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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


• 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


1 Software Engineering

2 Physical ModelingGoverning EquationsThermochemistryTurbulence Modeling

3 ResultsViscous Reacting FlowAdaptive Mesh RefinementTurbulent Flow

4 Related Efforts & Ongoing WorkHigh-Temperature ThermochemistryVerificationNear-term Effort


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


http://www.ices.utexas.edu/~benkirk


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.

...



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.

...


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


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


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



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)



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.


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


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



10000950090008500800075007000650060005500500045004000350030002500200015001000

TemperatureT (K)

U∞ = 6 ,731 m/sρ∞ = 6.81×10-4 kg/m3

T∞ = 265 K



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



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



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



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



Speedup

Number of Processor Cores

Spe

edup

100 101 102 103100

101

102

103

IdealScaled-Size (Weak) ScalingFixed-Size (Strong) Scaling


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


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


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



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

)



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.



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

)


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


Related Efforts & Ongoing Work Near-term Effort

Thank you!

Questions?


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