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PECOSPredictive Engineering and Computational Sciences
Fully-Implicit Navier-Stokes (FIN-S)
Benjamin S. Kirk
NASA Lyndon B. Johnson Space Center
July 21, 2010
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 1 / 28
https://ntrs.nasa.gov/search.jsp?R=20100025561 2018-07-07T01:52:08+00:00Z
Acknowledgments
PECOS Collaborators & Support
• Todd Oliver
• Roy Stogner
• Marco Panesi
• Karl Schulz
• Paul Bauman
• Juan Sanchez
• Graham Carey
• Chris Simmons
• Bob Moser
NASA JSC• Adam Amar
• Brandon Oliver
• Jay LeBeau
• Randy Lillard
Sandia NationalLabs• Steve Bova
• Ryan Bond
NASA Ames• Michael Wright
• Todd White
• Joe Olejniczak
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 2 / 28
• FIN-S is a SUPG finite element code for flow problems under activedevelopment at NASA Lyndon B. Johnson Space Center and withinPECOS
I The code is built on top of the libMesh parallel, adaptive finite elementlibrary
I The initial implementation of the code targeted supersonic/hypersoniclaminar calorically perfect gas flows & conjugate heat transfer
I Initial extension to thermochemical nonequilibrium about 9 months agoI The technologies in FIN-S have been enhanced through a strongly
collaborative research effort with Sandia National Labs
• NASA has allowed me to work here with the PECOS team sinceSeptember
• FIN-S background and high-level overview was first presented to theDOE review team in October
• This talk will highlight some of new capabilities and discuss ongoingefforts
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 3 / 28
1 Software Engineering
2 Physical ModelingGoverning EquationsThermochemistryTurbulence Modeling
3 ResultsViscous Reacting FlowAdaptive Mesh RefinementTurbulent Flow
4 Related Efforts & Ongoing WorkHigh-Temperature ThermochemistryVerificationNear-term Effort
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 4 / 28
Software Engineering
Development Environment• Integration into PECOS Redmine development environment
I Source tree now housed under PECOS svn repositoryI Redmine ticket system is being used to track feature requests,
bugfixes, etc. . .I Automatic Buildbot regression testing
• Doxygen-based source code documentation
• Rigorous modeling document
• Example suite, unit tests, regression tests
• GNU automake build system
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 5 / 28
Software Engineering
FIN-S Code Reuse and Dependencies• autoconf
• automake
• libtool
• Boost• Cantera
I BLASI LAPACK
• libMesh
I MPII Intel R© Threading Building BlocksI PETSc
• BLAS• LAPACK• MPI
! LaTeX Error: Too deeply nested.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 6 / 28
Software Engineering
FIN-S Code Reuse and Dependencies• autoconf
• automake
• libtool
• Boost• Cantera
I BLASI LAPACK
• libMeshI MPII Intel R© Threading Building BlocksI PETSc
• BLAS• LAPACK• MPI
! LaTeX Error: Too deeply nested.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 6 / 28
Software Engineering
FIN-S Code Reuse and Dependencies• autoconf
• automake
• libtool
• Boost• Cantera
I BLASI LAPACK
• libMeshI MPII Intel R© Threading Building BlocksI PETSc
• BLAS• LAPACK• MPI
! LaTeX Error: Too deeply nested.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 6 / 28
Software Engineering
FIN-S Code Reuse and Dependencies• autoconf
• automake
• libtool
• Boost• Cantera
I BLASI LAPACK
• libMeshI MPII Intel R© Threading Building BlocksI PETSc
• BLAS• LAPACK• MPI
! LaTeX Error: Too deeply nested.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 6 / 28
Physical Modeling Governing Equations
Governing Equations• Extension from a single-species calorically perfect gas to a reacting
mixture of thermally perfect gases requires species conservationequations and additional energy transport mechanisms
∂ρ
s
∂t+ ∇ · (ρ
s
u) = 0
∂ρu
∂t+ ∇ · (ρuu) = −∇P + ∇ · τ
∂ρE
∂t+ ∇ · (ρHu) = −∇ · q + ∇ · (τu)
+ ∇ ·(ρ
ns∑s=1
hsDs∇cs
)
• Problem class may also require a multitemperature thermalnonequilibrium option
∂ρeV∂t
+ ∇ · (ρeV u) = −∇ · qV + ∇ ·(ρ
ns∑s=1
eV sDs∇cs
)+ ωV
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 7 / 28
Physical Modeling Governing Equations
Governing Equations• Extension from a single-species calorically perfect gas to a reacting
mixture of thermally perfect gases requires species conservationequations and additional energy transport mechanisms
∂ρs∂t
+ ∇ · (ρsu) = ∇ · (ρDs∇cs) + ωs
∂ρu
∂t+ ∇ · (ρuu) = −∇P + ∇ · τ
∂ρE
∂t+ ∇ · (ρHu) = −∇ · q + ∇ · (τu) + ∇ ·
(ρ
ns∑s=1
hsDs∇cs
)
• Problem class may also require a multitemperature thermalnonequilibrium option
∂ρeV∂t
+ ∇ · (ρeV u) = −∇ · qV + ∇ ·(ρ
ns∑s=1
eV sDs∇cs
)+ ωV
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 7 / 28
Physical Modeling Governing Equations
Governing Equations• Extension from a single-species calorically perfect gas to a reacting
mixture of thermally perfect gases requires species conservationequations and additional energy transport mechanisms
∂ρs∂t
+ ∇ · (ρsu) = ∇ · (ρDs∇cs) + ωs
∂ρu
∂t+ ∇ · (ρuu) = −∇P + ∇ · τ
∂ρE
∂t+ ∇ · (ρHu) = −∇ · q + ∇ · (τu) + ∇ ·
(ρ
ns∑s=1
hsDs∇cs
)
• Problem class may also require a multitemperature thermalnonequilibrium option
∂ρeV∂t
+ ∇ · (ρeV u) = −∇ · qV + ∇ ·(ρ
ns∑s=1
eV sDs∇cs
)+ ωV
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 7 / 28
Physical Modeling Thermochemistry
Thermodynamics & Transport Properties• Thermochemistry models have been extended for a mixture of
vibrationally and electronically excited thermally perfect gases
eint =etrans + erot + evib + eelec + h0
=
ns∑s=1
csetranss (T ) +
∑s=mol
cserots (T ) +
∑s=mol
csevibs (TV ) +
ns∑s=1
cseelecs (TV ) +
ns∑s=1
csh0s
Here we have assumed that T trans = T rot = T and T vib = T elec = TV
• The transport properties have been extended as requiredI Species viscosity given by Blottner curve fitsI Species conductivities determined from an Eucken relationI Mixture transport properties computed via Wilke’s mixing ruleI Mass diffusion currently treated by assuming constant Lewis number
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 8 / 28
Physical Modeling Thermochemistry
Chemical Kinetics• We consider r general reactions of the form
N2 +M 2N +M. . .
N2 + O NO + N
. . .
• The reactions are of the form
Rr = kbr
ns∏s=1
(ρsMs
)βsr− kfr
ns∏s=1
(ρsMs
)αsr
where αsr and βsr are the stoichiometric coefficients for reactants and products
• The source terms are then
ωs =Ms
nr∑r=1
(αsr − βsr) (Rbr −Rfr)
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 9 / 28
Physical Modeling Thermochemistry
Kinetic Rates• The forward rate coefficients are defined with a modified Arrhenius
law as a function of some temperature T
kfr(T)
= CfrTηr exp
(−Ear/RT
)where the rate constants are determined empirically.
• The corresponding backward rate coefficient can be found using theprinciple of detailed balance and the equilibrium constant Keq
Keq =kfrkbr
• In thermal equilibrium T = T . We are currently using CANTERA in thisregime.
• In thermal nonequilibrium T = T (T, TV ) and typical hackery ensues.
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 10 / 28
Physical Modeling Turbulence Modeling
Turbulence Models• Use standard closure assumptions and eddy viscosity models
• Spalart-Allmaras: µt = ρνsafv1
∂ρνsa∂t
+∇ · (ρuνsa) = cb1Ssaρνsa − cw1fwρ(νsad
)2+
1
σ∇ · [(µ+ ρνsa)∇νsa] +
cb2σρ∇νsa ·∇νsa
• k–ω (1988): µt = ρk/ω
∂ρk
∂t+ ∇ · (ρuk) = τ : ∇u− β∗ρkω + ∇ · [(µ+ σ∗µt)∇k]
∂ρω
∂t+ ∇ · (ρuω) = α
ω
kτ : ∇u− βρω2 + ∇ · [(µ+ σµt)∇ω]
• k–ω (2006) and SST soon to come
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 11 / 28
Results Viscous Reacting Flow
2D Extended Cylinder• Laminar flow in thermal equilibrium
• No-slip, adiabatic, noncatalytic wall
• Chemical nonequilibrium, 5 species air (78% N2, 22% O2)
U∞ = 6, 731 m/sec
ρ∞ = 6.81× 10−4 kg/m3
T∞ = 265 K
• Blottner/Wilke/Eucken with constant Lewis number Le = 1.4 fortransport properties
• Mesh, iterative convergence
• FIN-S/DPLR comparison
• Weak & Strong Scaling
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 12 / 28
Results Viscous Reacting Flow
10000950090008500800075007000650060005500500045004000350030002500200015001000
TemperatureT (K)
U∞ = 6 ,731 m/sρ∞ = 6.81×10-4 kg/m3
T∞ = 265 K
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 13 / 28
Results Viscous Reacting Flow
Mesh Convergence
x (m)
T(K
)
-0.02 -0.015 -0.01 -0.005 00
2000
4000
6000
8000
10000
12000
14000
400×400200×200100×100
x (m)
ρ(k
g/m
3 )
-0.02 -0.015 -0.01 -0.005 00.000
0.002
0.004
0.006
0.008
0.010
0.012
400×400200×200100×100
x (m)
P(N
/m2 )
-0.02 -0.015 -0.01 -0.005 00
5000
10000
15000
20000
25000
30000
400×400200×200100×100
x (m)
u(m
/s)
-0.02 -0.015 -0.01 -0.005 00
1000
2000
3000
4000
5000
6000
7000
400×400200×200100×100
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 14 / 28
Results Viscous Reacting Flow
Iterative Convergence
Time Step
Rel
ativ
eT
rans
ient
Res
idua
l,|∆U
/∆t| ∞
Tim
eS
tep
Siz
e,∆t,
(sec
onds
)
0 50 100 150 200 25010-10
10-8
10-6
10-4
10-2
100
10-10
10-8
10-6
10-4
10-2
100
100×100200×200400×400
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 15 / 28
Results Viscous Reacting Flow
Code-to-Code Comparison –Stagnation Line
x (m)
T(K
)
-0.025 -0.02 -0.015 -0.01 -0.005 00
2000
4000
6000
8000
10000
12000
14000
FIN-SDPLR
x (m)
Spe
cies
Mas
sF
ract
ion
-0.025 -0.02 -0.015 -0.01 -0.005 010-3
10-2
10-1
100
FIN-SDPLR
O2
NO
NO
N2
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 16 / 28
Results Viscous Reacting Flow
Code-to-Code Comparison –Flank Line
T (K)
y(m
)
3000 3500 4000 4500 5000 5500 6000 65000.1
0.12
0.14
0.16
0.18
FIN-SDPLR
Species Mass Fraction
y(m
)
10-6 10-5 10-4 10-3 10-2 10-1 1000.1
0.12
0.14
0.16
0.18
FIN-SDPLR
O2
NO
NO
N2
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 17 / 28
Results Viscous Reacting Flow
Speedup
Number of Processor Cores
Spe
edup
100 101 102 103100
101
102
103
IdealScaled-Size (Weak) ScalingFixed-Size (Strong) Scaling
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 18 / 28
Results Adaptive Mesh Refinement
(x/R)
T(K)×10
-3
-1.3 -1.25 -1.2 -1.15 -1.1 -1.05 -10
1
2
3
4
5
6
7
240×480
060×120090×180
Adaptive
120×240
Stagnation LineTemperature Profile
AMR – 13,079 node mesh, “spot on” with uniform 115,921 node mesh
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 19 / 28
Results Turbulent Flow
Initial Turbulent Results• Fully turbulent flow over a flat plate
• k-ω turbulence model; calorically perfect N2; adiabatic wall
• ReL ≈ 1× 106; M∞ ≈ 0.2
Boundary layer profiles at trailing edge
100
101
102
103
0
5
10
15
20
25
y+
u+
0 0.2 0.4 0.6 0.8 1 1.20
0.5
1
1.5
2
2.5
3
3.5
y/δ
k/ u
τ2
Code and solution verification activities are ongoingBenjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 20 / 28
Related Efforts & Ongoing Work High-Temperature Thermochemistry
HTChem• The high-temperature thermodynamic and transport models currently
implemented in FIN-S are one of several possible choices, and serveto provide the minimum set required for algorithm development
• It is expected that these simplified models will be invalidated forcertain problem classes and that more complex models will berequired
• Similar thermochemical models are required by other areas ofPECOS research, e.g. ablation and shock layer radiation
• The HTChem library is being developed to consolidate efforts andprovide a common source for requisite high-temperaturethermochemistry and transport property data
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 21 / 28
Related Efforts & Ongoing Work Verification
Manufactured Analytical Solution Abstraction Library
• Dearth of exact solutions necessitates method of manufactured solutions
• Some manufactured solutions exist for the calorically perfect Navier-Stokesequations
I Developed in large part by Sandia National LabsI Specific solutions for field, boundary condition order-of-accuracy verification
• Existing solutions provide a necessary but not sufficient test suiteI Will need to develop many more solutions to verify reacting flows with complex
transport models
• Manufactured solutions are a valuable resource that should be accessible toanyone
• PECOS is developing the Manufactured Analytical Solution Abstraction(MASA) library to provide well-defined manufactured solutions and sourceterms for a range of physics applications
Manufactured solutions are being constructed and will be incorporated into theFIN-S regression test suite
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 22 / 28
Related Efforts & Ongoing Work Verification
Manufactured Analytical Solution Abstraction Library
• Dearth of exact solutions necessitates method of manufactured solutions
• Some manufactured solutions exist for the calorically perfect Navier-Stokesequations
I Developed in large part by Sandia National LabsI Specific solutions for field, boundary condition order-of-accuracy verification
• Existing solutions provide a necessary but not sufficient test suiteI Will need to develop many more solutions to verify reacting flows with complex
transport models
• Manufactured solutions are a valuable resource that should be accessible toanyone
• PECOS is developing the Manufactured Analytical Solution Abstraction(MASA) library to provide well-defined manufactured solutions and sourceterms for a range of physics applications
Manufactured solutions are being constructed and will be incorporated into theFIN-S regression test suite
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 22 / 28
Related Efforts & Ongoing Work Verification
Manufactured analytical solutions (used by Roy, Smith, and Ober) for eachone of the primitive variables in Navier-Stokes equations are:
ρ (x, y) = ρ0 + ρx sin(aρxπx
L
)+ ρy cos
(aρyπyL
),
u (x, y) = u0 + ux sin(auxπx
L
)+ uy cos
(auyπyL
),
v (x, y) = v0 + vx cos(avxπx
L
)+ vy sin
(avyπyL
),
p (x, y) = p0 + px cos(apxπx
L
)+ py sin
(apyπyL
)
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 23 / 28
Related Efforts & Ongoing Work Verification
The method of manufactured solutions applied to Navier-Stokes equationsrequires modifying the governing equations by adding a source term to theright-hand side of each equation:
∂(ρ)
∂t+∂(ρu)
∂x+∂(ρv)
∂y= Qρ
∂(ρu)
∂t+∂(ρu2 + p− τxx)
∂x+∂(ρuv − τxy)
∂y= Qu
∂(ρv)
∂t+∂(ρvu− τyx)
∂x+∂(ρv2 + p− τyy)
∂y= Qv
∂(ρet)
∂t+∂(ρuet + pu− uτxx − vτxy + qx)
∂x+∂(ρvet + pv − uτyx − vτyy + qy)
∂y= Qet
so the modified set of equations has a known, analytical solution.Symbolic representations of requisite source terms and C-source codehave recently been generated for 2D and 3D calorically perfect gas flows.
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 24 / 28
Related Efforts & Ongoing Work Verification
Qρ =aρxπρxL
cos(aρxπx
L
) [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
]− aρyπρy
Lsin(aρyπy
L
) [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
]+auxπuxL
cos(auxπx
L
) [ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]+avyπvyL
cos(avyπy
L
) [ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]Qu −
apxπpxL
sin(apxπx
L
)+aρxπρxL
cos(aρxπx
L
) [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
]2− aρyπρy
Lsin(aρyπy
L
) [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
] [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
]+
2auxπuxL
cos(auxπx
L
) [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
] [ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]− πauyuy
Lsin(auyπy
L
) [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
] [ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]+πavyvyL
cos(avyπy
L
) [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
] [ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]+
4a2uxπ2µux
3L2sin(auxπx
L
)+
a2uyπ2µuy
L2cos(auyπy
L
)Qv =
apyπpyL
cos(apyπy
L
)+πaρxρxL
cos(aρxπx
L
) [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
] [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
]− aρyπρy
Lsin(aρyπy
L
) [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
]2+auxπuxL
cos(auxπx
L
) [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
] [ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]− avxπvx
Lsin(avxπx
L
) [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
] [ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]+
2avyπvyL
cos(avyπy
L
) [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
] [ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]+a2vxπ
2µvxL2
cos(avxπx
L
)+
4a2vyπ2µvy
3L2sin(avyπy
L
)Qet =− apxπpx
L
γ
γ − 1sin(apxπx
L
) [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
]+apyπpyL
γ
γ − 1cos(apyπy
L
) [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
]+aρxπρx
2Lcos(aρxπx
L
) [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
] [[ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
]2+[vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
]2]− aρyπρy
2Lsin(aρyπy
L
) [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
] [[ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
]2+[vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
]2]+auxπux
2Lcos(auxπx
L
){[px cos
(apxπxL
)+ py sin
(apyπyL
)+ p0
] 2γ
γ − 1+
+
[3[ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
]2+[vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
]2] [ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]}− auyπuy
Lsin(auyπy
L
) [ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
] [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
] [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
]− avxπvx
Lsin(avxπx
L
) [ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
] [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
] [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
]+avyπvy
2Lcos(avyπy
L
){[px cos
(apxπxL
)+ py sin
(apyπyL
)+ p0
] 2γ
γ − 1
+
[[ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
]2+ 3
[vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
]2] [ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]}
+a2pxπ
2k px cos(apxπx
L
)[ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]L2R
+a2pyπ
2k py sin(apyπy
L
)[ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]L2R
−2a2ρxπ
2k ρ2xL2R
cos2(aρxπx
L
) [px cos(apxπx
L
)+ py sin
(apyπyL
)+ p0
][ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]3 −a2ρxπ
2kρx
L2Rsin(aρxπx
L
) [px cos(apxπx
L
)+ py sin
(apyπyL
)+ p0
][ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]2−
2a2ρyπ2k ρ2y
L2Rsin2
(aρyπyL
) [px cos(apxπx
L
)+ py sin
(apyπyL
)+ p0
][ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]3 −a2ρyπ
2kρy
L2Rcos(aρyπy
L
) [px cos(apxπx
L
)+ py sin
(apyπyL
)+ p0
][ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]2+
4a2uxπ2µux
3L2sin(auxπx
L
) [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
]− 4a2uxπ
2µu2x3L2
cos2(auxπx
L
)+a2uyπ
2µuy
L2cos(auyπy
L
) [ux sin
(auxπxL
)+ uy cos
(auyπyL
)+ u0
]−
a2uyπ2µu2yL2
sin2(auyπy
L
)+a2vxπ
2µvxL2
cos(avxπx
L
) [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
]− a2vxπ
2µv2xL2
sin2(avxπx
L
)+
4a2vyπ2µvy
3L2sin(avyπy
L
) [vx cos
(avxπxL
)+ vy sin
(avyπyL
)+ v0
]−
4a2vyπ2µv2y
3L2cos2
(avyπyL
)− 2apxaρxπ
2k pxρxL2R
cos(aρxπx
L
)sin(apxπx
L
)[ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]2 − 2apyaρyπ2k pyρy
L2R
cos(apyπy
L
)sin(aρyπy
L
)[ρx sin
(aρxπxL
)+ ρy cos
(aρyπyL
)+ ρ0
]2+
4auxavyπ2µuxvy
3L2cos(auxπx
L
)cos(avyπy
L
)− 2auyavxπ
2µuyvxL2
sin(auyπy
L
)sin(avxπx
L
)
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 25 / 28
Related Efforts & Ongoing Work Verification
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 26 / 28
Related Efforts & Ongoing Work Near-term Effort
Additional Focus Areas1 Physics Modeling
I Weakly Ionized FlowsI Surface CatalycityI Additional Boundary Conditions
2 CouplingI RadiationI Ablation
3 AdjointsI Sensitivity analysisI Adaptivity
4 Scalability
• Push range of applicability of code through internal NASA-JSC usethis summer
• Perform PECOS full-system simulations using FIN-S as part of year 3deliverables
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 27 / 28
Related Efforts & Ongoing Work Near-term Effort
Thank you!
Questions?
Benjamin S. Kirk Fully-Implicit Navier-Stokes July 21, 2010 28 / 28