on nite volume discretization is

Ri, j =Wi,j Hi,j Fi + 12,j

+ Fi 12,j

Gi,j + 12+ Gi, j 12

(18)

Conservativevariables are updated from the evaluated increment atevery time-integrationlevel and subiterationstep:

Qn + 1, m + 1

i, j =Q

n + 1, m

i,j + DQ

n + 1, m

i, j (19)

In this time-integration scheme, n is the designation of time-integrationleveland mis that of the sub-iterationstep. Time march-

ing is performed by assuming that the initial guess of the conserva-tive variable at n +1 time levelQn + 1, 0 is Qn . In Eq.(17)c0,c1, andc2 are the functions of time step sizeD t

n , which is dened as

D tn =tn + 1 tn (20)

This time-integrationsche me is second-order-accuratefor arbitrarytime step size by selecting c0,c1 , andc2 as

c0 =1 r

(1 r)Dtn + Dtn 1, c1 =

r

(1 r)D tn + D tn 1

c2 =1

(1 r)Dtn + Dtn 1, r =

1 +

D tn 1

D tn

2

(21)

Equation(21

)can be rst-order accurate by choosing

c0 =1/D tn , c1 =c0, c2 =0 (22)

TheimplicitpartofEq. (17) constitutesa blockpentadiagonalsys-tem of equations. Alternating direction implicit schemes based onapproximate factorization are often used for solving these kinds ofpentadiagonalsystems, but their efciency degrades severely whenthe numberof equationsis verylarge. Thus, a lower upper (LU) re-laxation scheme12 is used in this study that involves only pointwisematrix inversion. By applying the upwind discretizationo f ux Ja-cobianmatrices,Eq.(17) can be rewritten in the followingfactorizedform:

D =c0I+ (A+ A + B+ B Z)i, j

L =D A+i 1, j B

+

i,j 1 , U =D + Ai +1, j

+B i,j +1

LD1UDQi, j =RHSi,j (23)

Here A and B are split Jacobian matrices of ux vectors F a ndG, and Z is the Jacobian matrix of chemical source vector W. Thepreceding LU-factored governing equation is actually inverted bythe followingtwo stepsof the symmetricGaussSeidel methodwitha well-knownLU-decomposition method for linear algebraic equa-tions:

Di,jDQi,j

=RHSi,j +A+

i 1, jDQ

i 1,j + B +

i,j 1DQ

i,j 1

Di,jDQi,j =A

i +1, j

DQn + 1, m

i + 1, j B

i,j 1DQ

n + 1, m

i,j 1

DQn + 1, m

i, j =DQi,j + DQ

i, j

(24)

Among several choices available in evaluating a split uxJacobian matrix, the Steger Warming ux splitting method andan approximate splitting method are used in this study. Becausethese methods involve only a pointwise Jacobian matrix and do not

involve neighboring points, evaluation of the left- and right-handsides(RHS)in Eq.(24)could be carried out in a much more sim-plied manner by the following manipulation of the ux Jacobianmatrix.

IncaseoftheSteger Warmingapproach,the ux Jacobianmatrixis split as

A=RK

R1, K

=(K jKj)/2 (25)

Here,Ran dR 1 are the left and right eigenvector matrix and K isthe eigenvector havingk1,k2, andk3 as different eigenvalues.Theeigenvaluesare dened as

k1 =U, k2,3 =U aqn2

x + n 2

y (26)

The ux Jacobian matrix is reconstructedin the following form:

A =RKR1 =A0 + k1I (27)

where

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A0 =266666666666666664

pq1yk

a2u

ykU

aw

peu

a2yku +

nx

aw

p e v

a2yku +

ny

aw

pe

a2yku

upq1

a2 nx Uu +

nxpq1 u U

a w n2

x pe u

2

a2 u + (2 c)u

anxw nxny

pe uv

a2 u + (nx v nype u)

aw

peu

a2u +

penx

aw

v pq1

a2 ny U u +

nypq1 v U

a

w

nxny

peuv

a2u +

(nx v nypeu)

aw

n2

y pev

2

a2u +

ny v (2 c)

aw

pe v

a2u +

peny

aw

H pq1

a2 U2 u +

U

pq1 H

aw

Unx

up eH

a2u +

(nxH u peU)

aw

Uny

v peH

a2u +

(nyH v peU)

aw

peH

a2 u +

peU

aw

377777777777777775

k =1, . . . , N and l =1, . . . ,N (28)

u =[(k2 + k3)/2] k1 , w =(k2 + k3)/2 (29)

Here, pe a nd pqk are the derivativesof pressurewith respect to totalenergy and partial density. All of the metrics in Eq.(29)are usedafter normalization.

With this reconstructed form of the ux Jacobian matrix, thedifference term of the Jacobian matrix in the diagonal matrix ofEq. (24) is simply evaluatedby replacingthe eigenvaluesin Eqs. (27)

and(29)by their absolute values:

A+ A =RjKjR 1 =A0 (jk1,2,3j) + jk1jI (30)

More benecially, the matrix and vector product terms in Eq.(24)

can be evaluated by the following formulation in vector form:

ADQ =A(jk1,2,3 j)DQ + jk1jDQ

=

2666666666666666664

y1E + F + jk1j...

yNE + F + jk1juu

a2+ nxw

aA +

unx +

uw

aB + jk1j

vu

a2+ nyw

aA +

uny +

vw

aB + jk1j

Hua

2+

Uw

aA + uU + Hw

aB +jw1 j

3777777777777777775

A =

NXk =1

pqkDqk + pe( uDqn1 vDqn2 + Dqn3)

D =U

NXk =1

Dqk, C =nxDqn1 + nyDqn 2

B =nxDqn1 +nyDqn2 U

NXk =1

Dqk =C D

E =u

a2A

w

aD, F =

w

aC (31)

This matrix vector product formulation is an exact one and savesa lot of computing time because this formulation is much moreefcient than the matrix vector product calculation.

MUSCL-Type Total Variation Diminishing Scheme Based

on Roes Flux Difference Splitting

The nitevolumecell-vertexscheme is usedfor spatialdiscretiza-tion of governing equations. The convective terms in Eq.(18)areexpressed as differencesof numerical uxes at cell interfaces.The

numerical uxes containing articial dissipationare formulated us-ing Roes ux difference splitting (FDS)method13:

Fi + 12

, j =1

2[F(QR ) +F(QL ) jA(QR , QL )j(QR QL )] (32)

where subscript L and R imply the extrapolated values at the leftand right grid points of cell interface.A(QR , QL ) implies that theJacobian matrix A is evaluated by the Roes average ofQR andQL . Although there are various derivations of Roes FDS methodfor multispecies chemically reacting ow such as HartenYee to-tal variation diminishing (TVD)schemes,14 and these methods alsoshow good shock-capturingcharacteristics, the characteristic vari-able approachof Grossman and Cinnella15 is used in this study andextended to two-dimensional curvilinear coordinates because onlythe characteristic variable approach satises the denition of theapproximate Riemann solver and presents numerical exibility ofarticial damping control. By following the primitive variable ap-

proach used by Grossman and Cinnella15 for the interface Jacobianmatrix, the articial dissipation term for chemically reacting owcan be written in two-dimensionalcurvilinearcoordinateas follows:

jA( Q)jdQ =RjKjR 1dQ =

4Xl =1

jdFjldw l (33)

where

jdFj1 =

26666666664

y1...

yN

u

v

H a2

pe

37777777775

jdFj2 =

266666666664

dy1...

dyN

du nxdU

dv nydV

(nxdv nydu)(nx v ny u) 1

pe

N

Xk =1pqkdyk

377777777775

jdFj3,4 =

266666664

y1...

yN

u nx a

v ny a

H a U

377777775

(34)

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dw1 =w(k)

dq

dp

a2, dw 2 =w(k1) q

dw3,4 =w(k2,3 )

dp qadU

2a2

(35)

where()implies the Roes averaged value at cell interface and d( )means the difference of the extrapolated valuesat the left and rightgrid points. Grid metrics nx and ny are used after normalization.

Because Roes FDS method does not satisfy the second law ofthermodynamics,itssolutionsometimesshowsunphysicalsolutionssuch as the carbuncle phenomena. As a remedy of such problems,theentropyxingfunctionby Montagneet al.16 is usedin thepresentstudy. In Eq.(35), w(k) is the corrected eigenvalueas follows:

w(k) =

jkj, jkj > e

(k2 +e2)/2e, jkj < e(36)

e =e[jrnkUj + jrgkVj + ( a/2)(jrnj + jrgj)]i + 12

, j (37)

Small number e, which is typically used in the range of 0.01

e0.1, is known to affectthe articial dissipation,the shock thick-ness,andtheconvergencerate,16 butalargevaluee =0.4wasusedasa referencevalue in the presentstudyto ensurethe solutionstability.

As a higher-order extension of the upwind scheme, the MUSCLscheme is used for the extrapolation of primitive variables at thecell interface.17 A limiter function is also used to overcome thesevere dispersionerror introducedby the higher-orderextrapolationand to preserve the TVD property.17 Using a minmod limiter, theextrapolatedvariables at the cell interfaces are expressed as

qL =qi + 1

4

(1 g)ri 1

2+ ( 1 + g)D i + 1

2

qR =qi + 1

1

4(1 + g)ri + 1

2+ ( 1 g)Di + 3

2 (38)

where

D i + 12=minmod

d

i + 1

2, bdi 1

2

ri + 1

2=minmod

d

i + 1

2,bdi + 3

2

d

i + 1

2=qi +1 q i (39)

minmod(x , y ) =sign(x ) max[0, min{jx j, y sign(x )}] (40)

whereqi is any primitive variable at grid point(i, j ). The minmodfunction preserves the TVD property in the range of

1b(3 g)/(1 g) (41)

where g is the spatial accuracy control parameter. Ifg=1 thesolution is second-order accurate in space and ifg=1

3 the solu-

tion is third-orderaccurate. In this study, the third-order scheme isused for all cases. The minmod limiter shows different characteris-tics accordingto the high-order extensionparameterb. The smallervalue ofbpreserves the monotonicitybut introduces the larger nu-merical dissipation. Because a prior one-dimensional shock-tubestudy showed stable shock-capturing characteristic with b =1.25in Mach number range 4 5, this value is used as a standard valuefor the present study.

General Aspects of a Baseline Method

To nd accurate and efcient numerical methods for the analysisof unsteadyshock-induced combustion, some comparative numeri-

cal experimentsare performedas follows.As a rststep, simulationsof experimentalresults were conductedto validate the performanceof the baseline method. A constant time step is used equivalentto aCFL number of three for incoming ow, and four subiterations areperformed.The computationaldomainis composedof a grid systemwith 200 grid points along the projectilesurfaceand 300 gridpointsnormal to the surface. The inow boundary is set some distance infront of the oscillating bow shock. In addition to the simulation ofexperimental cases, numerical experiments were performed for one

of the cases using the rst- and second-ordertime-accurateschemeswith a different number of subiterationsto nd the optimum value.Use of an approximate Jacobian splitting method is also examinedto understand the adaptabilityof this scheme to unsteady problems.

Simulation of Experimental Results

For the purpose of validating the developed computational uiddynamics code, Lehrs experiments of periodically oscillating

shock-induced combustion3 were numerically simulated. A stoi-chiometric hydrogenair mixture was used in the experiment at320 mmHg and 403 m/s of sonic velocity.This sonic velocity corre-spondsto the temperature of 293 K. The diameter of the hemispher-ical projectile used in the experiment is 15 mm. The consideredcases are oscillating results at Mach numbers 4.18, 4.48, and 4.79.Experimentally observed oscillation frequencies are 148, 425, and712 kHz, respectively. Figure 2 is the experimental shadowgraphfor the cases of Mach number 4.18 and 4.79.

For the efciency of computation, 2000 steps of time integrationwere performed initially by a rst-order scheme without subitera-tion. After 1000stepsof integration,the oscillationbeginsto appearand goes to a regular cycle. The nal solution after the 2000 stepsof integration are used as an initial condition of the second-order

integration. This rst-order solution is also used as an initial condi-tionfor most of the numerical experiments performed in the presentstudy.

The nal solution of the second-orderintegrationis obtainedafter1000more steps with foursubiterations.The error normof a subiter-ation was lessthan1% of theinitialerrorafterthe subiterationsin ev-ery time step. Figures 35 show the local Mach numberdistributionof the oweld and the temporal variationof densityalong the stag-nationstreamlinefor threeMach numbers. The Machnumber distri-butionis selectedfor plottingbecauseit showsmostof theimportantow characteristics of shock-induced combustion very clearly.

Fig. 2 Experimental shadowgraph of periodically oscillating shock-

induced combustion at Mach numbers 4.18 and 4.79(Ref. 3).

Fig. 3 Local Mach number distribution and the temporal variation ofdensity along stagnation line for the case ofM= 4:18.

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Table 2 Comparison of the oscillation frequency of unstableshock-induced combustion

Frequency, kHz M = 4.08 M = 4.48 M = 4.79

Experiment 148 425 712

Present result 155 426 707

Yungster and Radhakrishnan8 163 431 701

Matsuo and Fujiwara5 160 725

Wilson and Sussman7 530

Hosangadi et al.18 450

Fig. 4 Local Mach number distribution and the temporal variation of

density along stagnation line for the case ofM= 4:48.

Fig. 5 Local Mach number distribution and the temporal variation of

density along stagnation line for the case ofM= 4:79.

The plots of Mach numberdistributionagree well with the exper-imentalresults,and the densityhistoryshows allof the detailsof theinstability mechanism. The instability mechanism will not be dis-cussedanymore becauseit is beyondthe scope of thispaper and hasbeen discussed in previous papers.5 7 The frequency of oscillationis obtained from these results and compared in Table 2 with ex-perimental and numerical values obtained by previous researchers.The results of the present simulation show satisfactory agreementwith experimental results even though they do not exactly agree.The chemistry mechanism used in this study is same as the oneused by Yungster and Radhakrishnan8 and Matsuo and Fujiwara,5

and the differentchoice of chemistrymechanism has beenknown tobe responsible for frequency disagreement in the results presentedby Wilson and Sussman7 and Hosangadi et al.18 The accuracy of

the chemistry mechanism will not be discussed anymore because itseems to be satisfactory, judging by the simulation of experimentalresults. Anyway, the baseline method could be conside red to havereliable solution accuracy accordingto the aforementionedcompar-isons of the results.

On the basis of the reliability of the baseline method, the ef-fect of different choices of numerical approaches will be discussedin the following sections. The case of Mach number 4.48 will beused as a referencecase becauseit shows the oweld most clearly

amongthe three cases and also showsa sufcientnumberof wigglesin the reaction front and a sufciently long induction distance be-tween the bowshockand the reactionfront,whichit can be resolvedwith the present grid system.

Order of Integration and Number of Subiterations

The effect of the order of temporal integration and the conver-gence of subiteration were examined for the case of Mach number

4.48.A rst-ordertime-accuratesolutionand second-order-accuratesolutions with 0, 2, 4, and 10 subiterations were obtained using thesame initial conditions for comparison. The initial condition is thesame as the one discussed in the preceding section. All other com-putational conditions are xed to that of the baselinemethod.

Figure 6 is the local Mach number distribution and temporalvariation of density along the stagnation streamline obtained bythe rst-order-accuratetime-integrationmethod. The Mach numberdistribution shows relatively smooth variation in comparison withthe second-order time-accurate solution in Fig. 4. This smoothnesssmears the sharpness of the waves, and some contact surfaces be-tween the normal shock and the reaction front are not discerniblein the supersonic ow region, which can be identied in second-order solution in Fig. 4. Such a trend of numerical smearing also

appears in the density variation. The rst-order solution seems tosmear some waves but shows a very periodic oscillation, whereasthe second-order solution shows sharp but numerically oscillatingbehavior in the postshock region.

In Fig. 7, the rst-ordersolution shows a smooth variation of thepressure at the stagnation point whereas the second-order solutionshows some overlaidhigh-frequencypeakson the main pressure os-cillation. However, the period of oscillations eems to be unchangedin all cases with the change of the temporal accuracy of integration,exceptin the case of a second-ordersolutionwith zero subiterations.

Fig. 6 Local Mach number distribution and the temporal variation ofdensity along stagnation line for the case ofM= 4:48 with rst-ordertime integration.

Fig. 7 Temporal variations of pressure at stagnationpoint by differentorders of time integration and numbers of subiteration.

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Fig. 8 Temporalvariations of pressure at stagnationpoint by differentorders of time integration and numbers of subiteration; magnied plot

of dotted box in Fig. 7.

In cases when subiterations are performed, the solutionsshow con-

vergence as the number of subiterationsis increased, although thesolution is nearly unchanged after two subiterations.This trend isshown in Fig. 8, the magnied plotof the dotted box in Fig. 7. Aftertwo subiterations,the subiterationerror normwas lessthan 1% of theinitial variation for all of the time integration steps. As a reference,the error norms were less than 1% and 0.01% for 4 subiterationsand 10 subiterations. Thus, two subiterations, or the convergencecriterion of 1%, is considered sufcient in this case because thecomputational time increases nearly proportionally to the numberof subiterations.

Flux Jacobian Splitting Methods

Instead of the Steger Warming ux Jacobian splitting method,an approximate ux Jacobian splitting could be a simple and ef-

cient way to carry out matrix inversion of LU-factored implicitformulations.12 This is especially true for a steady-state problembecause the latter requires just one N Nmatrix inversion in alower sweep and a scalar diagonal inversion in an upper sweep,whereas the use of an exactly split Jacobian matrix requires two(N+3) (N +3) matrix inversions in lower and upper sweeps.By the latter method, the split Jacobianmatrix in Eqs.(23) and(24)

is dened as

A = 12

[A r(k)]

r(k) =j

jUj + a

qn2

x +n 2

y

, j 1 (42)

The difference of split Jacobian matrices is just a scalar value, andmatrix inversion can be done much more efciently, even though

this procedure could presentsome difculties in unsteadyproblemsdue to the approximation.Moreover,the upper partof the matrix in-version becomes a scalar inversionby includingthe chemicalsourceJacobianmatrixZ only inlowerpartof Eq. (23), and the lower couldbecarriedoutbyN Nmatrix inversionby partitioningthe matrix.The matrix and vector product in Eq.(24)can be also evaluated bythe following simplied formulation:

ADQ =

1

2

26666666664

y1A + BDq1...

yNA + BDqN

u A + BDqN1 +nx C

v A+

BD

qN2+ny C

H A + BDqN3 + U C

37777777775

A =nxdqN1 +nydqN2 U

NXk=1

Dqk, B =U r(k)

C =

NXk=1

pqkDqk pe (udqN1 + v dqN2 dqN3) (43)

Fig.9 Local Machnumberdistributionand temporalvariationof den-

sity along stagnation line for the case ofM= 4:48, with approximateJacobian matrix splitting method and rst-order time integration.

Fig.10 Temporal variationsofpressure atstagnationpointby different

ux Jacobian splitting methods and orders of time integration.

Becausethis kind of simplied formula takes only 10% of the com-putational time of the matrix evaluation and matrix vector product,thisformulasavesup to 50%of total computingtimein thebest caseof a nonreacting ow calculationthat is used as an initial conditionfor the unsteadys imulation.

The result with the approximate Jacobian and rst-order timeintegrationis plotted in Fig. 9. In this case, the time step equivalentto a CFL number of two is used because the CFL number of three

shows an instability problem. With the exception of the time step,all of the computational conditions are xed to that of the baselinemethod. The Mach number distribution shows a similar diffusivemanner as that in Fig. 6. Worse yet is that the periodicity begins tobreak in densityvariation. A temporally laggedsolution is shown atthe nal distributionof the Mach number, that is, the wiggles of thecombustionfront are locatedat differentpositionsthen those shownin Figs. 4 or 6. This trend can be also found in Fig. 10, which showsthe pressure variation at the stagnation point.

However, the solution is improved by using second-order timeintegrationwith four subiterations.A time step equivalentto a CFLnumber of three was available in this case, and the solution showsnearly the same quality as that obtained using the StegerWarmingux Jacobian splitting method, as shown in Fig. 11. Thus, it is

understood from this result that the approximationof the left-hand-side matrix would not be a signicant matter if sufcient conver-gence could be attained. However, it is hard to say that approx-imate splitting is more efcient than the exact splitting methodbecause two subiterations seem to be sufcient in the case of theSteger Warming Jacobian splitting method. The time needed forthe case of the Steger Warming Jacobian with two subiterations is34.24 s/iteration, and the time needed for the case of the approxi-mate Jacobian with four subiterations is 34.90 s/iteration on a DEC

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Fig. 11 Temporal variations of pressure at stagnation point by dif-ferent ux Jacobian splitting methods and orders of time integration;

magnied plot of dotted box in Fig. 10.

Fig. 12 Comparison of temporal variations of pressure at stagnationpoint for different grid systems; reference time is translated to match

the time of second pressure peaks.

personal workstation. The computer is composed of a 500-MHz Al-pha 21164 microprocessorwith 1 MB of L2 cache and 128 MB ofmain memory.

Grid Renement StudyA grid renement study was carried out for ve grid systems

as an extension of the considerations on spatial discretization.Thecomputations were performed by the rst-order time integrationwith the same time step size used in the baseline method. All otherconditions were xed at those of the baseline method as well. Theresulting pressure history is plotted in Fig. 12 for comparison. Be-cause it is not possibleto use thesame initialconditionsfor differentgridsystems, the pressure curves are translatedto match the time ofthe second pressure peaks.

The periodicity of the solution is reasonably captured in all g ridsystems. The frequencyof oscillation is smallest for the 400 600grid, and other results show the trend of convergence to the re-sults of the 400 600 grid system. The deviation of the oscillation

frequency is within 2% error. This trend says that the coarse gridresolution may result in a high oscillationfrequency, which may beconsidered to be in connection with articial damping studied inthe limiter function section in part 2 of this paper,19 in which thesteepest method (having smallest numerical damping) shows thesmallest frequency. Even though the result of the 100 150 gridsystem shows a reasonable solution in Fig. 12, the 150 200 gridsystem is the smallest grid system that is adequate for capturingthe oscillatory behaviorof shock-induced combustion,as shown in

Fig. 13 Temporalvariations of pressure at stagnationpoint by using a150 200 grid system.

Fig. 14 Temporalvariations of pressure at stagnationpoint by using a100 150 grid system.

Fig. 13. The periodicity of solution breaks down in the 100 150grid system, as plotted in Fig. 14, if more time passes.

Conclusions

A fully implicit and time-accurate computational uid dynamiccode is developed for the analysis of unsteady reactive ow sys-tems. As a baseline method, an iterative fully implicit algorithm is

used for timeintegration,and a third-order-accurateupwind schemeis used for spatial discretization. Periodically oscillating shock-inducedcombustionphenomenawereconsideredas validationprob-lems, and comparison with experimental results shows the validityof the baseline method.

A comparativestudyof numerical methodsis made as a develop-ing process of an efcient computationaluid dynamics code. Therst-order-accurate time-integration method results in a spatiallydiffusive solution, and the periodicity of the solution is preservedwell. For the iterative solution of second-orderaccuracy, two subit-erations seem to be sufcient with the StegerWarming formulationof the ux Jacobian matrix. The approximate splitting of the uxJacobian matrix resultsin a nearly same solutionas thatof the base-line method, although the approximate splitting shows a diffusive

and time-lagging solution without or with a small number of subit-erations. Thus, the approximation of the implicit part would not bea matter of importance if a sufcient number of subiterations werecarried out for convergence.

From the grid renementstudy, the 100 150 grid system seemsto be insufcient for the reproduction of the periodic solution. A150 200 grid system solution would be the smallest for the sta-ble periodic solution, although the use of a ner grid exhibits aconverging trend to a smaller frequency solution.

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Acknowledgments

This research is supported by the Turbo and Power MachineryResearchCenter and the Korea Science and EngineeringFoundationunder Grant 971-1005-031-2.

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K. KailasanathAssociate Editor

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