Sensitivity Analysis for DSMC Simulations of High-

Temperature Air Chemistry

James S. Strand and David B. GoldsteinThe University of Texas at Austin

Sponsored by the Department of Energy through the PSAAP Program

Predictive Engineering and Computational Sciences

Computational Fluid Physics Laboratory

Motivation – DSMC Parameters• The DSMC model includes many parameters related to gas dynamics at the molecular level, such as: Elastic collision cross-sections. Vibrational and rotational excitation cross-sections. Reaction cross-sections. Sticking coefficients and catalytic efficiencies for gas-

surface interactions. …etc.

DSMC Parameters

• In many cases the precise values of some of these parameters are not known.• Parameter values often cannot be directly measured, instead they must be inferred from experimental results.• By necessity, parameters must often be used in regimes far from where their values were determined.• More precise values for important parameters would lead to better simulation of the physics, and thus to better predictive capability for the DSMC method.

MCMC Method - Overview

• Markov Chain Monte Carlo (MCMC) is a method which solves the statistical inverse problem in order to calibrate parameters with respect to a set or sets of experimental data.

MCMC MethodEstablish

boundaries for parameter space

Select initial position

Run simulation at current position

Calculate probability for

current position

Select new candidate position

Run simulation for candidate position parameters, and

calculate probability

Accept or reject candidate

position based on a random number draw

Candidate position is accepted, and becomes

the current chain position

Candidate position becomes

current position

Current position remains

unchanged.

Candidate automatically

accepted

Candidate Accepted

Candidate Rejected

Probcandidate

< Probcurrent

Probcandidate

> Probcurrent

Omega

Dref(in

meters)

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 13E-10

3.5E-10

4E-10

4.5E-10

5E-10

5.5E-10

6E-10

Omega

Dref(in

meters)

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 13E-10

3.5E-10

4E-10

4.5E-10

5E-10

5.5E-10

6E-10

Previous MCMC Results – Argon VHS Parameters

Omega

Dref(in

meters)

0.5 0.6 0.7 0.8 0.9 13E-10

3.5E-10

4E-10

4.5E-10

5E-10

5.5E-10

6E-10

Omega

Dref(in

meters)

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 13E-10

3.5E-10

4E-10

4.5E-10

5E-10

5.5E-10

6E-10

P. Valentini, T. E. Schwartzentruber, Physics of Fluids (2009), Vol. 21

Sensitivity Analysis - Overview

• In the current context, the goal of sensitivity analysis is to determine which parameters most strongly affect a given quantity of interest (QoI). • Only parameters to which a given QoI is sensitive will be informed by calibrations based on data for that QoI.• Sensitivity analysis is used here both to determine which parameters to calibrate in the future, and to select the QoI which would best inform the parameters we most wish to calibrate.

Numerical Methods – DSMC Code

• Our DSMC code can model flows with rotational and vibrational excitation and relaxation, as well as five-species air chemistry, including dissociation, exchange, and recombination reactions.• Larsen-Borgnakke model is used for redistribution between rotational, translational, and vibrational modes during inelastic collisions.• TCE model provides cross-sections for chemical reactions.

Variable Hard Sphere ModelThe VHS model allows the collision cross-section to be dependent on relative speed, which is more physically realistic than the hard sphere model.

There are two relevant parameters for the VHS model, dref and ω.

Internal Modes

• Rotation is assumed to be fully excited. Each particle has its own value of rotational energy,

and this variable is continuously distributed.• Vibrational levels are quantized.

Each particle has its own vibrational level, which is associated with a certain vibrational energy based on the simple harmonic oscillator model.

• Relevant parameters are ZR and ZV, the rotational and vibrational collision numbers.

ZR = 1/ΛR, where ΛR is the probability of the rotational energy of a given molecule being redistributed during a given collision.

ZV = 1/ΛV

ZR and ZV are treated as constants.

Chemistry Implementation

Reaction cross-sections based on Arrhenius rates TCE model allows determination of reaction cross-

sections from Arrhenius parameters.

, the average number of internal degrees of freedom which contribute to the collision energy.

is the temperature-viscosity exponent for VHS collisions between type A and type B particles

𝜎 𝑟𝑒𝑓∧𝑇 𝑟𝑒𝑓 are both constants related ¿ the VHS collisionmodel𝜀=1 (𝑖𝑓 𝐴≠𝐵 )𝑜𝑟 2 (𝑖𝑓 𝐴=𝐵 )

σR and σT are the reaction and total cross-sections, respectively

k is the Boltzmann constant, mr is the reduced mass of particles A and B, Ec is the collision energy, and Γ() is the gamma function.

Reactions𝑘 (𝑇 )=𝑨𝑇 𝜼𝑒−𝑬𝒂 /𝑘𝑇

Reaction # Reaction Equation A η EA 1 N2 + N2 --> N2 + N + N 1.16E-08 -1.6 1.56E-18 2 N + N2 --> N + N + N 4.98E-08 -1.6 1.56E-18 3 O2 + N2 --> O2 + N + N 4.98E-08 -1.6 1.56E-18 4 O + N2 --> O + N + N 4.98E-08 -1.6 1.56E-18 5 NO + N2 --> NO + N + N 4.98E-08 -1.6 1.56E-18 6 N2 + O2 --> N2 + O + O 3.32E-09 -1.5 8.21E-19 7 N + O2 --> N + O + O 3.32E-09 -1.5 8.21E-19 8 O2 + O2 --> O2 + O + O 3.32E-09 -1.5 8.21E-19 9 O + O2 --> O + O + O 3.32E-09 -1.5 8.21E-19 10 NO + O2 --> NO + O + O 3.32E-09 -1.5 8.21E-19 11 N2 + NO --> N2 + N + O 8.30E-15 0 1.04E-18 12 N + NO --> N + N + O 8.30E-15 0 1.04E-18 13 O2 + NO --> O2 + N + O 8.30E-15 0 1.04E-18 14 O + NO --> O + N + O 8.30E-15 0 1.04E-18 15 NO + NO --> NO + N + O 8.30E-15 0 1.04E-18 16 N2 + O --> NO + N 9.45E-18 0.42 5.93E-19 17 O2 + N --> NO + O 4.13E-21 1.18 5.53E-20 18 NO + N --> N2 + O 2.02E-17 0.1 0 19 NO +O --> O2 + N 1.40E-17 0 2.65E-10

T. Ozawa, J. Zhong, and D. A. Levin, Physics of Fluids (2008), Vol. 20, Paper #046102.

Reaction Rates – Nitrogen Dissociation

Temperature (K)

Rea

ctionRate(#/m

3 -s)

0 5000 10000 15000 20000 25000

2.0E+27

5.0E+27

8.0E+27

1.1E+28

1.4E+28

1.7E+28

2.0E+28

N2 + N2 --> N2 + N + N (Arrhenius)N2 + N2 --> N2 + N + N (DSMC)N + N2 --> N + N + N (Arrhenius)N + N2 --> N + N + N (DSMC)

Reaction Rates – O2 and NO Dissociation

Temperature (K)

Rea

ctionRate(#/m

3 -s)

5000 10000 15000 20000 250002.0E+27

5.2E+28

1.0E+29

1.5E+29N2 + O2 --> N2 + O + O (Arrhenius)N2 + O2 --> N2 + O + O (DSMC)N + NO --> N + N + O (Arrhenius)N + NO --> N + N + O (DSMC)

𝝈𝑹≮� 𝝈𝑽𝑯𝑺

Temperature (K)

Rea

ctionRate(#/m

3 -s)

5000 10000 15000 20000 250002.0E+27

5.2E+28

1.0E+29

1.5E+29

2.0E+29

N2 + O --> NO + N (Arrhenius)N2 + O --> NO + N (DSMC)O2 + N --> NO + O (Arrhenius)O2 + N --> NO + O (DSMC)NO + N --> N2 + O (Arrhenius)NO + N --> N2 + O (DSMC)NO + O --> O2 + N (Arrhenius)NO + O --> O2 + N (DSMC)

Reaction Rates – NO Exchange Reactions

𝝈𝑹≮� 𝝈𝑽𝑯𝑺

Parallelization

• DSMC: MPI parallel. Ensemble averaging to reduce stochastic noise. Fast simulation of small problems.

• Sensitivity Analysis: MPI Parallel Separate processor groups for each parameter. Large numbers of parameters can be examined

simultaneously.

0-D Relaxation, Pure Nitrogen

• Scenarios examined in this work are 0-D relaxations from an initial high-temperature state.• 0-D box is initialized with 100% N2.

Initial number density = 1.0×1023 #/m3. Initial translational temperature = ~50,000 K. Initial rotational and vibrational temperatures are

both 300 K.• Scenario is a 0-D substitute for a hypersonic shock at ~8 km/s.

Assumption that the translational modes equilibrate much faster than the internal modes.

0-D Relaxation, Pure Nitrogen

Time (s)

Tempe

rature

(K)

Den

sity

(kg/m

3 )

0 5E-07 1E-06 1.5E-06 2E-060

10000

20000

30000

40000

50000

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0.0050

Ttrans - N2Trot - N2Tvib - N2Ttrans - N - N2 - N

Quantity of Interest (QoI)

{𝑸𝒐𝑰 }={𝑸𝒐𝑰𝟏𝑸𝒐𝑰𝟐𝑸𝒐𝑰𝟑

⋮𝑸𝒐𝑰𝒏

}J. Grinstead, M. Wilder, J. Olejniczak, D. Bogdanoff, G. Allen, and K. Danf, AIAA Paper 2008-1244, 2008.

Sensitivity Analysis - QoI

ZR,min ZR,maxZR,nom

ZV,min ZV,maxZV,nom

ωmin ωmaxωnom

dref,min dref,maxdref,nom

Sensitivity Analysis – Type 1

ZR,min ZR,maxZR,nom

ZV,min ZV,maxZV,nom

ωmin ωmaxωnom

dref,min dref,maxdref,nom

ω = ωmin

dref = dref,nom

ZR = ZR,nom

ZV = ZV,nom

ωmin

Sensitivity Analysis – Type 1

ZR,min ZR,maxZR,nom

ZV,min ZV,maxZV,nom

ωmin ωmaxωnom

dref,min dref,maxdref,nom

ω = ωmax

dref = dref,nom

ZR = ZR,nom

ZV = ZV,nom

ωminωmax

Sensitivity Analysis – Type 1

ωmin ωmaxωnom

Δω = ωmax – ωmin

ωmin

ωmax

Sensitivity Analysis – Type 1

ωmin ωmaxωnom

Δω = ωmax – ωmin

ΔQoI2

ΔQoI1

ΔQoI3

ΔQoIn

𝑺𝒆𝒏𝒔𝒊𝒕𝒊𝒗𝒊𝒕𝒚={𝜟𝑸𝒐𝑰 }𝑻 {𝜟𝑸𝒐𝑰 }

{𝜟𝑸𝒐𝑰 }={𝜟𝑸𝒐𝑰 𝟏𝜟𝑸𝒐𝑰 𝟐𝜟𝑸𝒐𝑰 𝟑

⋮𝜟𝑸𝒐𝑰𝒏

}

Sensitivity Analysis – Type 2

ωmin ωmaxωnom

Δω = (ωmax – ωmin)×0.10 𝑺𝒆𝒏𝒔𝒊𝒕𝒊𝒗𝒊𝒕𝒚={𝜟𝑸𝒐𝑰 }𝑻 {𝜟𝑸𝒐𝑰 }

{𝜟𝑸𝒐𝑰 }={𝜟𝑸𝒐𝑰 𝟏𝜟𝑸𝒐𝑰 𝟐𝜟𝑸𝒐𝑰 𝟑

⋮𝜟𝑸𝒐𝑰𝒏

}

Pure Nitrogen – ParametersParameter

Number Parameter

Name Meaning Minimum Nominal Maximum

1 ω (N2-N2) Temperature-viscosity exponent for N2-N2 collisions 0.5 0.68 1.0

2 ω (N2-N) Temperature-viscosity exponent for N2-N collisions 0.5 0.665 1.0

3 ω (N-N) Temperature-viscosity exponent for N-N collisions 0.5 0.65 1.0

4 dref (N2-N2) VHS reference diameter for N2-N2 collisions 2.00E-10 (m) 3.58E-10 (m) 5.00E-10 (m)

5 dref (N2-N) VHS reference diameter for N2-N collisions 2.00E-10 (m) 3.35E-10 (m) 5.00E-10 (m)

6 dref (N-N) VHS reference diameter for N-N collisions 2.00E-10 (m) 3.11E-10 (m) 5.00E-10 (m)

7 ZR Rotational collision number 1 5 10 8 ZV Vibrational collision number 1 10 50

9 α1 𝟏𝟎𝜶𝟏 = 𝑨𝟏, the pre-exponential constant for the reaction N2 + N2 --> N2 + N + N

-8.94 (A1 = 1.16E-9)

-7.94 (A1 = 1.16E-8)

-6.94 (A1 = 1.16E-7)

10 α2 𝟏𝟎𝜶𝟐 = 𝑨𝟐, the pre-exponential constant for the reaction N + N2 --> N + N + N

-8.30 (A2 = 4.98E-9)

-7.30 (A2 = 4.98E-8)

-6.30 (A2 = 4.98E-7)

0.00

0.05

0.10

0.15

0.20

0.25

1 2 3 4 5 6 7 8 9 10 11 12 13

Nor

mal

ized

Sens

itivi

ty

Parameter

≈Pure Nitrogen – Results

Sensitivity Analysis Type 1

1.00

Numerical Parameters

dref (N2-N2)

ω (N2-N2)

ZR

ZV

α1

(N2 + N2 N2 + N + N)α2

(N + N2 N + N + N)

≈0.77

0.00

0.05

0.10

0.15

0.20

0.25

1 2 3 4 5 6 7 8 9 10 11 12 13

Nor

mal

ized

Sens

itivi

ty

Parameter

Pure Nitrogen – Results

Sensitivity Analysis Type 2

1.00

Numerical Parameters

dref (N2-N2)ω (N2-N2)

ZR

ZV

α1

(N2 + N2 N2 + N + N)α2

(N + N2 N + N + N)≈ ≈0.53

Pure Nitrogen – Results

Time (s)

|QoI|(K)

5E-07 1E-06 1.5E-06 2E-060

500

1000

1500

(N2-N2)dref (N2-N2)ZRZV12RF Seed

Sensitivity Rank Sensitivity Type 1 Sensitivity Type 2 1 α2 α2 2 α1 α1 3 ω (N2-N2) ZV 4 dref (N2-N2) ω (N2-N2) 5 ZV dref (N2-N2) 6 ZR ZR 7 dref (N2-N) dref (N2-N) 8 ω (N2-N) ω (N2-N)

0-D Relaxation, Five-Species Air

• Another 0-D relaxation from an initial high-temperature state.• 0-D box is initialized with 79% N2, 21% O2.

Initial bulk number density = 1.0×1023 #/m3. Initial bulk translational temperature = ~50,000 K. Initial bulk rotational and vibrational temperatures are

both 300 K.• Scenario is a 0-D substitute for a hypersonic shock at ~8 km/s.

Assumption that the translational modes equilibrate much faster than the internal modes.

Time (s)

Den

sity

(kg/m

3 )

0 5E-07 1E-06 1.5E-06 2E-060

0.001

0.002

0.003

0.004

0.005

BulkN2NO2ONO

Five-Species Air – Densities

Time (s)

T trans(K)

0 5E-07 1E-06 1.5E-06 2E-060

10000

20000

30000

40000

50000

BulkN2NO2ONO

Five-Species Air – Translational Temperatures

Five-Species Air - Parameters𝑘 (𝑇 )=𝑨𝑇 𝜼𝑒−𝑬𝒂 /𝑘𝑇 10𝛼=𝑨Reaction # Equation αmin αnom αmax Anom η EA 1 N2 + N2 --> N2 + N + N -6.94 -7.94 -8.94 1.16E-08 -1.6 1.56E-18 2 N + N2 --> N + N + N -6.30 -7.30 -8.30 4.98E-08 -1.6 1.56E-18 3 O2 + N2 --> O2 + N + N -6.30 -7.30 -8.30 4.98E-08 -1.6 1.56E-18 4 O + N2 --> O + N + N -6.30 -7.30 -8.30 4.98E-08 -1.6 1.56E-18 5 NO + N2 --> NO + N + N -6.30 -7.30 -8.30 4.98E-08 -1.6 1.56E-18 6 N2 + O2 --> N2 + O + O -7.48 -8.48 -9.48 3.32E-09 -1.5 8.21E-19 7 N + O2 --> N + O + O -7.48 -8.48 -9.48 3.32E-09 -1.5 8.21E-19 8 O2 + O2 --> O2 + O + O -7.48 -8.48 -9.48 3.32E-09 -1.5 8.21E-19 9 O + O2 --> O + O + O -7.48 -8.48 -9.48 3.32E-09 -1.5 8.21E-19 10 NO + O2 --> NO + O + O -7.48 -8.48 -9.48 3.32E-09 -1.5 8.21E-19 11 N2 + NO --> N2 + N + O -13.08 -14.08 -15.08 8.30E-15 0 1.04E-18 12 N + NO --> N + N + O -13.08 -14.08 -15.08 8.30E-15 0 1.04E-18 13 O2 + NO --> O2 + N + O -13.08 -14.08 -15.08 8.30E-15 0 1.04E-18 14 O + NO --> O + N + O -13.08 -14.08 -15.08 8.30E-15 0 1.04E-18 15 NO + NO --> NO + N + O -13.08 -14.08 -15.08 8.30E-15 0 1.04E-18 16 N2 + O --> NO + N -16.02 -17.02 -18.02 9.45E-18 0.42 5.93E-19 17 O2 + N --> NO + O -19.38 -20.38 -21.38 4.13E-21 1.18 5.53E-20 18 NO + N --> N2 + O -15.69 -16.69 -17.69 2.02E-17 0.1 0 19 NO +O --> O2 + N -15.85 -16.85 -17.85 1.40E-17 0 2.65E-10

Five-Species Air - Results

0.00.10.20.30.40.50.60.70.80.91.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Nor

mal

ized

Sens

itivi

ty

Parameter

QoI = Ttrans,N

• We used only sensitivity analysis type 2 for the five species air scenario.

Numerical Parameters

Nitrogen Dissociation

Reactions Oxygen Dissociation

Reactions

NO Dissociation

Reactions

NO Exchange Reactions

N2 + O NO + N

NO + N N2 + O

0.00.10.20.30.40.50.60.70.80.91.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Nor

mal

ized

Sens

itivi

ty

Parameter

Five-Species Air - Results

QoI = ρNO

• We also tested sensitivity with respect to a second QoI, the mass density of NO.

Numerical Parameters

Nitrogen Dissociation

Reactions

Oxygen Dissociation

Reactions

NO Dissociation

Reactions

NO Exchange Reactions

N2 + O NO + N

NO + N N2 + O

Five-Species Air - ResultsSensitivity

Rank QoI = Ttrans,N QoI = ρNO

Equation Reaction # Equation Reaction # 1 N2 + O --> NO + N 16 N2 + O --> NO + N 16 2 NO + N --> N2 + O 18 NO + N --> N2 + O 18 3 N + N2 --> N + N + N 2 N2 + NO --> N2 + N + O 11 4 N2 + NO --> N2 + N + O 11 O2 + N --> NO + O 17 5 N2 + N2 --> N2 + N + N 1 N + NO --> N + N + O 12 6 O + N2 --> O + N + N 4 O + NO --> O + N + O 14 7 N2 + O2 --> N2 + O + O 6 N + N2 --> N + N + N 2 8 N + NO --> N + N + O 12 N2 + O2 --> N2 + O + O 6 9 O2 + N --> NO + O 17 O + N2 --> O + N + N 4

10 O + NO --> O + N + O 14 N2 + N2 --> N2 + N + N 1 11 O2 + N2 --> O2 + N + N 3 O2 + N2 --> O2 + N + N 3 12 N + O2 --> N + O + O 7 N + O2 --> N + O + O 7 13 O2 + O2 --> O2 + O + O 8 NO + NO --> NO + N + O 15 14 O + O2 --> O + O + O 9 O + O2 --> O + O + O 9 15 NO + N2 --> NO + N + N 5 NO + N2 --> NO + N + N 5 16 - - NO +O --> O2 + N 19 17 - - O2 + O2 --> O2 + O + O 8 18 - - O2 + NO --> O2 + N + O 13

Conclusions

Pure nitrogen scenario: Sensitivities to reaction rates dominate all others. ZR, ZV, and VHS parameters for N2-N2 collisions are

important in the early stages of the relaxation.Five-species air scenario: Sensitivities for the forward and backward rates for the

reaction N2 + O ↔ NO + N are dominant when using either Ttrans,N or ρNO as the QoI.

NO dissociation reactions are moderatly important for either QoI.

Nitrogen and oxygen dissociation reactions are important only for the Ttrans,N QoI.

Future Work

• Perform calibration with synthetic data for the 0-D relaxation scenarios.• Perform synthetic data calibrations for a 1-D shock with chemistry.• Perform calibrations with real data from EAST or similar facility.