NASA Technical Reports Server (NTRS) 20040111229: Dual-Code Solution Strategy for Chemically-Reacting Hypersonic Flows

16 Pages·0.49 MB·English

by  NASA Technical Reports Server (NTRS)

Dual-Code Solution Strategy for Chemically-Reacting
Hypersonic Flows
(cid:3)
William A. Wood
NASA Langley Research Center, Hampton, VA 23681
and
y
Scott Eberhardt
University of Washington, Seattle, WA 98195
Abstract Nomenclature
c mass fraction
A new procedure seeks to combine the thin-
Cp frozen speci(cid:12)c heat at constant pressure
layer Navier-Stokes solver LAURAwith the parabolized
Da Damko(cid:127)hlernumber
Navier-StokessolverUPSfortheaerothermodynamicso-
Et total energy
lutionofchemically-reactingair(cid:13)ow(cid:12)elds. Theinterface
h enthalpy
protocol is presented and the method is applied to two
h0 reference enthalpy at the reference temperature
slender, blunted shapes. Both axisymmetric and three-
H total enthalpy
dimensional solutions are included with surface pres-
k conductivity
sure and heat transfer comparisons between the present
kb backward reaction rate constant
method and previously published results. The case of
kf forward reaction rate constant
Mach 25 (cid:13)ow over an axisymmetric six degree sphere-
keq equilibriumconstant
cone with a non-catalytic wallis considered to 100 nose
Le Lewis number
radii. A stability bound on the marching step size was
M Mach number
observed with this case and is attributed to chemistry
M molecularweight
e(cid:11)ects resulting from the non-catalytic wall boundary
n normaldistance fromthe wall
condition. A second case with Mach 28 (cid:13)ow over a
N number of moles
sphere-cone-cylinder-(cid:13)are con(cid:12)guration is computed at
P pressure
both two and (cid:12)ve degree angles of attack with a fully-
q heat transfer rate
catalytic wall. Surface pressures are seen to be within
R(cid:22) universal gas constant, 8314.3J/kg-mole-K
(cid:12)ve percent with the present method compared to the
Rn nose radius
baseline LAURA solution and heat transfers are within
Re Reynolds number
10 percent. The e(cid:11)ect of grid resolution is investigated
s distance measured along body surface
in both the radial and streamwise directions. The pro-
T temperature
cedure demonstrates signi(cid:12)cant, order of magnitude re-
u, v, w cartesian velocity components
ductions in solution time and required memory for the
X, Y, Z cartesian coordinates
three-dimensionalcaseincomparisontoanallthin-layer
Yb body location
Navier-Stokes solution.
Yshock shock location
(cid:11) angle of attack
(cid:18)b body angle
(cid:3)Aerospace Technologist, Aerothermodynamics Branch, Gas (cid:22) viscosity
DynamicsDivision. MemberAIAA.
yAssociate Professor, Department of Aeronautics and Astro- (cid:24); (cid:17); (cid:16) curvilinear coordinates
(cid:26) density
nautics. MemberAIAA.
Copyright (cid:13)c1995 by the American Institute of Aeronautics (cid:28) characteristic time
and Astronautics, Inc. No copyright is asserted in the United
States under Title 17, U.S. Code. The U.S. Government has
Subscripts:
a royalty-free license to exercise all rights under the copyright
claimedhereinfor governmentpurposes. All otherrightsare re- l LAURA
servedbythecopyrightowner. s species
1
u UPS time and memory requirements relative to TLNS algo-
wall value at the wall rithms. The principle di(cid:14)culty in applying the PNS
1 freestream value equations to the class of vehicles considered here is
that commonlythe algorithms cannot solve blunt-body
(cid:13)ow(cid:12)elds, and most reentry vehicle designs incorporate
bluntednose andleadingedgeregions inorder toreduce
Introduction
peak heating rates.
International interest in a space station, the possi- The present study looks to combine two well-
bilities for human exploration to other planets, and the established computationalcodes, one aTLNS algorithm
advancingage ofthe space shuttle (cid:13)eet have allbrought and the other PNS, for the solution of chemically-
the issue of advanced launch and reentry vehicles to the reacting, hypersonic (cid:13)ow(cid:12)elds. The technique is suc-
forefront. A critical design point for these vehicles is cessfully applied to blunted, multi-conic geometries at
during hypersonic reentry, when peak heating rates oc- both zero and non-zero angles of attack. Di(cid:11)erent sets
curandaerodynamiccontrole(cid:11)ectivenessmaybealtered of freestream conditions are considered, and the e(cid:11)ects
dueto(cid:13)ow(cid:12)eldphenomenonuniquetothehigh-altitude, of wall catalycity are investigated. Challenges and ob-
high-velocity conditions. The high temperatures and stacles to the consistent integrationofthe two codes are
high convective velocities relative to reaction times cre- observed and comments regarding the applicabilityand
ate an environment where chemical nonequilibrium ef- limitationsof the procedure are documented.
fects can be signi(cid:12)cant. Accurate aerothermodynamic
predictions duringthispartofthereentry trajectory are
Prelude to the Present Method
essential for sizing both the thermal protection system
and aerodynamic control surfaces. Ground based tests Recently, e(cid:11)orts have been made to combine the
simulatingthese(cid:13)ightconditions,includingconsiderable TLNS and PNS approaches in order to get timely, ac-
nonequilibrium e(cid:11)ects, are di(cid:14)cult to perform. Flight curate hypersonic viscous solutions while circumventing
tests can be prohibitivelyexpensive. some of the above mentioned limitations. Weilmuenster
3
Two popular computationalapproaches for obtain- and Gno(cid:11)o proposed a multi-blocksolution procedure,
ing aerothermodynamic predictions on these classes of inwhichthedomainisdividedintoblocksordered inthe
vehiclesaretosolvethethin-layerNavier-Stokes(TLNS) streamwisedirection. Thegeneralideaistomarchthese
equations orthe parabolizedNavier-Stokes (PNS) equa- blocks downstream, analogous to the PNS approach of
tions. TLNSisderivedfromthefullNavier-Stokesequa- marchingtwo-dimensionalplanes, and to solve the inte-
tions by neglecting viscous terms in the streamwise and riorofeachblockwithTLNS. Thisprocedureprincipally
cross(cid:13)ow directions. The assumptions inherent in the attacks the memory requirements inherent in obtaining
TLNS equations are often acceptable for a wide class a full-body TLNS solution by splitting the domain,but
of conditions and con(cid:12)gurations, including cases of hy- does not decrease the time required to obtain the solu-
personic, chemically reacting (cid:13)ow. Excessive compu- tion since TLNS remains the governing equations.
tational requirements can become a drawback to using The TLNS code used by Weilmuenster is the
TLNS as the entire solution domain is relaxed in time. Langley Aerothermodynamic Upwind Relaxation Algo-
1 4{7
Complexcon(cid:12)gurations cantaxcomputermemorywith rithm (LAURA). LAURA is a (cid:12)nite-volume, shock-
millionsof grid points, and solution times may be mea- capturing, hyperbolic equation solver with second-order
sured in CPU days. In addition, solving for chemical spatial accuracy for the steady-state solution of viscous
nonequilibriumcan, for some algorithmsperformingex- or inviscid hypersonic (cid:13)ows. The scheme employs a
act matrix inversions, increase the computer memory point implicitrelaxation strategy with the upwind (cid:13)ux-
8
and time requirements by the cube of the number of di(cid:11)erence splitting of Roe. The right-hand-side (RHS)
2 9
species considered. of the equations are formulated according to Yee with
10
The PNS equations are obtained from the full the entropy condition of Harten. Perfect gas, equilib-
Navier-Stokes equations by neglecting the time deriva- rium air, and nonequilibriumair calculations can all be
tives and the streamwiseviscous derivatives. Limitedto performed.
11
(cid:13)ow(cid:12)elds with streamwise supersonic (cid:13)ow outside the Greene hasextendedtheLAURAcodeintoaPNS
boundary layer, no streamwise separation, and weak version. With this method, LAURA-TLNS is used on
streamwise pressure gradients in the subsonic region, theblunt-noseportionofahypersonicvehicle. Atapoint
PNS algorithmsare well suited for solving sharp-nosed, in the (cid:13)ow(cid:12)eld consistent with the PNS equations, the
slender-body supersonic/hypersonic con(cid:12)gurations. Be- transferismadetoLAURA-PNS,whichisthenmarched
ing space marching and steady state, PNS formulations downthe remainderofa slender vehicle afterbody. This
can realize appreciable decreases in both computational particular formulation, being a TLNS extension, is lo-
2
callyiterativeinpseudo-timesteps, andits performance Modi(cid:12)cations for Compatibility
su(cid:11)ers from arriving as a PNS solver via a TLNS algo- Changes made to the LAURA pre-processor for
rithm, rather than being a code that was optimized as compatibility with UPS focus mostly on grid genera-
a PNS solver from inception. Thus, while this method tion. The grid on the cone portion of a sphere-cone was
signi(cid:12)cantly reduced the memory required to obtain a changed from being body normal to being axis normal
solution, it was not able to reduce solution time to the soastofacilitatespacemarchingonslender bodies. The
level desired. spacing normalto the body in the initialgrid was mod-
Upwind Parabolized Navier-Stokes Solver i(cid:12)ed so as to better capture the bow shock for vehicles
12{20
(UPS) is an upwind, (cid:12)nite-volume, state-of-the-art withveryslenderafterbodies. Thenumberofcellssolved
PNS code with chemical nonequilibrium capability. It on spherical nosecaps was reduced to 12.
is second-order accurate in the cross(cid:13)ow plane and (cid:12)rst The wall boundary conditions in LAURA were
order accurate inthemarchingdirection. Theequations changedtocorrespond withtheUPSwallboundarycon-
are approximately factored and solved implicitly, with ditions by switching fromthe standard LAURA bound-
21
the approach of Vigneron et al. employed to suppress aryconditionstothe primaryalternateboundarycondi-
departure solutions. tions. ThestandardLAURAviscouswallboundarycon-
UPS was identi(cid:12)ed as a code that, when com- ditionsapplythewallvalues,i.e.,zerovelocity,(cid:12)xedwall
bined with LAURA, might provide the tremendous re- temperature, etc., at the center of an image cell below
duction in vehicle solution time originally sought with thevehiclesurface. TheUPSapproachistousere(cid:13)ected
the LAURA-TLNS/LAURA-PNS method. The present boundaryconditionsfortheimagecell,soastoapplythe
methodseeks tocombineLAURAandUPSforaconsis- boundary conditions to be at the cell face de(cid:12)ning the
tentsolutionprocedure forair(cid:13)owsinchemicalnonequi- wall. The UPS approach is considered to be a higher-
librium. order methodthan the defaultLAURA boundaryimpo-
sition. However, the LAURA default boundary condi-
Previously, UPS has been joined with the TLNS
15 tions were found to be more robust than the re(cid:13)ected
code CNS by Lawrence et al. forperfect gas computa-
boundaryconditions,sothe LAURAsolutionswere (cid:12)rst
tions. NonequilibriumsolutionsarepresentedbyBuelow
22 23 partially converged with the standard boundary condi-
etal. andMuramoto usingUPSwiththeTLNScode
tions,and then switched to the re(cid:13)ective boundary con-
TUFF. LAURAhastheadvantageoverTUFFinthatit
ditions during the later stages of convergence after the
can handle generic, three-dimensionalgeometric shapes,
(cid:13)ow(cid:12)eld had stabilized. This switch is usually made at
as are encountered with real vehicle con(cid:12)gurations, and
thesametimespatialsecond-order accuracy isenforced.
is an upwind, (cid:12)nite volumemethod, like UPS.
Of the (cid:12)ve kinetic modelsavailablein LAURA,the
29
15 reaction model of Kang et al. was chosen as being
28
the closest match with the rates of Blottner in UPS.
Present Method Twofurther parametersweretoggledfromthedefaultin
LAURAtobetterdealwithslender-bodycon(cid:12)gurations.
A combined LAURA-UPS solution procedure has The eigenvalue limiter was set to be scaled by the cell
24
been implemented by Wood and Thompson for per- aspect ratioandthe upwindingofthe surface properties
fect gas and equilibriumair (cid:13)ows. That study included was turned o(cid:11).
detailed solutions for an axisymmetric perfect gas case The principle change made to UPS involved the
and a three-dimensional equilibriumair solution for the restart (cid:12)le. A jump in properties was observed during
25
Reentry F vehicle, including turbulence. Generally nonequilibriumrestarts. This was tracked to the use of
goodresultswere seenwiththecombinedmethod,anda afreestreamvalueofthemixturemolecularweightwhen
very signi(cid:12)cant reduction in solutiontimewas achieved. initiallydecoding the temperature from the energy and
The extension of this procedure to nonequilibrium air species concentrations, prior to marching. The remedy
calculations, however, is not straightforward, because was to read the local mixture molecularweight into the
while both UPS and LAURA use the same equilibrium standard restart (cid:12)le.
aircurve(cid:12)ts,theydonotusethesamechemistrymodels
for nonequilibriumair. Remaining Di(cid:11)erences Between the Codes
The TLNS LAURA solutions were sought using Some di(cid:11)erences in the chemistry models remain
a chemical nonequilibrium, thermal equilibrium kinetic between LAURA and UPS. Algorithmically, LAURA
model with a seven-species air model. The PNS solu- solves the chemistry equations with a fully-coupledpro-
tions were obtained with UPS using the seven species, cedure while UPS uses a loosely-coupled approach, but
seven reaction nonequilibriumair chemistry modelwith with the option for local subiterations to get a close ap-
28
the reaction rates of Blottner et al. proximation to a fully-coupled scheme. The two codes
3
500 x10-6 500 x10-6
400 400
m , m ,
kg 300 kg 300
m-s m-s
200 200
LAURA LAURA
UPS UPS
100 Gupta 100 Gupta
x103 x103
0 0
0 5 10 15 20 25 30 0 5 10 15 20 25 30
T, K T, K
Fig. 1 Comparisonof molecularoxygen species Fig. 2 Comparisonof nitric oxide species viscosities.
viscosities. 500 x10-6
compute the species enthalpies with fundamentally dif- 400
ferent approaches, asLAURAuses polynomialcurve (cid:12)ts m ,
while UPS uses interpolated table look-ups. This pre- kg 300
ventstheexactmatchingofspeciesconcentrations,inter-
m-s
nal energy, and temperature between the codes, though
200
LAURA
the magnitudeofthedi(cid:11)erence isconsidered tobe small
UPS
enough to not prohibit the interfacing of the codes.
100 Gupta
x103
Further di(cid:11)erences exist in the wayeach code com- 0 5 10 15 20 25 30
putesthebulkthermodynamicandtransportproperties. T, K
Thisleadstosmallmismatchesbetweenthecodesforpa-
rameterssuchasviscosityandspeedofsound. Oneques-
tionthisraises is whether tomatchthe non-dimensional Fig. 3 Comparisonof molecular nitrogen species
freestream quantities Mach number and Reynolds num- viscosities.
ber between the codes, or to match the dimensional
freestreamvelocityanddensity. ForhighReynoldsnum-
ber, hypersonic applications where Mach number inde- Interface Protocol
pendence has been reached, the decision made here is
The interface procedure between LAURA and UPS
to match the dimensionalfreestream conditions. As ex-
begins with the standard LAURA restart (cid:12)le for a
amplesofthe di(cid:11)erences in the transport property com-
converged chemical-nonequilibrium solution. From the
putations, plots of viscosity versus temperature are pre-
LAURArestart(cid:12)le,across(cid:13)owdataplaneisextractedto
sented for molecular oxygen, Fig. 1, and nitric oxide,
become the UPS starting plane. Currently, this plane is
Fig.2. The computationsof both LAURA and UPS are
chosen at least three cells upstream ofthe (cid:12)nalLAURA
presented along with the recommendations of Gupta et
solution plane in order to avoid possible contamina-
30
al., who conducted one ofthe mostrecent studies into
tionfromtheextrapolatedout(cid:13)owboundaryconditions.
transport property computations. Generally, the other
The variables available in the LAURA restart (cid:12)le are:
species viscosities match fairly well over the tempera-
the three velocity components, temperature, the seven
ture range 1000{30,000 K, with the molecular nitrogen
species densities, and the (cid:12)nite volumegrid,
viscosity computations presented in Fig. 3 as a typical
example. Sample computations of thermal conductivi-
ties performed for typical near wall conditions resulted [u;v;w;T;(cid:26)s;x;y;z]l
in a 4{5 percent higher value from UPS than LAURA.
It is di(cid:14)cult to predict a priori what e(cid:11)ect these dif- The variables needed by UPS to start are: mixture
ferences would have relative to solutions from the two density, the three momentumcomponents, total energy,
codes. mixture molecular weight, species mass fractions, and
4
the starting plane in (cid:12)nite volumeform, the Vigneron condition’s limitationon the pressure gra-
dientnear the wallset uposcillationsthat restricted the
[(cid:26);(cid:26)u;(cid:26)v;(cid:26)w;Et;M;cs;x;y;z]u stability of the marching UPS solution. The (cid:12)x to this
problem was to pass the LAURA temperatures directly
In the equations which follow, a subscripted \l" is used
through to the UPS species enthalpy interpolated table
toindicateaLAURAvariableorquantity,whilethesub-
look-ups,then to complete the computationof the total
script \u" refers to the corresponding UPS parameter.
energy as described above,
The variables required by UPS are obtained from
the LAURAvariablesinthe followingmanner. The grid 1 2 2 2 Pu
is transformed according to the transformation of the Et;u = (ul +vl +wl)+Hu(cid:0) (7)
2 (cid:26)u
physical coordinates as,
where,
xu =(cid:0)zl ; yu =xl ; zu =yl (1) X
Hu = hs;ucs;u (8)
The total density is found from summing the species s
densities,
X and,
(cid:26)u = (cid:26)s;l (2)
s hs;u =Cp;uTl+h0;s;u (9)
The three components of momentumare obtained from
Since both LAURA and UPS are (cid:12)nite volumefor-
the velocity components and the total density,
mulations, the UPS starting-plane grid is taken at a
streamwiselocationcorresponding tothe locationofthe
(cid:26)uu =(cid:26)u(cid:1)((cid:0)wl) ; (cid:26)vu =(cid:26)u(cid:1)ul ; (cid:26)wu =(cid:26)u(cid:1)vl (3)
cell-centered LAURA data. Converting the nondimen-
Species mass fractions are found by dividingthe species sionalizations so that the UPS velocities are normal-
densities by the total density, ized by the freestream speed of sound, rather than the
freestreamvelocityasisdoneinLAURA,andperforming
(cid:26)s;l the curvilinear transformationbetween the two codes,
cs;u = (4)
(cid:26)u
(cid:24)u =(cid:0)(cid:17)l ; (cid:17)u =(cid:16)l ; (cid:16)u =(cid:24)l (10)
The mixture molecular weight is found by applying the
perfect gasequationofstatetothemixturetemperature
completes the interface process.
and pressure,
(cid:26)uR(cid:22)Tl
Mu = (5)
Pu
Results
where the mixture pressure was determined from sum-
mingthe species partial pressures, The present method is successfully applied to two
primarycon(cid:12)gurationsand(cid:13)owconditions. Case1isan
Xcs;uR(cid:22)Tl axisymmetricsphere-cone,chosentocorrespondwiththe
Pu =(cid:26)u (6) 31
Ms resultsofGuptaetal. Thenoseradiusis0.0381mand
s
thebody angleis sixdegrees. The freestreamconditions
a step consistent with the assumption,commonto both areforMach25atanaltitudeof53.34km(175kft.). The
codes, that the working(cid:13)uid is a mixtureof idealgases. wall temperature is held (cid:12)xed at 1260 K, with a non-
TheUPStotalenergynowremainstobecomputed. catalytic chemistry boundary condition. Case 2 is for
Initially,thee(cid:11)ortwasmadetotakethetemperatureand Mach28(cid:13)owoverthesphere-cone-cylinder-(cid:13)are con(cid:12)gu-
32
species densities from the LAURA solution, pass them rationstudiedbyBhuttaetal. atbothtwoand(cid:12)vede-
through the LAURA enthalpy curve (cid:12)ts, add in the ve- gree angles ofattack. Thiscon(cid:12)guration has a0.1524m
locity and species property information, and obtain a spherical nosecap followed by a nine degree cone. Af-
total energy that would be passed directly to UPS. A ter 10 nose radii the cone is followed by a cylinder and
problem was encountered when UPS took this energy then a (cid:12)ve degree (cid:13)are, each of 10 nose radii length.
and decoded temperature and pressure. The di(cid:11)erences The freestream conditions correspond to an altitude of
between the UPS and LAURA enthalpy computations 83.8 km, (275 kft.), at a Reynolds number per meter of
lead to di(cid:11)erences between the decoded UPS tempera- 6148. The wall temperature for this case is 833 K and
tures and pressures and the original LAURA tempera- a fully-catalytic boundary condition is employed. Ta-
tures and pressures. These variations, in combination ble 1 presents a summaryof the nominalconditions for
with a (cid:12)xed wall-temperature boundary condition and thetwocases. Forallcalculationsthe freestreamspecies
5
0.15
M = 25
Table 1 Nominalconditions for Cases 1 and 2. R = 0.0381 m
n
q = 6 deg
b
Case 1 Case 2 0.10
Con(cid:12)guration sphere-cone blunted multi-conic
M1 25 28 Y, m
(cid:0)1 5
Re (m ) 3.95 (cid:2)10 6148
Altitude (km) 53.34 83.8 0.05
Rn (m) 0.0381 0.1524
(cid:18)b (deg) 6 9-0-5 (10 Rn each) UPS starting point
Twall (K) 1260 833
0.00
Wallcatalycity none fully 0.00 0.05 0.10 0.15 0.20
(cid:11) (deg) 0 2, 5 X, m
Fig. 4 Case 1 LAURA computationalgrid, showing
concentrations were set at,
every fourth body-normalpoint.
2 3 2 3
cN2 0:767
66 cO2 77 66 0:233 77
6666 ccNO 7777=6666 67::271578(cid:2)(cid:2)1100(cid:0)(cid:0)290 7777 (11) ianctteuraplolUaPteSdginridt.heTshtirseaims wthiesestdairnedcatirodnUtPoSobatpapinroathche
4 cNO 5 4 4:981(cid:2)10(cid:0)5 5 for handling external grids. A signi(cid:12)cant overlap of the
cNO+ 4:567(cid:2)10(cid:0)24 solution domains was deliberately chosen for this case
to allow for a direct code-to-code comparison between
The seventh specie, electrons, are found from a charge LAURA and UPS. In general, an overlap of this size is
balance with the ionized nitric oxide, not required by the combined procedure.
The UPS solution was carried out 100 nose radiito
Ne(cid:0) =NNO+ (12)
3.81 m by extending the external grid downstream in a
conicalextrapolation. The gridwasmoderatelyadapted
tothe solutioninthe body-normaldirection asthe solu-
Case 1 tion proceeded, in such a way as to maintain the origi-
Aviscous,second-order accurateTLNSLAURAso- nalgrid spacing at the wallwhile linearlystretching the
lution was obtained for Case 1 with chemical nonequi- outer 60 percent of the grid. This adaption routine is
librium, thermal equilibrium, and a non-catalytic wall currently not fully integrated into the version of UPS
condition, implementedin both codes as, used here, and relies uponthe user toprovidethe neces-
(cid:12) sary stretching parameters.
@cs(cid:12)(cid:12) =0 (13) Figure 5 displays Mach contours from the LAURA
(cid:12)
@n
wall and UPS solutions, covering the overlap region to (cid:12)ve
nose radii. The location of the UPS starting plane is
i.e., the mass fractions of the image cells are set equal
indicated,andtheUPSMachcontoursareoverlaidupon
to the mass fractions of the (cid:12)rst cell outside the wall.
the LAURA Mach contours downstream of that point.
The axisymmetricLAURA computationalgrid contains
Excellentagreementisseen between thepresent method
64 cells normal to the body and 28 cells in the stream-
and the LAURA-onlysolution.
wise direction, extending (cid:12)ve nose radii to 0.19m. This
gridwasadaptedusingthestandard LAURAgridadap- Figures 6 and 7 plot surface pressures, normalized
tion routine. Figure 4 displays the (cid:12)nal LAURA grid, by twice the freestream dynamic pressure, versus the
for clarity showing only every fourth point in the body- streamwisedistancemeasuredalongthesurface,normal-
normal direction. For consistency, Fig. 4 and all subse- ized by the nose radius. The viscous shock-layer (VSL)
31
quent (cid:12)gures use the UPS coordinate system. The loca- solutions of Gupta are included for comparison. The
tion where the UPS starting plane was extracted from VSL equations employ a further approximation to the
the LAURA solution is indicated in Fig. 4. That por- governingequations beyond the PNS equations to allow
tionoftheLAURAgriddownstreamoftheUPSstarting solutionmarchinginboththestreamwiseandcircumfer-
planewassuppliedasanexternalgridtoUPS. Sincethe ential directions. Figure 6 is a close-up on the interface
UPS marching step size was smaller than the LAURA region, extending to 10 nose radii. The Gupta-VSL so-
cell sizes shown in Fig. 4, the LAURA grid was linearly lution extends the full 10 nose radii, while the LAURA
6
107
0.15 M = 25
M = 25 LAURA
R = 0.0381 m
Rn = 0.0381 m UPS q, n
q = 6 deg Non-catalytic wall
b W a = 0 deg
0.10
m2
Y, m 106 LAURA
UPS
Gupta-VSL
0.05
UPS starting point
105
0.00 0 1 2 3 s 4 5 6
0.00 0.05 0.10 0.15 0.20
R
X, m n
Fig. 8 Case 1 surface heating|the interface region.
Fig. 5 Mach contours: UPS solution overlaid upon
LAURA solution.
solution was terminated at six nose radii in this plot.
The UPS solution was initiated at two nose radii and
extends to 10 nose radii. Excellent agreement is seen
100
between the UPS and LAURA solutions. The Gupta-
M = 25
VSLsolutionisseen toagree verywellwiththeLAURA
R = 0.0381 m
n and UPS solutions outside of the region of sphere-cone
P Non-catalytic wall
wall tangency, where Gupta-VSL predicts higher pressures,
r u2 a = 0 deg probably due to the surface curvature smoothing em-
¥ ¥
10-1 LAURA ployed in this VSL code. A slight pressure bump ap-
pears in the UPS solution at (cid:12)ve nose radii. The cause
UPS
forthis isnotknownatthistime,andmaybe aresidual
Gupta-VSL
of the LAURA-UPS interface. However, the e(cid:11)ect is lo-
calizedanddoesnotappeartoin(cid:13)uencethedownstream
solution.
10-2
0 2 4 s 6 8 10 Figure7extendsthesurfacepressure plotoutto100
R nose radii, capturing the overexpansion and recompres-
n
sion regions. There is a maximum di(cid:11)erence between
the UPS and Gupta-VSLsolutionsof3{4percent inthe
Fig. 6 Case 1 surface pressures|the interface region. recompressionregion. AsinFig. 6,theLAURAsolution
was terminated at six nose radii.
100 Surface heat transfer results forLAURA,UPS, and
M = 25 Gupta-VSL are presented in Figs. 8 and 9. Figure 8
Rn = 0.0381 m plots the interface region out to a distance of six nose
Non-catalytic wall radii. Similartrends areseen intheheatingaswere seen
P 10-1 a = 0 deg for the pressure in this region. The heating at the in-
wall
terface between the LAURA and UPS codes picks up
r ¥ u¥2 smoothly, but there is a bump in the UPS heating be-
10-2 tweenfourand(cid:12)venoseradii,correspondingtothepres-
LAURA sure bump discussed earlier. The Gupta-VSL heating is
UPS elevated above the LAURA-UPS heating in the region
Gupta-VSL of the sphere-cone juncture.
10-30 20 40 s 60 80 100 Figure9carriesthepresentmethodandGupta-VSL
heating out to 100 nose radii. Note that the LAURA
R
n heatingterminatesat sixnose radii. A noticeable di(cid:11)er-
ence exists between the UPS and Gupta-VSL solutions
Fig. 7 Case 1 surface pressures to 100 nose radii. that persists from the overexpansion region on down-
stream. The Gupta-VSL results are consistently 18{22
7
107
M = 25
Table 2 Reaction rates for ionized nitric oxide.
R = 0.0381 m
q, n
Non-catalytic wall
W a = 0 deg kf kb
m2 106 LAURA KBlaontgtner 91::043(cid:2)(cid:2)110069TT10:5:5eexxpp(cid:0)(cid:0)3312T9T40000 16::87(cid:2)(cid:2)11001291TT(cid:0)(cid:0)11::05
UPS
Gupta-VSL
Kangreaction sets are similaroridenticalformostreac-
tions,asigni(cid:12)cantdi(cid:11)erence inthe equilibriumconstant
105
0 20 40 s 60 80 100 can occur in the equation controllingproduction of ion-
R ized nitric oxide,
n
N +O *)NO++e(cid:0) (14)
Fig. 9 Case 1 surface heating to 100 nose radii.
Table2liststheforwardandbackwardrates forEqn.14
1.0 from the two kinetic models. At a temperature of 1280
M = 25
K, an average temperature for a Case 1 surface cell, the
Non-catalytic wall
0.8 Blottner equilibriumconstant,
X / R = 5
n
a = 0 deg kf
Y-Yb 0.6 keq = kb (15)
Y
shock (cid:0)16
0.4 for this reaction is 2:33(cid:2) 10 , while the Kang equi-
librium constant is two orders of magnitude lower at
LAURA
(cid:0)18
6:58(cid:2)10 . Underthe(cid:13)owconditionsforthiscase,both
0.2 UPS
atomic oxygen and atomic nitrogen concentrations at
the surface are large,withthe (cid:13)owconsisting ofroughly
0.0
0.00 0.05 0.10 0.15 0.20 0.25 equalpartsatomicoxygen,atomicnitrogen,andmolecu-
c larnitrogennearthewall. ThenetresultisthattheUPS
O
solution produces signi(cid:12)cantly more ionized nitric oxide
relative to the starting solution provided by LAURA,
Fig. 10 Case 1 atomicoxygen massfraction pro(cid:12)les andatafastrate. Thiscreates amarchingstep-size sta-
at X/Rn = 5. bility restriction characterized by the Damko(cid:127)hler num-
ber,
(cid:28)flow
Da = (16)
percent lowerthan the UPS heating. More investigation (cid:28)reactions
isrequiredtounderstandwhythere isthislevelofdi(cid:11)er- see Ref. 33 pp. 149{154, which was found to be exacer-
encebetweenthesolutions,but,whileFigs.1and2show bated by a tight grid spacing near the wall.
goodagreementbetween theUPSandGuptaviscosities, AcompromisewassoughtwherebytheLAURAgrid
there aredi(cid:11)erences inother aspects ofthe kineticmod- was modi(cid:12)edto double the cell size of the (cid:12)rst grid cell,
els which maybe contributing to the heating disparity. which sets a nominal cell Reynolds number of two at
Looking speci(cid:12)cally at reacting chemistry e(cid:11)ects, the wall. This was found to still allow accurate reso-
Fig. 10 pro(cid:12)les the atomic oxygen mass fraction ver- lution of gradients at the wall while somewhat relaxing
sus normal distance from the surface, as a fraction of the Damko(cid:127)hler imposed marching stability restriction.
the shock layer, at an axialposition (cid:12)ve nose radii from In this case the non-linearity inherent in the chemical
the nosetip. The pro(cid:12)les fromthe LAURAand UPS so- reactions allows for marching steps more than twice as
lutions are seen to be similar,with a di(cid:11)erence in mass large as were possible with a wall cell Reynolds number
fractionatthesurfaceoftwopercent. Themassfraction of one. Larger grid spacings at the wall were found to
gradients at the surface are seen to be zero, as de(cid:12)ned be too coarse to provide suitable LAURA solutions.
by the non-catalytic wall assumption. The LAURA solution for this case was converged
This di(cid:11)erence in oxygen mass fraction at the sur- through an L2 norm of the residual of seven orders of
face becomes critical in realizing the di(cid:14)culties encoun- magnitudein 2200 iterations. The total CPU time on a
tered in obtaining the combined LAURA-UPS solution Cray 2 was 1411 seconds. Figure 11 contains the con-
forthisparticularcon(cid:12)guration. WhiletheBlottnerand vergence history of the LAURA solution. One caveat to
8
101
actly matchthe other was not attempted,being beyond
100
al the scope of the present study.
du 10-1 TheUPSsolutioninstabilityforthiscaseistypically
esi 10-2 manifested by a divergence of the cell temperatures at
of r 10-3 thewall. Itwasthoughtthatthere(cid:13)ectedboundarycon-
m ditions, where the wall temperature is enforced only as
r 10-4 thegeometricaverageofthe imagecelltemperature and
o
n the temperature of the (cid:12)rst cell above the wall, might
10-5
L2 be contributing to the instability because the wall tem-
10-6 perature is not explicitly enforced. The UPS boundary
10-7 conditionswere altered to applythe wallboundary con-
0.0 0.1 0.2 0.3 0.4 0.5
ditionsofnoslip,nopenetration, and(cid:12)xed walltemper-
CPU, hours ature at the image cell center, and a new solution was
obtainedwith both LAURAand UPS using this bound-
ary condition, but no appreciable improvement in the
Fig. 11 Case 1 LAURA convergence history.
stability of the present method was observed. An ef-
fort to enforce a limiter on the Newton iteration used
this performance is that the solution was begun from a
to decode the temperature and pressure given the to-
converged solution for a similar,but not identical, case.
tal energy, mixture density, and species concentrations
The sharp spikes occurring early in Fig. 11 are the re-
alsofailedto produce a useful relaxationofthe stability
sultofgridadaptations,whilethelaterspikesaredue to
restriction on the marching step size.
shock ringing.
Some attempts at solution smoothing and solution
The UPS solution was obtained on a Cray YMP
modulation were tried with the present method. Sev-
with a (cid:12)nal marching step size of 0.25 mm. This is
eral approaches were attempted, beginning by trying to
a small step size in relation to other cases which have
march the UPS solution one step, modifying the origi-
been run with the present method, but is a result of
nal interface plane with an under relaxation scheme by
thepreviouslymentionedmarchingstabilityrestrictions.
23 adding some fraction of the di(cid:11)erence between the ini-
Muramoto reports using the same marching step size
tial starting plane data and the (cid:12)rst step solution, and
for a Mach 20,seven-degree sphere-cone nonequilibrium
repeating in a locally iterative procedure. The idea was
case, with a modi(cid:12)ed version of UPS, and Tannehill
18 to allowthe solution to relax without creating excessive
et al. report usingastep size of0.2mmonan axisym-
transients. The next attempt tried to march the UPS
metric cone. The full UPS solution to 100 nose radii
solution while modulatingit with the LAURA solution,
required 4198 seconds.
so that the (cid:12)rst step was 10 percent UPS and 90 per-
cent LAURA, the second step 20 percent UPS and 80
Attempts at Larger Marching Steps
percent LAURA, and so on. While these attempts had
While the axisymmetricgeometry of Case 1 was able to some small success in delaying or postponing the insta-
be solvedbythe present methodinareasonableamount bilitywithlarge step sizes, they were unable tosuppress
of computer time with the small marching step size, the instability enough to solve a signi(cid:12)cant portion of
there is concern that a full-sized, three-dimensional ve- thegeometrywithlargemarchingsteps. Moreexoticso-
hicle might require excessive computationalresources if lutionmodulationmethods were tried whereby the UPS
conditions were such that the stability restriction ob- domainwassplittoallowtheinviscid,viscous,andnear-
served here applied. Several attempts were made to wallregionstorelaxfromthe LAURAsolutionatdi(cid:11)er-
enhance the stability of the UPS marching solution entrates,buttheresultwasstillthesame|themarching
for the Case 1 conditions. Local chemistry iterations step-size was limitedto the millimeterrange or less.
33
were added, second- and fourth-order subsonic smooth- AnattemptatasolutionwasmadeusingthePark
ing terms were turned on, the safety factor applied to kineticmodelinLAURA,withnomorejusti(cid:12)cationthan
theVigneronconditionwasadjusted,andtheeigenvalue thatitisareadilyavailableoption. Perhapspredictably,
stability parameters EPSA and EPSS in the UPS in- this did not produce any improvementin stability. The
put(cid:12)lewerechanged. Somesmallstabilityimprovement locationoftheinterfacepointwasvariedaswell,without
was found by increasing the values of the second-order producing a change in the behavior of the solution with
implicitsmoothingterm,theVigneronsafetyfactor,and the present method.
the stabilityparameterEPSA, but not enough to allow Changes tothe gridincluded trying 40,64,and 128
order-of-magnitude larger step sizes. Direct reprogram- points in the body normal direction with nominal cell
mingof either code’s complete chemistry package to ex- Reynolds numbers at the wall of 0.5, 1, 2, 5, and 10.
9
108
M = 28
M = 25 2
R = 0.1524 m
R = 0.0381 n
n a = 2 deg
Fully-catalytic wall
q,107 a = 0 deg 1
W Y, m
m2 UPS
0
106 Gupta-VSL
-1 UPS starting point
105
0 20 40 s 60 80 100 0 1 2 3 4 5
R X, m
n
Fig. 12 Case 1 heat transfers for a fully-catalyticwall. Fig. 13 LAURA Case 2 symmetryplane grid, showing
every eighth radialpoint.
The numberof points did not seem to alter the solution
appreciably for this con(cid:12)guration, but as discussed ear- species concentrations equal to their freestream values,
lierthe cellsize atthewallprovedtobe very important.
The tradeo(cid:11) had to be made between a tight clustering cs;wall =cs;1 (17)
at the wallfor good gradient resolution and a morerea-
Since the species concentration gradients are no longer
sonable cell aspect ratio to allow feasible marchingstep
zero at the wall for this case, as they were for the non-
sizes.
catalytic solutions, a computation of the di(cid:11)usive heat-
A (cid:12)nalparametric onthe basic Case 1solution was ing rate was added to the UPS surface property output
performed by employinga fully-catalyticwallinstead of routine as,
thenon-catalyticboundarycondition. Thesurfaceheat- (cid:12)
transfer results for this case are presented in Fig. 12. qdiffusive= kLe X hs @cs(cid:12)(cid:12)(cid:12) (18)
For this solution a march larger step size was possible Cp s @n wall
withUPS,because thefully-catalyticwallconditioncre-
ates a di(cid:11)erent gas composition in the near-wall region Computationsof the di(cid:11)usive heating for the cases con-
which does not involvethe ionized nitric oxide reaction, sidered in the present study showed its contribution to
Eqn. 14, to the same degree as the non-catalytic solu- the totalheat transfer to be a very smallpercentage. In
tion. However, as can be seen in Fig. 12 the heating the calculations of Ref. 32 a variable wall temperature
fromthepresentmethodimmediatelydownstreamofthe was employed, but for the present calculations a (cid:12)xed
interface region does not look good, although the heat walltemperature of833K wasused. This waschosen as
transfers agree well from 15{100 nose radii with the re- a rough average to use for comparison with the results
sults ofGupta for the samecon(cid:12)guration. Interestingly, of Bhutta.
the samebehaviorinthe UPSheatingnear the interface
point is reported by Muramoto in Fig. 11 of Ref. 23 for Two Degrees Angle of Attack
an axisymmetric,seven degree sphere-cone with a fully-
The LAURA symmetry plane grid for this case is dis-
catalytic boundary condition. In discussing his result,
22 played in Fig. 13, for clarity showing only every eighth
Muramoto further cites Buelow as another investiga-
pointin the radialdirection. The fullLAURA grid con-
tor who has seen a similar heating behavior with UPS.
tains 51 streamwise cells, 18 circumferential cells, and
128 radial cells. The UPS starting plane was extracted
from the (cid:12)fteenth streamwise cell in the LAURA solu-
tion.
Case 2
Figure14containsbothwindsideandleesidesurface
Three-dimensional (cid:12)nite-rate chemistry solutions pressures, normalized by twice the freestream dynamic
were sought for the Case 2 con(cid:12)guration for Mach 28 at pressure, versus axial distance, normalized by the nose
twoand(cid:12)ve degree anglesofattack. The wallboundary radius, for both the full-body LAURA solution and the
conditionwasset tobe fullycatalytic,whichinboththe coupled LAURA-UPS solution of the present method.
LAURA and UPS versions employed here sets the wall The agreement is very good, with the most noticeable
10