McDaniel, Keith S. Three Dimensional Simulation of Time-Dependent Scramjet Isolator
/Combustor Flowelds Implemented on Parallel Architectures, ( Under the directions of
Dr. J. R. Edwards)
The development of a parallel Navier-Stokes solver for computing time-dependent,
three-dimensional reacting owelds within scramjet (supersonic combusting ramjet)
en-gines is presented in this work. The algorithm combines low-diusion upwinding
meth-ods, time accurate implicit integration techniques, and domain decomposition strategies
to yield an eective approach for large-scale simulations. The algorithm is mapped to a
distributedmemoryIBMSP-2architecture andasharedmemoryCompaqES-40
architec-ture using the MPI-1 message-passing standard. Two and three-dimensional simulations
oftime-dependenthydrogenfuelinjectionintoa modelscramjet isolator/combustor
con-guration at two equivalence ratios are performed. These simulations are used to gain
knowledge of engine operability, inlet performance, isolator performance, fuel air mixing,
ame holding, mode transition, and engine unstart. Results for an injection at a ratio
of 0.29 show qualitative agreement with experiment for the two-dimensional case, but
re-vealed a slow progression toward engine unstart for the three-dimensional case. Injection
at an equivalence ratio of 0.61 resulted in engine unstart for both two-dimensional and
three-dimensional cases. Engine unstart for the three-dimensional case occurs as a
re-sponse tothe formation and growth of large pockets of reversed owalong the combustor
side wall. These structures develop at an incipient pressure above 154 kPa and result in
signicant blockage of the core ow, additionalcompression, and chemicalreactionwithin
Architectures
by
Keith Scott McDaniel
A thesis submitted tothe Graduate Faculty of
North Carolina State University
in partialfulllmentof the
requirementsfor the Degree of
Master of Science
Department of Mechanical and Aerospace Engineering
Raleigh, North Carolina
December 19,2000
Approved By:
I dedicate this thesis to my grandmother, Emily C. Dotson, and my brother,
Don Mark McDaniel, who have stood behind me through many years of hard work.
Their support of my goalsand aspirations have kept me strong. Without them, my road
Biography
Keith Scott McDaniel was born March 18, 1975 in Winston-Salem, North Carolina. He
grewup nearLibertyAirportinWinston-Salemwherehisfascinationwithaircraftandthe
driving forces behind them sprang from. After graduation from Carver High School, he
began his collegiateactivities at North CarolinaState University. He received aBachelor
ofScience inAerospace EngineeringinMay of1998 and iscurrently attainingaMaster of
Science in Aerospace Engineering from the same university. Keith willbegin employment
Acknowledgements
First,I wanttogivelovingthankstomymother,DonnaFayMcDaniel, andfather, Danny
Jay McDaniel, for they have given me a life full of opportunities. I thank Dr. Jack
Edwards for being a teacher and an advisor during my studies at North Carolina State
University. I acknowledgethe AirForceOÆceof Scientic Research(AFOSR) forsupport
of the current research activities through AFOSR GRANTF499620-98-1-0238,monitored
by Dr. Len Sakell. I also want to thank all my friends and schoolmates for all the help
Contents
1 Introduction 1
2 Governing Equations 4
2.1 Reference Equations . . . 4
2.2 Averaging . . . 7
2.2.1 Reynolds Averaging. . . 8
2.2.2 FavreAveraging. . . 9
2.3 ChemicallyReacting Equation Set. . . 11
2.3.1 Chemical Kinetics. . . 13
3 Numerical Formulation 18 3.1 Turbulent Closure . . . 18
3.2 Upwinding . . . 20
3.3 Time Advancement . . . 23
4 Multiblock Parallel Implementation 25 4.1 Algorithm . . . 25
4.2 ParallelPerformance . . . 28
5 Application 32 5.1 FlowModel . . . 32
5.2 Boundary Conditions . . . 33
5.3 Fuel Injection . . . 34
6.2.1 EquivalenceRatio 0.29 . . . 39
6.2.2 EquivalenceRatio 0.61 . . . 42
7 Conclusions 45
8 Figures 47
References 73
List of Tables
1 Abridged JachimowskiChemistry Model . . . 16
2 Balakrishnan etal. ChemistryModel . . . 17
3 Execution Times onScramjet . . . 30
4 Execution Times onIBM SP . . . 31
List of Figures
1 Upwindingstencil ( -direction ) . . . 47
2 Block decomposition of domain . . . 48
3 Parallelalgorithmow chart . . . 48
4 NAL isolator / combustor conguration . . . 49
5 NAL fuel strut . . . 49
6 NAL isolator / combustor congurationgrid, (every secondpoint shown) . 50 7 Hydrogen mass ow rate normalized by equivalence ratio versus time . . . 50
8 Floweld contour plots , =0:0 . . . 51
9 Wallpressure atseveral characteristic times =0:29 . . . 52
10 Floweld contour plots , =0:29 . . . 53
11 Wallpressure atseveral characteristic times , =0:61 . . . 54
12 Mach number contour plot atseveral time levels, =0:61 . . . 55
13 Static pressure at several time levels, =0:61 . . . 56
14 OH contours atseveral time levels, =0:61 . . . 57
15 Centerline Machnumberversus X, =0:61 . . . 58
16 U-velocity contours ( m s )near fuel strut ( t= 13.25 ms, =0:29 ) . . . 59
17 U-velocity contours ( m s ),=0:29 . . . 60
18 Static pressure (Pa)contours, =0:29) . . . 61
19 Temperature contours (K), =0:29 . . . 62
20 Hydrogen mass fraction, =0:29) . . . 63
21 Hydroxyl mass fraction contours, =0:29) . . . 64
22 Wallpressure atseveral characteristic times , =0:29 . . . 65
23 Time averaged wallpressure ,=0:29 . . . 65
24 U-velocity contours ( m s ) near fuel strut ( t= 0.25 ms, =0:61 ) . . . 66
26 Temperature contours (K), =0:61 . . . 68
27 Static pressure (Pa)contours, =0:61 . . . 69
28 Hydrogen mass fraction, =0:61 . . . 70
29 Hydroxyl mass fraction contours, =0:61 . . . 71
30 Wallpressure atseveral time levels ,=0:61 . . . 72
List of Symbols
A Pre-exponential factor
~
A ~
G Combined inviscidand viscous uxJacobians
^
A System Jacobian matrix
a Speed of sound
b Reaction order
C
p
CoeÆcient of specic heats at constantpressure
~ c
j
Species Concentration
D Diusion coeÆcient
e
t
Total internal energy
E,F,G Flux vectors (x, y, z,directions respectively)
E Inviscidux component (x -direction)
Ep Parallel eÆciency based ontime
E
v
Viscous ux component (x -direction)
F Inviscidux component (y -direction)
F
T
Time averaged owvariable
F
v
Viscous ux component (y -direction)
f Instantaneous owvariable
G Inviscidux component (z-direction)
G
v
Viscous ux component (z-direction)
~
H Combined Jacobianmatrix at point ijk
h Stagnation enthalpy per unit mass
h o
f
m
Heat of formationfor species m
J Jacobian
J
mj
Diusion Flux Tensor
k
f ;k
b
Forward and backward reactionequilibriumconstants
k Turbulent kinetic energy(section 3)
k Specic reactionrate (section2.3.1)
l
k
Kolmogorov length scale
l
mfp
Mean freepath length scale
M Mach number(section 3)
M Collision partner(section 2.3.1)
M
j
Molecular weightfor species j
Pr Prandtl number
p Pressure
q
j
Heat uxvector
q
t
j
Reynolds heat ux vector u 0 0
i h
0 0
R Universal gas constant
R R
f ;R R
b
Forward and backward reactionrates
S Source vector
Sp Parallel speedup
Sc Schmidt number
t Time
T Temperature
T
a
Activation Temperature
T Integral time scale
TB Third body eÆciencies
T
1
Turbulentintegraltime scale
T
2
Mean property variationtime scale
W Primitivevariable vector
U Conservativevariablecolumn vector
u Velocity component instreamwise direction
u
j
Velocity vector with componentsu, v, w
v Velocity component inwallnormal direction
v 0
m;n
StoichiometriccoeÆcients for reactant m inreactionn
v 00
m;n
StoichiometriccoeÆcient for product m inreaction n
w Velocity component inspanwise direction
x Cartesian coordinate(generallystreamwise component)
Y
m
Species mass fraction
y Cartesian coordinate(generallynormal component)
z Cartesian coordinate(generallyspanwise component)
_ w
m
Species production rate
Q
j
Turbulentheat ux tensor
T
ij
Turbulentstress tensor
U;V;W Contravariant velocity components
V
j
Species diusion velocity vector
U m
;V m
;W m
Species contravariant diusion velocitycomponents
Y
mj
Turbulentspecies mass uxcomponent
Greek Letters:
Æ
ij
Kronecker Delta
Density
;; Computationalcoordinate
; Arbitrary variables
Thermal conductivity
t
Turbulentthermal conductivity
Molecular viscosity
! Specic dissipation rate
ij
Laminarstress tensor
kinematicviscosity
Subscripts:
1 Freestream quantity
o Stagnation (total)quantity
i;j;k Spatialcoordinates, cellindices
L;R Left and RightStates
m Chemical species
n Time level orchemicalreaction
t Totalor turbulentquantity
mix Mixture quantity
v Viscous quantity
1
2
Interface data
Superscripts:
0
Reynolds uctuating component
00
Favre uctuating component
c Convective component
p Pressure component
Accents:
Averagedvariable
~ Favre-averaged component
^ Linear system ux Jacobianin computationalspace
Abbreviations:
CFD ComputationalFluidDynamics
CS Control Surface
CV Control Volume
DNS DirectNumerical Simulation
HXLV Hyper-X LaunchVehicle
HXR V Hyper-X Research Vehicle
LES Large EddySimulation
R ANS Reynolds AveragedNavier StokesEquations
NACA National Aeronautics and Space Administration
NAL National Aerospace Laboratory
NASA National Aeronautics and Space Administration
NASP National AeroSpace Plane
NS Number of species
NR Number of reactions
PNS Parabolized Navier-Stokes
1 Introduction
The scramjet( supersonic combusting ramjet) engineis a key toairbreathing hypersonic
technology. ThistypeofpropulsiondevicehasbeenstudiedbyNACA/NASAfornearly60
years [1]. A review of the scramjet technology in the 1980's indicates a rm fundamental
basis for the performance potential of dual mode scramjets. Dual mode scramjets are
similar to conventional ramjets but do not contain a second minimal area. This allows
both supersonic (scramjet) and subsonic (ramjet) combustion modes. For example, a
tactical missile(an expendable, lowcost, lowweight, xed geometrydevice) must operate
as a ramjet at low ight Mach numbers and as a scramjet at high ight Mach numbers
(M > 6:5). If adequate combustor-inlet isolation is provided, the scramjet will operate
in a ramjet mode with slightly lower eÆciencies than that of a conventional ramjet [2].
Knowledge of engine operability,inlet performance, isolator performance, fuel air mixing,
ame holding, combustion eÆciency, mode transition, and engine unstart is crucial for
scramjet applications. A large amount of scramjet combustion experimental data was
collected in ground based test facilities during the National Aero Space Plane (NASP)
Program. After termination of NASP in 1995, the focus of NASA Langley shifted to the
Hyper-X program. The Hyper-X program will demonstrate the free-ight operation of
an airframe-integrated, hydrogen-fueled, scramjet owpath in atmospheric ight at Mach
7 and 10 by 2002 [3]. \One shot" test experiments, such as Hyper - X, and scramjet
experiments in ground-based facilitiesare limited in the amount of oweld information
which can be recorded.
When astage inadevelopment cycle is reached wherewind tunnelor ight testingis
required, computationaltechniques can beextremelyuseful asadiagnostic tooltoexplain
certainunexplainedowphenomenaobservedinmeasureddata. Thisisparticulartruefor
performance. ThepotentialofCFDtechniquesasadiagnostictoolisdirectlyrelatedtothe
fact that computational solutions provide complete ow eld data, whereas wind tunnel
tests generally measure global characteristics and surface data. In the past, less general
equation sets have been used for the simulation of scramjet owelds. An example is the
use of the Parabolized Navier-Stokes equations (PNS). For supersonic ow, the
Navier-Stokesequationscanbemadeparabolicinthestreamwisedirections byassumingthat the
streamwise viscous terms are negligible and by a special treatment of the pressure in the
subsonicnearwallregionoftheboundarylater. Thesteadyequationsarethensolvedusing
astreamwisemarchingnite-dierencescheme[4]. Theseschemesare eÆcientforasteady
state solution butare only accurate if asupersoniccore ow exitsand are invalidfor time
dependent simulations. The solution of the complete Navier-Stokesequations isnecessary
tocomputescramjetcombustionowswere largesubsonic orseparatedregionsofowmay
exist. Theseregionsofowmayoccurduringatransitionfromscramjettoramjet modeof
operationand duringunsteady modes ofoperation, suchastime dependent fuelinjection.
Also, these subsonic regions as well as shock / boundary layer interactions play a major
role indetermining the inlet ow structure and the resultinginlet performance.
An understanding of dynamic modes of scramjet operation requires, at a minimum,
time-dependent solutions of the Favre-Averaged Navier-Stokes equations, extended for
chemicallyreactingows. Thisequation system can capture complex owstructures such
as expansion fans, boundary layers, shear layers, reaction fronts, subsonic separated-ow
regions, and shock-boundary layer interactions. The use of Favre-Averaging for an
un-steady meanowimpliesthe averagingof theNavier-StokesequationsoveratimeTmuch
larger than the longest turbulent time scale but smaller than the time scale characteristic
of the bulk uid motion. This separates the turbulent uctuating time scales from the
of operation and to explain results obtained from ground-based experiments. One such
mode of operation is the time dependent injection of fuel into the combustor. Time
de-pendentfuelinjectionleadstoconcerns ofowpathstability,scramjet/ramjettransition,
ame stabilization, and engine unstart. This directly corresponds to the proposed Hyper
-X Launch Vehicle (HXLV) ight test sequence. The HXLV will separate from a B-52
and will be boosted to the predetermined stage separation point. The Hyper-X Research
Vehicle(HXRV)willthenseparatefromtheHXLV.Aftera5secondunpoweredcontrolled
ight, sequenced fuel injection will commence [5]. Simulation of these dynamic modes of
engineoperationrequirescompleteoweldcalculationsofthecombustorandinletto
cap-ture the complextimedependentowstructures discussedpreviously. Thisrequiresof the
order of 1:0x10 6
grid points for 3-D resolution of a model scramjet combustor, assuming
that all eects of turbulence on the unsteady mean ow are modeled. Noting a solution
of the reactingoweld using a 7 species 7 reaction mechanism and the k ! turbulence
modelwillrequirethesolutionof14simultaneousequationsateverygridpoint,asolution
of roughly 14 million simultaneous equations is required at every time level. Obviously,
a parallel algorithm is needed to solve a problem of this size. This work develops a
par-allel Navier-Stokes solver for conducting three-dimensionalsimulations of time-dependent
scramjet owelds.
Sections2-3present the governing Navier-Stokesequations,their closureusing
turbu-lencemodelingconcepts,theirdiscretizationusinglow-diusionupwindmethods,andtheir
integration using time-accurate implicit methods. An approach for parallelizingthe
algo-rithm using a domain-decomposition / message-passing paradigm is discussed in section
4. The application of the algorithmto a model scramjet isolator / combustor
congura-tion tested atthe National Aerospace Lab, Japan [6]is discussed inSection 5,and results
2 Governing Equations
2.1 Reference Equations
This section will state the governing uid dynamic equations for viscous, unsteady
com-pressibleows. Theseequationsare extended toaccountfor niteratechemicalreactions.
The extended equations are then Favre-Averaged to model the eects of all turbulent
uctuations on the mean ow.
The compressible Navier-Stokes equations in Cartesian coordinates without body
forces orexternal heat additioncan bewritten as[7]:
@ @t () + @ @x j (u j
)=0 (2.1)
@ @t (Y m ) + @ @x j (Y m u j )= @ @x j (J mj )+w_
m (2.2) @ @t (u i ) + @ @x j (u i u j )= @ @x j (pÆ ij )+ @ @x j ( ij ) (2.3) @ @t (e t ) + @ @x j (h t u j )= @ @x j [q j + ji u i ] (2.4)
The diusion ux tensor J
mj
, diusion velocities V
j
, laminar stress tensor
ij
, and the
laminar heat ux vector q
j
are given by:
J mj = m V (m) j (2.5) V (m) j = D Y m @(Y m ) @x j (2.6) ij = @u i @x j + @u j @x i 1 2 Æ ij @u k @x k q j = @T @x j NS X m=1 h m Y m V j (2.7)
whereisthelaminarviscosity,isthethermalconductivity,DisthediusioncoeÆcient.
and Y
m =
m
The total enthalpy and energy are dened as [8]: h t = h mix + u i u i 2 e t = h t p (2.8) whereh mix
isthe mixturestatic enthalpy
The general eects of gravitational forces and radiative heat transfer are neglected
in this formulation, and thermal equilibrium at a common temperature is assumed. The
thermodynamicstate ofthegasmixtureisspeciedbythe followingauxiliaryrelations[9]:
= NS X m=1 m (2.9) Y m = m (2.10)
p = R
mix T (2.11) R mix = R NS X m=1 Y m M m (2.12) h mix = NS X m=1 h m Y m (2.13) h m = h o fm + Z T T ref C p m
(T)dT (2.14)
a 2 = C p mix C pmix R mix p (2.15) D = 1 Sc (2.16) C p mix = NS X m=1 Y m C p m
(T) (2.17)
Mixture values for the specic gas constant R
mix
are dened in equation 2.12, where
R is the universal gas constant, M
m
is the molecular weight of species m, The specic
heat at constant pressure for species m is C
pm
. Equation 2.11 denes the state equation
for a mixture of ideal gases. The species enthalpy h o
m
is equal to the species enthalpy of
formationh o
plustheintegralofC
pm
fromT
ref
the mixtureenthalpy. Curve-t polynomialsfromMcBrideetal[10]are usedforobtaining
C
p
m
as a function of temperature. The molecular viscosity and thermal conductivity
are determined fromSutherland's viscosity lawwhile Wilke's law[11] isused todetermine
mixture values. The molecular diusion coeÆcient D is modeled using Fick's law. By
assuminga constantSchmidt number of0:5for allspecies, one diusioncoeÆcientcan be
dened as inequation 2.16.
The four governing equations represented (equations 2.1-2.4) in vector form are the
ConservationofMass,ConservationofMomentum(Newton's2ndLaw),andthe
Conserva-tion of Energy (1st Law of Thermodynamics)laws appliedtoan arbitrarycontrolvolume
(CV). They are referred to as the Continuity, Momentum, and Energy equations
respec-tively. Therst terminthe continuityequationisthe rateofincrease of themass perunit
volume in the CV, and the second term represents the rate of mass ux per unit volume
passing through thecontrolsurface (CS).The rst terminthe momentumequationis the
rate ofofincrease ofmomentumperunitvolumeintheCVand thesecondtermrepresents
the rateofmomentumlossperunitvolumethroughtheCSduetoconvectionandstresson
the uidinternaltothe CV.Therst termintheenergy equationistherate ofincrease of
total energy per unit volume inthe CV and the second termrepresents the rate of energy
lost toconvection ,conduction,and dissipation through the CS. The equationsformulated
above can bewritten inCartesian vector form as
@U
@t +
@(E)
@x +
@(F )
@y +
@(G)
@z
=S (2.18)
If equation 2.1 is replaced by the auxiliary equation 2.9, then solution of the species
written as
E=E E
v = 0 B B B B B B B B B @ 1 u . . . NS u u 2 +p vu wu h t u 1 C C C C C C C C C A 0 B B B B B B B B B @ 1 U (1) . . . NS U (NS) xx yx zx u xx +v xy +w xz +q x 1 C C C C C C C C C A (2.19) 2.2 Averaging
Turbulentreacting owelds atlarge Reynolds numbers are inherentlythree dimensional
and cover alarge spectrumof turbulent lengthand time scales [12]. The smallestof these
scales are theKolmogorov scales. One ofthe premisesofKolmogorov'sequilibriumtheory
is that the smaller eddies are in a state at which the rate of their receiving energy from
the larger eddies is nearly equal to the rate at which the smallest eddies dissipate energy
to heat. The smallest turbulent scales can be shown to be acontinuum phenomenon. For
example, the ratio of the Kolmogorov length scale tothe mean free path of airmoving65
MPH over a driver's windowat 68 o
Fis approximately 72.
l
mfp
1:810 4
inch
2:510 6
inch
72 (2.20)
Therefore the Kolmogorov length scale is much bigger than the mean free path of air,
which inturn, is typically10 times the moleculardiameter.
To resolve all the turbulent length scales of a high Reynolds number ow using the
discretized Navier-Stokes equationswould requireextremely small spatialgrid spacing. In
addition,large integration timesare needed toresolve the average oweld over the time
scales. There have been several approaches used for solving the turbulent Navier-Stokes
equations. One method, known as Direct Numerical Simulation (DNS), is to solve the
governing equations directly. DNS attempts to resolve all important scales down to the
Anothermethod isLargeEddy Simulation(LES),whichattempts toresolveonly the
largereddiesintheoweldwhilemodelingtheeectsofunresolved(subgrid)eddies. LES
is alsoquite expensive for high Reynolds number ows.
Another method is to model the eects of all turbulent uctuations on the mean
ow properties. In this the governing equations are averaged over a time T that is much
larger than all the turbulent uctuating time scales T
1
. Forunsteady ows, such as those
investigated inthis work, the averaged time T must be smaller than the time scale of the
slowmean property variationsT
2
. Therefore, the turbulent uctuationsare averagedover
time T, while maintaining the time dependent mean property variations in the ow eld.
This implies that a large separation of time scales between the mean property variations
and the turbulent uctuationsmust existin the ow eld forthis approach to be valid.
2.2.1 Reynolds Averaging
The following sectionspresent Reynolds and Favre-averagingtechniques as appliedto the
Navier-Stokes equations. The averaging concepts introduced by Reynolds [13] generally
assume a wide variety of forms. The type most often used in CFD is time averaging.
The following is a derivation of the Reynolds Averaged Navier-Stokes (RANS)equations.
Although the RANSequations are derived fromtime averaging,the formof the equations
generalizes totime,spatial,and ensembleaveraging. The timeaveragedowvariablescan
beexpressed as :
f(x
i
)= lim
T!1 1
T Z
t+T
t f
i (x
i
;t)dt (2.21)
where f is the instantaneous ow variable. This variable can be decomposed into mean
and uctuatingcomponents,
f =f +f 0
(2.22)
ofturbulentuctuationsbutsmallerthanthesmallestperiodofslowpropertyvariationsin
the meanoweld. Thistypeofdecompositionresultsinthetimeaverageofauctuating
componentbeing equaltozero. Italsoresultsinthe time averageof theproducts ofmean
componentsto be equaltothe productsthemselves.
= ; (2.23)
0
= 0
=0 (2.24)
Equations 2.23-2.24 are known as the Reynolds postulates. They do not imply however,
thatthe timeaverageoftheproductsofuctuatingcomponentsareequaltozero[14]. The
average of the productof two uctuatingproperties is:
= +
0
+ 0
= +
0
+ 0
+ 0 0
= +
0 0
(2.25)
2.2.2 Favre Averaging
Density and temperature uctuations must be taken into account for compressible ow
elds. A density weighting procedure (Favre Averaging)isused tosimplifythe derivation
of the time averagedequations [13]. The Favre(mass) averaged componentis dened as
~
f(x
i )=
1
lim
t!1 1
t Z
t+t
t
f(x
i
;t)dt= f
(2.26)
The instantaneous function f can now be decomposed intoFavre mean and Favre
uctu-atingcomponents as:
Favre averaging is employed by decomposing p, , q
i
in the normal Reynolds averaging
fashion and decomposing the remainingvariables using the Favre decomposition.
= +
0
u
j =u~
j +u 00 j u (m) j =u~
(m) j +u (m) 00 j
p = p+p 0 ij = ij + 0 ij T = ~
T +T 00 (2.28) q j = q j +q 0 j Y m = ~ Y m +Y 00 m _ w m =w_
m +w_
0
m
e
t
= ~e
t +e 0 0 t h t = ~ h t +h 0 0 t
The equations are then mass-averaged using the denition of the Favre averaged
compo-nent inequation2.26. Usingthis typeof decomposition,the compressibleFavre-Averaged
Navier-StokesEquations (RANS) extended for species mass transportcan be written as
@ @t () + @ @x j (~u j
)=0 (2.29)
@ @t ~ Y m + @ @x j ~ Y m ~ u j = @ @x j h m ~ V (m) j +Y mj i
+w_
m (2.30) @ @t (~u i ) + @ @x j (~u i ~ u j )= @ @x j (pÆ ij )+ @ @x j [ ij +T ij ] (2.31) @ @t (~e t ) + @ @x j ~ h t ~ u j = @ @x j q j +u~
i ji +Q j (2.32) Y mj = Y 0 0 m u 00 j (2.33) T ij = u 00 i u 00 j (2.34) Q j = u 00 j h 00 t ~ u i T ji =q t j u i T ji (2.35) ~e t = NS X m=1 ~ Y m ~ h m ~ T p+ 1 2 NS X j=1 ~ u j ~ u j (2.36) ~
The averagedlaminar stress tensor,heat ux, and diusion velocities are given by:
ij
=
@u~
i @x j + ~ @u j @x i ! 1 2 Æ ij
@u~
k @x k q j = @ ~ T @x j NS X m=1 ~ h m ~ Y m ~ V (m) j ~ V (m) j = 1 ~ Y m D @ ~ Y m @x j (2.38)
2.3 Chemically Reacting Equation Set
TheFavreaveragedequationsetdescribedaboveistransformedintocurvilinearcoordinates
by the steady transformation[12]:
= (x;y;z)
= (x;y;z) (2.39)
= (x;y;z)
to yieldthe following equationset:
@U @t + @ ^ E ^ E v @ + @ ^ F ^ F v @ + @ ^ G ^ G v @
=S (2.40)
The conservative variable vector U and the source term vector S are:
U = 1 J 0 B B B B B B B B B @ 1 . . . NS ~u ~v w~ ~e 1 C C C C C C C C C A
; S =
The inviscidux vectors are given by: ^ E = 1 J 0 B B B B B B B B B @ U . . . NS U
~uU +
x p
~vU +
y p
wU~ +
z p ~ h t U 1 C C C C C C C C C A ; ^ F = 1 J 0 B B B B B B B B B @ V . . . NS V
~uV+
x p
~vV+
y p
wV~ +
z p ~ h t V 1 C C C C C C C C C A ; ^ G= 1 J 0 B B B B B B B B B @ W . . . NS W
~uW+
x p
~vW+
y p
wW~ +
z p ~ h t W 1 C C C C C C C C C A (2.42)
The viscous uxvectors are given by:
^ E v = 1 J 0 B B B B B B B B B B B B B @ 1 U (1) . . . NS U (NS) x ( xx +T xx )+ y ( xy +T xy )+ z ( xz +T xz ) x ( yx +T yx )+ y ( yy +T yy )+ z ( yz +T yz ) x ( zx +T zx )+ y ( zy +T zy )+ z ( xz +T zz ) x (~u xx +~v
xy +w~
xz +q x +Q x ) + y ~ u yx +v~
yy +w~
yz +q y +Q y + z (~u zx +v~
zy +w~
zz +q z +Q z ) 1 C C C C C C C C C C C C C A (2.43) ^ F v = 1 J 0 B B B B B B B B B B B B B @ 1 V (1) . . . NS V (NS) x ( xx +T xx )+ y ( xy +T xy )+ z ( xz +T xz ) x ( yx +T yx )+ y ( yy +T yy )+ z ( yz +T yz ) x ( zx +T zx )+ y ( zy +T zy )+ z ( xz +T zz ) x (~u xx +v~
xy +w~
xz +q x +Q x ) + y ~ u yx +~v
yy +w~
yz +q y +Q y + z (~u zx +~v
zy +w~
zz +q z +Q z ) 1 C C C C C C C C C C C C C A (2.44) ^ Q v = 1 J 0 B B B B B B B B B B B B B @ 1 W (1) . . . NS W ( NS) x ( xx +T xx )+ y ( xy +T xy )+ z ( xz +T xz ) x ( yx +T yx )+ y ( yy +T yy )+ z ( yz +T yz ) x ( zx +T zx )+ y ( zy +T zy )+ z ( xz +T zz ) x (~u xx +v~
xy +w~
xz +q x +Q x ) + y ~ u yx +~v
yy +w~
yz +q y +Q y + z (~u zx +v~
zy +w~
zz
+q +Q
, where w_
m
is the averaged species production rate for species m, U;V;W are the
con-travariant velocity components and U m
;V m
;W m
are the contravariant diusion velocities
forspecies m dened by:
U x ~ u+ y ~ v+ z ~ w V x ~ u+ y ~ v+ z ~ w (2.46) W x ~ u+ y ~ v+ z ~ w U m x ~ V m 1 + y ~ V m 2 + z ~ V m 3 V m x ~ V m 1 + y ~ V m 2 + z ~ V m 3 (2.47) W m x ~ V m 1 + y ~ V m 2 + z ~ V m 3
and J isthe Jacobianof the coordinate transformation.
2.3.1 Chemical Kinetics
The chemical mass productionrate perunit volume w_
m
for species m is given by the law
ofmass action:
_ w m = NR X n=1 h v 00 mn v 0 mn i (R R fn R R bn ) = NR X n=1 h v 00 mn v 0 mn i k f n NS Y j=1 (~c j ) v 0 j k b n NS Y j=1 ( ~c
j ) v 0 0 j !! (2.48) and v 0 j
isthe stoichiometriccoeÆcient of the reactants,v 00
j
is the stoichiometriccoeÆcient
of the products, NR is the total number of reactions involved, NS isthe total number of
speciesinvolved, R R
fn
andR R
bn
are theforward andbackward reactionratesrespectively
forreactionn, where ~c
j = j M j
isthe molardensity for species j. The specic reactionrate
k is writtenbelowinArrhenius form:
where A is the pre-exponential constant, and T
a
is the activation temperature [15]. The
variables A;b; andT
a
are found in Tables 1 and 2 for the two reaction mechanisms
con-sidered in this study. The reaction mechanisms are composed of two types of reactions,
exchangeand thirdbody. Theexchangereactiondescribestheinterdependentforwardand
backward reactions between products and reactants. Forthe followingexchange reaction:
A+B *
)C+D (2.50)
The forward and backward reactionrates are given by the lawof mass actionas:
R R
f
= k
f (T)c~
v 0 A A ~ c v 0 B B R R b = k b (T)c~
v 0 0 C C ~ c v 0 0 D D (2.51)
Third body reactions occur when two species react in the presence of a collision partner
M. The law of mass action can be rewritten for a third body reaction by considering
thatathird bodyreactionisactuallyNS reactionswhere the reactionrate depend onthe
collisionpartnerM.
_ w m = NR X n h v 00 mn v 0 mn i NS X i=1 (R R f ni R R b ni ) ! (2.52)
Forthe followingthird body reaction
AB +M *
)A+B +M (2.53)
the third body reactionrate R
n
for reactionn is
R n = NS X i=1 (R R fni R R bni ) (2.54)
and the forward and backward reactionrate in equation 2.54 are
If one of the specic reaction rate coeÆcients is chosen as a reference value, denoted by
k 0
f
n
then the third body eÆciency can be written as the ratio of the particular species
coeÆcientA
f
ni
tothe reference coeÆcient A 0 f n (TB) i = A f ni A 0 f n (2.57)
Usingthis relation,the forward and backward specic reactionrates can bewritten as
k f ni = k 0 fn (TB) i k b ni = k 0 bn (TB) i (2.58)
Using equation 2.58 in equation 2.55 and 2.56, then the third body reaction rate can be
writteninterms of the third body eÆciencies as
R n = k 0 fn ~ c v 0 AB AB k 0 bn ~ c v 0 A A ~ c v 0 B B NS X i=1 (TB) i ~ c v 0 M i M i (2.59)
The 7 species 7 hydrogen-oxidation reaction mechanism by Eklund [16] is used for
the chemistry modeling in the 3-D scramjet combustor simulations. This involves six
reactingspecies, H
2 , O
2 , H
2
O,OH,H,O,plus inertN
2
,thatinteractthrough aseven-step
reactionmechanism. This reactionmechanism waschosen forthe 3-Dscramjetsimulation
to maintain a minimal problem size. Also as indicated by the 2-D results, the ame is
stabilizedatthe fuel injector exit and therefore the inclusionof ignitionchemistry should
provide littlevariationin the amedownstream of the nozzle exit. This should have little
eect onthe thermal choking of the combustor.
The 9species 21reactionmechanismproposed by Balakrishnan, etal. [17] isused to
model the hydrogen oxidation for the 2-D scramjet combustor simulations. This involves
the same reaction species as in the reaction mechanism by Eklund plus H
2 O 2 and HO 2 .
These radicals have been shown to important during ignition of the hydrogen-oxygen
Table 1: AbridgedJachimowski ChemistryModel
# Reaction A b T
a
1: H + OH+ M *
)H
2
O +M .221E + 23 -2.0 0.0
2: H +H + M *
)H
2
+ M .730E + 18 -1.0 0.0
3: H
2 +O
2 *
)
OH +OH .170E + 14 0.0 24157.0
4: H +O
2 *
)OH +O .120E + 18 -0.91 8310.5
5: OH+ H
2 *
)H
2
O+ H .220E + 14 0.0 2591.8
6: O+ H
2 *
)OH + H .506E + 05 2.67 3165.6
7: OH +OH *
) H
2
O+ O .630E + 13 0.0 548.6
TBEs: H
2
O 16.0, H
2
2.5, and 1.0 forall other species
A, b,and T
a
are given inunits of moles ,cm,s,K, and cal/mole,
k=AT b
e (
Ta
T )
Modelingproceduresthat accountfortheeects ofturbulentuctuationsonchemical
reactionratesarediÆculttodevelopduetothelargenumberofunclosedcorrelationswhich
arise. Two alternativemethodsdiscussed by Roy[12],the probability density formulation
(PDF)andEddyDissipationConcept(EDC),wereshowntohavedecienciesincalculating
supersonic combustion ames. The more attractive (PDF) method also had negligible
eects for calculating attached supersonic ames. Therefore, the above formulation for
the species production rates is evaluated using mean quantities. The eects of turbulent
uctuationson rate coeÆcientsand onmolar density are neglected, thus
k
f
(T) ! k
f T
(2.60)
c
j (
j
) ! c
j (
j
) (2.61)
high-Table 2: Balakrishnan et al. ChemistryModel
# Reaction A b E
1 a
H + OH +M *
)H
2
O+ M 2.2e22 -2.0 0.0
2 b
H +H +M *
)H
2
+ M 1.e18 -1.0 0.0
3 a
H + O+ M *
)
OH +M 6.2e16 -0.6 0.0
4 a
O +O +M *
) O
2
+ M 6.17e15 -0.5 0.0
5 a
H + O
2 + M
*
)HO
2
+ M 6.76e19 -1.42 0.0
6 c H 2 O 2 +M *
) OH+ OH +M 1.2e17 0.0 45500.0
7 O+ OH +M *
) HO
2
+ M 1.0e16 0.0 0.0
8 H +O
2 *
)OH + O 3.52e16 -0.7 17070
9 OH +H
2 *
) H
2
O+ H 1.17e9 1.3 3626
10 O +H
2 *
)
OH+ H 5.06e4 2.67 6290
11 OH + OH
*
) H
2
O+ O .70e14 0.0 56767
12 HO 2 +H * ) H 2 +O 2
4.28e13 0.0 1411.0
13 HO
2 + H
*
)OH + OH 1.70e14 0.0 874.0
14 HO 2 + O * ) O 2
+ OH 2.00e13 0.0 0.0
15 HO 2 + OH * ) H 2
O+ O
2
2.89e13 0.0 -497.0
16 HO 2 + HO 2 * ) H 2 O 2 +O 2
3.02e12 0.0 1390.0
17 H +H
2 O 2 * ) H 2 +HO 2
4.79e13 0.0 7950.0
18 H +H
2 O
2 *
)OH +H
2
O 1.00e13 0.0 3590.0
19 H 2 O 2 + H * )H 2
O+ OH 7.08e12 0.0 1430.0
20 H +HO
2 *
)O +H
2
O 3.10e13 0.0 1720.0
21 H
2 +O
2 *
)OH + OH 1.70e13 0.0 44780.0
a
TBEs: H
2
O12.0, H
2 2.5
b
TBEs: H
2
O 6.5, O
2 0.5, N 2 0.4 c TBEs: H 2
O 15.0, H
2 2.5
A, b, and E are given in unitsof moles ,cm,s,K, and cal/mole,
b
( E
3 Numerical Formulation
3.1 Turbulent Closure
The process of averaging the Navier-Stokes equations results inextra terms that must be
modeled. These termsarise because of the nonlinearityof the equations. By investigation
of equations 2.33 -2.35, three unknown terms appear due to the averaging. These include
the Reynolds shear stress T
ij
, Reynolds heat ux q
t
j
, and the turbulent species diusion
Y
mj
. Inordertoobtainturbulentclosure,modelsforeachofthe termsmust bedeveloped.
Mostturbulencemodelsinusetodayemploythe Boussinesqeddy-viscosity approximation
fortheReynoldsstresstensor,whichassumesthattheReynoldsstressbehavesinamanner
similarto the laminarstress tensor [18]:
T
ij =
T
@u~
i
@x
j +
@u~
j @x i 2 3 Æ ij @u~
k
@x
k
(3.3.1)
The Reynolds heat ux is modeled using a gradient-diusion approximation, yielding an
expression similar toFourier's law:
q j t = t @ ~ T @x j NS X m=1 ~ h m ~ Y m ~ V (m) tj (3.3.2) t = T C p Pr t (3.3.3) ~ V (m) t j = 1 ~ Y m D t @ ~ Y m @x j (3.3.4) D t = t Sc t (3.3.5)
The turbulent species diusion is modeled by a gradientdiusion modelas
Y mj =Y 00 m u 00 j = T Sc t @ ~ Y m @x j (3.3.6)
The closure problem is reduced to specication of the eddy viscosity. If we assume that
the eddy viscosity can be computed in a similar fashion as the molecular viscosity, then
the eddy viscosity is alsodependent ona velocity scale and lengthscale.
whereu
mix andl
mix
denotevelocityandlengthscales,respectively,thatareassociatedwith
the mixing of turbulent eddies. These scales must be provided by the turbulence model.
Theturbulencemodelusedinthisworkisthezonaltwoequationk !modeldevelopedby
Menter[19]. ThismodelisacombinationoftheWilcoxk !andthestandardk models.
The k ! model is used in the laminar sublayer of the boundary layer and logarithmic
regionofthe boundarylayer. Thismodelhas beenshown tobesuperior tothek model
inadverse pressure gradientows and incompressibleows. The zonalmodelapproaches
a k model in the outer parts of the boundary layer and in free -shear regions. This
reduces the sensitivity of the calculation to free-stream turbulence quantities and allows
reasonablepredictions of free-shearows. The transitionfromk ! tok is facilitated
byablendingfunction F
1
,whichiszero inthe wake regionandfree shearlayerand onein
the wallregion. This modelis writtenbelowas:
t = a 1 k max(a 1 !;F 2 ) (3.3.8) @ @t (k)+ @ @x j (ku j ) = ij @u i @x j !k+ @ @x j (+ k t ) @k @x j (3.3.9) @ @t (!)+ @ @x j (!u j ) = t ij @u i @x j ! 2 + @ @x j (+ ! t ) @! @x j
+ 2(1 F
1 ) 2 1 ! @k @x j @! @x j (3.3.10) Let 1
represent anyconstantinthe originalk ! and
2
anyconstantinthe transformed
k model. Then is dened by the blendingfunction:
=F
1
1
+(1 F
1 ) 2 (3.3.11) The 1
constants of the (Wilcox) k ! modelare :
k1 = 0:85; !1 =0:5; 1
= :0750; a
1
=0:31 (3.3.12)
= 0:09; =0:41;
The constants
2
of the standard (k )are:
k2
= 1:0;
!2
=0;856;
2
=0:0828; a
1
=0:31 (3.3.14)
= 0:09; =0:41;
2 = 2 !2 2 p (3.3.15)
The blendingfunctions are dened by Menter as:
F
1
= tanh arg 4
1
; where (3.3.16)
arg 1 = min " max p k 0:09!y ; 500 y 2 ! ! ; 4 !2 k CD k! y 2 #
; and (3.3.17)
F
2
= tanh arg 2
2
; where (3.3.18)
arg
2
= max 2 p k 0:09!y ; 500 y 2 ! ! (3.3.19) where, CD kw = max 2 ! 2 1 ! @k @x j @! @x j (3.3.20)
Equation3.3.8satisesBradshaw's assumption[19]thatthe shearstressisproportionalto
theturbulentkineticenergyintheboundarylayer,whilepreservingtheoriginalformulation
that
t
=k=! for the rest of the ow.
3.2 Upwinding
The invisciduxes inthe Navier-Stokesequations are discretized using the LowDiusion
Flux Splitting Scheme (LDFSS) of Edwards [20]. This scheme attempts to combine the
accuracy of ux dierence splitting methods with the numerical eÆciency of ux vector
splitting methods. The LDFSS algorithm provides accurate resolution of stationary and
moving contact discontinuities while preserving solution monotonicity and entropy
satis-faction. The LDFSS algorithmrequires O(N) operationsfor N equations compared with
O(N 2
) operations of ux dierence schemes. [21] For clarity, the algorithm is applied
to the three dimensional instantaneous equations in the direction. Similar results are
advection and pressure components, as ^ E = ^ E c + ^ E p = jrj J UE c +pE p
; where U = 1 jrj ( x u+ y v+ z w) (3.3.21)
is the normalized contravariantvelocity in the direction,
E c = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 Y 1 Y 2 . . . Y NS u v w h t k ! 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; ~ E p = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 0 . . . 0 x y z 0 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (3.3.22) x = x jrj ; y = y jrj z = z jrj (3.3.23)
The upwinding is performed at the cell interfaces, denoted by 1
2
in Figure 1, using the
propertiesatthetheleft(L)andtheright(R)states. Thediscretizationstoresinformation
atthenodecellvertex,withthemeshcellsdenotedbydashedlines. Themetricderivatives
areevaluatedatthecellinterfaces. TheLDFSSschemeappliedtothecellinterfacesresults
intheinvisciduxbeingsplitintoconvectiveandpressurecomponents,withtheconvective
uxwritten as
E c 1+ 1 2 = jrj J L a1 2 C + ~ E c L + R a1 2 C ~ E c R (3.3.24)
and the pressure ux dened by
^ E p 1 2 = jrj J ~ E p 1 2 D + L p L +D R p R (3.3.25)
where a1
2
is the interface sound speed, written as
and C
and D
are functions of the Mach number, dened as
D + = + L (1+ L ) L P + L (3.3.27) D = R (1+ R ) R P R (3.3.28) C + = + L (1+ L )M L L M + L L R ~ M + (3.3.29) C = + R (1+ R )M R R M R + L R ~ M (3.3.30) where + L = 1 2
[1+sign(1:0; M
L )] (3.3.31) + R = 1 2
[1 sign(1:0; M
R
)] (3.3.32)
L
= max[0:0; 1:0 int(jM
L
j)] (3.3.33)
R
= max[0:0; 1:0 int(jM
R j)] (3.3.34) M + L = 1 4 (M L +1) 2 (3.3.35) M R = 1 4 (M R 1) 2 (3.3.36) P + L = 1 4 (M L +1) 2 (2 M L ) (3.3.37) P R = 1 4 (M R 1) 2 (2 M R ) (3.3.38) ~ M + = 1 p L p R p L +p R +2 jp L p R j p L M 1 2 (3.3.39) ~ M = 1+ p L p R p L +p R 2 jp L p R j p L M 1 2 (3.3.40) M1 2 = 1 4 r 1 2 (M 2 L +M 2 L ) 1:0 ! 2 (3.3.41)
and the left and right state Mach numbers are dened according to the interface sound
In the denitions of ~ M + and ~
M (Equations 3.3.39-3.3.40), the rst term involving the
pressuresprovidesmonotonecapturing ofdiscontinuities[12], whilethe secondterm
coun-teractsthe \carbuncle"-typeodd-even decoupling discussed in Reference [22]
3.3 Time Advancement
The3-D code uses athree-pointbackward Euler timeadvancementprocedurethat results
ina large sparselinear system, formulatedfor aparticular grid pointas:
" 3 2 M t + ~ A @ ~ S @ ~ W # n ijk ~ W ijk + ~ B n ijk ~ W i;j 1;k + ~ C n ijk ~ W i;j+1;k + ~ D n ijk ~ W i;j;k 1 + ~ E n ijk ~ W i;j;k+1 + ~ F n ijk ~ W i 1;j;k + ~ G n ijk ~ W i+1;j;k = ~ R n ijk + 1 2 ~ U n ijk ~ U n 1 ijk t (3.3.43) where ~ A ~
GarefunctionoftheinviscidandviscousuxJacobians,asoutlinedinReference
[23]. M is the transformation matrix @ ~ U @ ~ W , @ ~ S @ ~ W
is the Jacobian of the source term vector,
and t isthe physical time step.
The basicimplicit operatoris formed by orderingequation 3.3.43overthe gridnodes
and incorporatingimplicitrepresentations for the boundaryconditions [12]. The resulting
system Jacobian matrix is denoted as ^
A. This system is approximately solved using a
symmetric planar Gauss-Seidel (SPGS) algorithm. Denoting h 3 2 M t + ~ A @ ~ S @ ~ W i as ~ H, a
planarGauss-Seidelpartitioning[12,20,24,25]isdenedbyrstgroupingmatrixelements
correspondingto ani= constantplane as D
D= h ~ H+ ~ B + ~ C+ ~ D+ ~ E i (3.3.44)
and groupingo-planeelementsinto L and U respectively, where
L = ~
F (3.3.45)
Inthis formulationallmatrixelementscomposingofD,L,andU arestoredoverthe entire
mesh. The partitioning is then dened by
(D+L) n D n 1
(D+U) n ~ W n+1 = ~ R n + 1 2 ~ U n ~ U n 1 t (3.3.47)
This Gauss-Seidel partitioning is an approximation to the matrix ^
A, with factorization
error LD 1
UÆW. The factorization is nearly exact for streamwise supersonic ows, but
thefactorization errorincreases asthethe numberof nonzeroelementsinUincreases [25].
This meansthat asthe amount of streamwise subsonic ow increases, the performance of
the algorithm will degrade. The SPGS algorithm is implemented in two steps,a forward
and backward step in the direction:
forward sweep: ~ W n+ 1 2 i =(D n i ) 1 h ~ R n i L n i 1 ~ W n+ 1 2 i 1 i (3.3.48) backward sweep: ~ W n+1 i = ~ W n+ 1 2 i (D n i ) 1 U n i+1 ~ W n+1 i+1 (3.3.49)
The approximate inversion of the block pentadiagonal matrix D
i
at each i index is
ac-complished using an Alternating Direction-Implicit algorithm developed in earlier works
4 Multiblock Parallel Implementation
The calculation of ows within complex domains using structured grids usually requires
some type of domain-decomposition strategy. Domain decomposition involves the
parti-tioningof acomplexdomainintosimplersubdomains,whichmaybesolvedinsequence or
inparallel. The following subsections discuss the domaindecomposition strategy used for
time-dependent, 3-D scramjet calculations. The parallel implementation of the solution
algorithmis alsodiscussed. A combineddecomposition isused overthe three dimensional
domain. The combined decomposition contains a primary decomposition in the and
directions and a secondary decomposition in the direction. The Multiplicative Schwarz
method[26] is used to solve the primary decompositions. MultiplicativeSchwarz methods
solve each subdomain problem in sequence by using the most recent articial boundary
data from the adjacent blocks. The Additive Schwarz method is used to solve the
sec-ondary decompositions. Additive Schwarz methods solve each subdomain problem using
the lastarticial boundary data from the adjacent blocks. Parallelismcan be achieved in
the secondary direction because of the use of the Additive Schwarz algorithm, where no
dependence onupdateddatafromadjacentdomainsexists. Therefore, theprimary
decom-position is calculated in a serial fashion, while the secondary decomposition is calculated
inparallel. The 2-Dcalculations use onlythe primarydecomposition andwere run onthe
Cray T-90,avectorarchitecture. The3-Dcalculationswere runinparallelonthe IBM-SP
attheNorth CarolinaSuperComputingCenter(NCSC)andonaCompaqES-40atNorth
CarolinaState University (NCSU).
4.1 Algorithm
The development of this algorithmwas governed by the ow modelbeing used. The time
advancementschemediscussed earliermaybewrittenforthe entire domainasthelinear
system ~
A n
W n+1
= R
n
, where ^
A is Jacobian matrix of the nonlinear system, ~
thecorrection vector,and R isthe residual vector. The system can bedecomposed overa
set of subdomainsas follows.
^ A = 0 B B @ ^ A 1 ^ A n1 ::: ::: ::: ::: ^ A n mm ^ A mm 1 C C A (4.4.1) where ^ A k
denotes elements of the Jacobian corresponding strictly to subdomain k and
^
A
n
k
denotes elements that connect the solution within block k to that within other
blocks. Byremovingthe explicitdependence onthe ^
A
n
k
elements,asuccessive iteration
procedure for solving the system can be used, with the boundary data updated through
thecalculationof theresidualvectorR on
k
. Consideringatwoblock decompositionthe
MultiplicativeSchwarz algorithmcan be written asa two step process.
W n+ 1 2 W n + ^ A 1 1 0 0 0 R n (4.4.2) W n+1 W n+ 1 2 + 0 0 0 ^ A 1 2 R n+ 1 2 (4.4.3)
The n + 1
2
notation denotes the updating of information from adjacent to subdomain
interfaces. This updated information is incorporated into the calculation of the residual
operator as soon as it becomes available. The matrix inverse indicated in 4.4.2 and 4.4.3
actually represents the application of the planar Gauss-Seidel procedure 3.3.47 on each
subdomain. If an Additive Schwarz type decomposition is also applied in the block then
theresultisafourblockdecompositionwithahybridMultiplicativeandAdditiveSchwarz
scheme. Equation 4.4.2-4.4.3 becomes:
W n+ 1 2 W n + 0 B B @ 0 B B @ ^ A 1 1;1
0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 C C A + 0 B B @
0 0 0 0
0 ^ A 1 1;2 0 0
0 0 0 0
0 0 0 0
1 C C A 1 C C A R n (4.4.4) W n+1 W n+ 1 2 + 0 B B @ 0 B B @
0 0 0 0
0 0 0 0
0 0 ^ A 1 2;1 0
0 0 0 0
1 C C A + 0 B B @
0 0 0 0
0 0 0 0
0 0 0 0
This is a two step Multiplicative Schwarz method with Additive Schwarz sub-steps. This
requires the update of the residual vector after each Multiplicative step for the primary
decomposition while only updating the residual vector for the secondary decomposition
afteralltheMultiplicativeSchwarzstepsarecomplete. ThisallowseachMultiplicativestep
tobeen solved in parallel. Fornominally two dimensional problems with a uniform mesh
inthe thirddimension, itisconvenient toapply AdditiveSchwarz inthe uniformdirection
and to apply Multiplicative Schwarz decomposition in the non-uniform directions. The
abovemethod cangeneralized forthe actualScramjet combustorgeometrydecomposition
as shown in Figure 2. The primary decomposition is in the and directions while the
secondarydecompositionisinthedirection. Figure3illustratesthesolutionprocedurefor
the combined Multiplicativeand Additive Schwarz methods for two processors. Consider
the,, decompositionofblocks1through 5inFigure2,withreferencetotheowchart
inFigure3. Eachprocessorreceivesandecompositionforblocks1-5,wheretheseblocks
aredecomposedintheand directions. Theneachinnerlooponthe owchartbeginson
block 1 and sequences through to block 5i.e. (m=1...mb). First,each processor allocates
theprimitivevector W
m
overthesubdomainforthe currentblock, includingthe articial
boundaries. Then the residual vector R
m
is formed for the subdomain. The system of
linear equations ^
A
m W
n+1
m
= R
m
is then approximately solved for the update vector
W n+1
m
. After the interior points are updated, a serial transfer of the articial boundary
informationto adjacent blocks in the and directions satisfying [W
m \W
m ]
; takes
place. After all the blocks are sequenced through, a parallel data transfer updates the
solutionvariableson the articialboundaries inthe direction, satisfying
W
m \W
m
takesplace. Therefore,theparallelalgorithmisindependentofthetimeintegrationmethod
usedtosolvethesystem. Theequationsetin4.4.2-4.4.3isdependentontheproperupdate
vector R
The parallel transfers were carried out by using the Message Passing Interface[27 ]
(MPI-1)standard. A sampleprogramof the MPI block transferis given in theAppendix.
A buer is used to store the data being transfered for the secondary decomposition. This
buer is sent using blocking, combined send-receive commands in the direction in the
appendix. Two waves of send-receive are performed to transfer data for the two articial
facesateachinterface. Thistypeofmessagepassingonlycompleteswhenalltheprocessors
involved in the transfer have posted their individual send-receive calls. This ensures that
data is transferred at the right time level of the integration and also prevents system
buering that could lead to slow performance as the system could be overloaded with
buered data. This type of message passing also prevents deadlock as waves of data are
transferedfor eachface.
Thearticialboundaryinformationoneach planeismappedintoanonedimensional
array before the parallel transfer to adjacent blocks. This allows an explicit packing and
unpacking of the buer, instead of using implicit derived data types generated by MPI.
Derived data types may be dependent onthe system being used, as dierent systems use
dierent memory layouts and sizes. The explicit buering is system-independent, but
requires ((
max
+6)(
max
+6)6) operations to pack and unpack the buer. The time to
perform these operations is negligible compared with the time required to perform one
iterationof the ow solver.
4.2 Parallel Performance
S(1;p) = t
1
t
p
(4.4.6)
E(1;p) = t
1
pt
p
(4.4.7)
(4.4.8)
The speedup S in equation 4.4.6 will not exceed the number of processors (p) used in a
particularcalculation, asindicatedby the SpeedupFolklore Theorem[28]. Ifthe speedup
S is greater than the number of processes, then a faster serial algorithm can be made
be simulating the parallel one, provided that it can be simulated. In the case where
cache performance gets better with smaller subdomains via domain decomposition, the
serial program can not simulate the parallel program and therefore super-linear speedup
is observed. The eÆciency E in equation 4.4.7 is simply a measure of the time cost
optimization,described belowas:
If E(1;p)<1 then the parallel algorithmisnot time cost optimal
If E(1;p)=1 then the parallel algorithmistime cost optimal,
provided that the sequential algorithmistime optimal
If E(1;p)>1 then afaster sequential algorithmcan be obtained
by simulatingthe parallel one
(4.4.9)
Tables3through4providethetimedatafortheparallelalgorithmappliedtoaLargeEddy
Simulation(LES) calculationona4-processorCompaqES-40 sharedmemoryarchitecture
and an IBM-SP distributed memory architecture. The LES calculation on 33x65x65 and
65x65x65 grids is that of a plane Poiseuille channel ow. This was used to validate the
parallelinterface boundary transfers and to gain knowledge of parallel performance. The
timesT1, T2, T3, are real,cpu, and system timesinunits of seconds. The 33x65x65node
casesolvedfortwentyiterationsisshowninTable3. Theexpectedtrendthatincreasingthe
isincreased,asthecompileronlyoptimizestheserialportionofthealgorithm. Thetimeto
perform twenty iterationsusingfour processorstook343 seconds withno optimization. A
71percent decrease intime was accomplishedby using alevelfour compileroptimization,
butthisonlyresultedina25percentdecreaseintheeÆciencyofthealgorithmascompared
tothe eÆciency of the case withoutoptimization. Using the Speedup Folklore theorem of
4.4.9,the most optimalcondition for the current algorithmis with fourprocesses and full
optimization, with a speedup of 3.04. This algorithm did not have the highest speedup
forthe fourprocessor cases, but had the highest speedup forthe most time optimalserial
program.
Mostoftheperformancelossisdirectresultoftheparallelalgorithmrequiringanorder
of O (
max
max (
max
+p 1)) 2
operations. As the domain is broken into up smaller
segments,thentheoveralldomainsizeincreasesbyp 1planesofdata. Withanoverlap
of 6 planes of data between domains, each iterationrequires 6p planes of data transfer
and anadditionalcomputationof p 1 planes of data.
T1 T2 T3 node % n options speedup eÆciency
1250.0 1249.75 .37 99 1 -O0 1.0 1.00
623.0 622.41 1.31 199 2 -O0 2.01 1.01
343.0 341.92 2.25 399 4 -O0 3.64 0.91
750.0 749.61 .77 99 1 -O1 1.00 1.00
400.0 396.74 6.58 199 2 -O1 1.88 0.94
215.0 214.01 4.31 399 4 -O1 3.49 0.87
342.0 342.16 .39 99 1 -O2 1.00 1.00
171.0 171.42 .80 199 2 -O2 2.00 1.00
107.0 106.62 2.64 398 4 -O2 3.20 0.80
309.0 309.01 0.40 99 1 -O3 1.00 1.00
193.0 189.72 6.68 199 2 -O3 1.60 0.80
101.0 100.64 1.29 399 4 -O3 3.06 0.77
295.0 294.75 0.40 99 1 -O4 -fast 1.00 1.00
97.0 96.62 2.78 399 4 -O4 -fast 3.04 0.77
IBMSPattheNorth CarolinaSupercomputingcenter for 1000iterationruns. These runs
were calculated using level 3 optimizationon IBM SP, which is comparable tothe level 4
optimizationon the Compaq ES40. The speedup and eÆciency results here demonstrate
thatincreases in thenumberof subdomainsand the attendant communication and
execu-tioncosts degradethe programperformance forprocessornumbers largerthan eight. This
can be explained by the fact that the ratio of the number of computational planes to
communication planes is 5
6
, and the problem size has increased by 21 percent. The extra
iterationwork canbe avoidedby usingacell-centered nitevolumediscretization,but the
communication costs donot disappear.
T1 T2 T3 n speedup eÆciency comment
21258.58 21222.52 24.30 1 1.0 1.0 LES(274625 nodes)
13153.38 13129.44 15.43 2 1.616 .808 LES(274625 nodes)
5690.53 5542.18 11.40 4 3.7357 .934 LES(274625 nodes)
3091.07 2989.94 5.15 8 6.8774 .860 LES(274625 nodes)
1898.43 1792.48 3.44 16 11.198 .700 LES(274625 nodes)
5 Application
5.1 Flow Model
The National Aerospace Lab (NAL) nominally two-dimensional scramjet isolator /
com-bustorconguration issimulatedin this work [29]. This congurationas shown in Figure
4 consists of a wedge shaped fuel strut on the centerline of the combustor. The strut
wedge angle is5 o
and the isolator length is0.140 m. The total lengthof the combustor is
0.540mwith the strut placed0.043 mbehindthe inowplane. Pressure taps were placed
10mm apart on the centerline of the top wall for the rst 0.340 m and 0.020 m apart
for the rest of the combustor. The fuel strut shown in Figure 5 is 94.3 mm wide in the
spanwise direction. The injector throat was designed to maintain supersonic injection of
the fuel stream. The 5 degree wedge angle was determined by Masuya[30 ] to not induce
boundarylayerseparation onthe ductwallduetotheimpingingshockfromthefuelstrut.
Thefuel strutis precededbythe inlet sectionthat receivesvitiated airfromthe upstream
combustor. The inow conditions are shown in Table 5. The scramjet combustor inlet
FlowProperty Experiment Simulation
T
o
2214 2214
h
o
2.79 2.79
p
o
(kPa) 1012 1012
M
1
2.44 2.43
p
1
59.0 59.117
T
1
1258 1258
v
1
1753 1752.14
m
O2
0.333 0.333
1
1.28 1.2845
R
1
321.0 322.8
Y
O
2
0.2615 0.2615
Y
N2
0.5288 0.5288
Y
H
2 O
0.2097 0.2097
conditions were obtained from from the exhaust gas of a Vitiated Air Generator (VAG).
Thistypeoftestgas containsalargewater massfractionof0.2097asseeninTable 5. The
nominal Mach number and stagnation temperature are 2.45 and 2214 respectively,
lead-ingto astatic temperature that is well above the auto-ignition temperature for hydrogen
combustion. Thethree-dimensionalblockdecompositionisshowninFigure2-6. The
nom-inallytwo-dimensional geometry of the NAL combustor is modeled precisely in the and
directions and is extended in the direction. Block 1 contains 17x41x33 points, block
2 contains 17x9x33 points, block 3 contains 49x41x33 points, block 4 contains 61x33x33
points,and block 5 contains 133x73x33points,for a total of 481,173 points. Due to
com-putermemoryrestrictions, the resolution inthe and directions isa fourthof that used
in the 2-D simulations of Reference [6]. The two-dimensional grid is shown in Figure 6
withevery othergrid pointshown. Mesh clustering to allsolid surfaces isused tocapture
viscous layergradients. Also the grid wasstretched inthe directiontoresolvethe shock
structurearoundthe wedgeleadingedge,the Prandtl-Meyerexpansionnear thecenterline
strut,the recirculation regionbehind the fuelstrut, and the free shearlayer that develops
aftof the fuelstrut.
5.2 Boundary Conditions
Both two and three dimensional turbulent channel ow calculations were performed to
obtain the inow data for the simulations. All ow variables are extrapolated from the
interiorofthe combustorsectiontothe outowplaneof thecombustorusing secondorder
extrapolation. This methodofextrapolationisquestionablewherethermalchokingresults
and the combustor is subsonic, but without a knowledge of the back pressure, no better
method can be found. No-slip and adiabatic wall conditions were applied to all solid
decomposition discussed in section 4.3 introduces interface boundaries between adjacent
blocks. The ow variables along interface boundaries in the direction are not updated
during the Multiplicative Schwarz solution steps, while the interface boundaries in the
and directions are updated from data contained on adjacent blocks. The interface
boundaries are updated using adjacent interface data during the parallel transfer.
5.3 Fuel Injection
Allcaseswere initializedwithaconvergedsolutionofthe baselinecaseinwhichthe
equiv-alence ratio () is set 0.0. This corresponds to no hydrogen fuel injection. The fuel ow
rate was ramped over time to match the equivalence ratio for each case, then maintained
forthedurationof the simulation. Thefuelowrate through the geometricthroatA
t was
calculatedfrom:
_ m
H2
A
t =
m_
O2
A
t M
H2
M
O
2
=51:388 (5.5.1)
The fuelow rate was ramped using the following procedure:
_ m
H2
A
t
=min
1:0; t
:5t
c
_ m
H2
A
t
(5.5.2)
where t
c
is the ow-through time based on the free-stream velocity. The fuel is ramped
over onehalfof aow-through time(approximately0.00015seconds). Afterthis time, the
fuel ow rate isheld constant. The conditionsat the subsonic inowboundary of the fuel
6 Results and Discussion
6.1 Two Dimensional Results
Threefuelequivalenceratiocasesof0.0(nofuelinjection),0.29,and0.61weresimulatedin
two dimensions. The =0:0and =0:29 cases are nominally steady, whilethe =0:61
case resulted in a unsteady unstart. All reactive ow simulations were initialized from a
converged =0:0 solution.
The Machnumber, temperature, and pressure contours are shown inFigure8 for the
=0:0case. This case isuseful inasareference pointfor describingchangesdue tomass
injectionandcombustionoffuelforthehigherequivalenceratiocases. Theboundarylayer
thickness isapproximately5.5mmthickandincreasesthroughthe ductduetofrictionand
shock boundary layer interactions to about 12 mm at the end of the combustor section.
A small separation region results from the interaction of the bow shock formed ahead
of the fuel strut with the top wall boundary layer, and a second separation zone results
from the recompression shock that forms behind the expansion fan aft of the fuel strut.
A large recirculation region exists behind the fuel strut. In this symmetric case, upper
and lower supersonic cores result in the combustor section, with a subsonic mixinglayer
between them. Temperature increases can be observed near the adiabatic walls, behind
the recompression shocks, and in the separation zones. A maximum static temperature
of 2250 K is observed in the stagnation region at the bow of the fuel strut. The pressure
contours illustrate the complex shock system in the isolator and combustor regions. The
obliqueshocksystem/expansionfaninteractionscontinue throughoutthe combustorwith
a total of six shock reections in the combustor. The total pressure ratio Pe
P1
along the
centerline is .4635. This ratio is slightly less than the .47911 total pressure ratio that
would result from a normal shock in the duct at the free-stream Mach number of 2.44.
rise across the recompression shock promoteame stabilization forthe higherequivalence
ratios. Thisowsolutionandits3-Danalogueisusedastheinitialconditionforthehigher
equivalence ratio cases.
Figure 9 displays surface pressure distribution for the = 0:0 and = 0:29 cases
at several characteristic time levels. A steady scramjet mode of combustion is obtained
after 22.66t
c
characteristic times (t
c
= :314 ms). For the = 0:0 cases, reasonably good
agreement with experimental data is found throughout the combustor except near the
impingementpointofthebowshockontheupperboundarylayer. Thepeakpressureriseis
signicantlyunderpredicted,andthesubsequentwaveinteractionsarelesspronouncedthan
shown in the experimental data. These discrepancies may result from several numerical
factors, such as insuÆcient grid resolution in the impingement region, the discretization
scheme's relatively low order of accuracy near discontinuities, and the possible tendency
of the turbulence modelto \over-separate" the boundary layer inresponse to the adverse
pressuregradient. Theexperimentaldata shows thatthe actualcombustionprocess inthe
combustor inuences the ow inthe isolatorsection to alarge extent, indicatingthat the
shocktrainestablishedtocontainthecombustion-inducedpressurerisepropagatesintothe
isolatorregionand isstabilizedthere. This isnotseen hereinthe twodimensionalresults,
but is observed for the three dimensional case as discussed later. The two dimensional
resultsalsoshowatransitiontosubsonicowthroughasuccessiveweakeningoftheoblique
shock structure at after x = 200mm, rather than a continuing oblique shock pattern
throughout the nozzle as observed in the experiment. The hydroxyl mass fraction (OH),
Machnumber, temperature, andpressure contours areshown inFigure10forthe =0:29
case. Contour levels for Mach number and temperature are restricted to jMj < 1 and
T >2000 respectively to better distinguish regionsof subsonic owand regions of strong
thatis stabilizedinthe recirculation regiondiscussed earlier. The gradualpressure rise in
thecombustorisdue tothe outward deectionof thereactiveshear layer, which decreases
the eective ow area occupied by the core uid. Small regions of high-temperature,
low-momentum uid occur on the upper boundary layer due to shock / boundary layer
interactions. The temperature in the mixingregion reaches a maximum value of around
2800Kelvin. Acomparisonofthehydroxyl contoursandtheMachnumbercontoursshows
that chemical reactions occur in the supersonic region of the ow. The majority of the
combustion occurs between the fuel strut and x = 200mm meters plane, as indicated by
the largelevelsof hydroxyl inthat region. This region isalso the expansionregion, which
willbecomea more signicantpart of the ow for the higherequivalence ratio cases. The
ame structure shown is typical of diusion ames, where the maximum concentrations
of hydroxyl radical delineate the ame front and correspond to the regions of highest
temperature.
Figures 11 - 15 illustrate the response of the isolator / combustor oweld to
time-dependentinjectionofhydrogenat=0:61. InFigure11wallpressure distributionsalong
the top wall of the scramjet combustor at several time levels are compared with
experi-mental pressure data time averaged over 40 ms. Three stages of the unsteady response
are observed for this case. The initial stage occurs before t < 4:63t
c
, where the injection
process raises the static pressure of the uid within the injector from around 50 kPa to
nearly 300 kPa, resulting in choked ow at the nozzle throat. The expansion behind the
fuel strut is nearly eliminated while the recompression shock is strengthened due to the
displacement eect of the hydrogen fuel mass injection (t = 1:54t
c
). OH contours shown
inthe close up viewof the injector region(Figure 14) reveal that the initialstages of fuel
-airmixingoccurthrough the growth of the vorticalstructures generated through
Kelvin-Helmholtzmechanisms. Thisresultsinthe intermittentpulsingofthe fuelowratebefore
