Application of Bayesian Statistical Methods for the Analysis of DSMC Simulations of Hypersonic Shocks

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 probabilities. 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. The ultimate goal of this work is to use experimental data to calibrate important DSMC parameters.

DSMC Method

Direct Simulation Monte Carlo (DSMC) is a particle based simulation method.

• Simulated particles represent large numbers of real particles.• Particles move and undergo collisions with other particles.• Can be used in highly non-equilibrium flowfields (such as strong shock waves). • Can model thermochemistry on a more detailed level than most CFD codes.

DSMC Method

Initialize

Move

Index

Collide

Sample

Performed on First Time Step

Performed on Selected Time Steps

Performed on Every Time Step

Create

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.

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 constantsrelated ¿ the VHS collisionmodel𝜀=1 (𝑖𝑓 𝐴≠𝐵 )𝑜𝑟 2 (𝑖𝑓 𝐴=𝐵 )

σR and σE are the reaction and elastic 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 )=𝚲𝑇𝜼𝑒−𝑬𝒂 /𝑘𝑇

R. Gupta, J. Yos, and R. Thompson, NASA Technical Memorandum 101528, 1989.

# Reaction Forward Rate Constants Backward Rate Constants

qreaction Λ η EA Λ η EA

1 N2 + N2 <--> N2 + N + N 7.968E-13 -0.5 1.561E-18 6.518E-47 0.27 0.0 -1.561E-18

2 N + N2 <--> N + N + N 6.9E-8 -1.5 1.561E-18 4.817E-46 0.27 0.0 -1.561E-18 3 O2 + N2 <--> O2 + N + N 3.187E-13 -0.5 1.561E-18 6.518E-47 0.27 0.0 -1.561E-18 4 O + N2 <--> O + N + N 3.187E-13 -0.5 1.561E-18 6.518E-47 0.27 0.0 -1.561E-18

5 NO + N2 <--> NO + N + N 3.187E-13 -0.5 1.561E-18 6.518E-47 0.27 0.0 -1.561E-18

6 N2 + O2 <--> N2 + O + O 1.198E-11 -1.0 8.197E-19 1.8E-47 0.27 0.0 -8.197E-19

7 N + O2 <--> N + O + O 5.993E-12 -1.0 8.197E-19 1.8E-47 0.27 0.0 -8.197E-19

8 O2 + O2 <--> O2 + O + O 5.393E-11 -1.0 8.197E-19 1.8E-47 0.27 0.0 -8.197E-19

9 O + O2 <--> O + O + O 1.498E-10 -1.0 8.197E-19 1.8E-47 0.27 0.0 -8.197E-19

10 NO + O2 <--> NO + O + O 5.993E-12 -1.0 8.197E-19 1.8E-47 0.27 0.0 -8.197E-19

11 N2 + NO <--> N2 + N + O 6.59E-10 -1.5 1.043E-18 2.0976E-46 0.27 0.0 -1.043E-18

12 N + NO <--> N + N + O 1.318E-8 -1.5 1.043E-18 2.0976E-46 0.27 0.0 -1.043E-18

13 O2 + NO <--> O2 + N + O 6.59E-10 -1.5 1.043E-18 2.0976E-46 0.27 0.0 -1.043E-18

14 O + NO <--> O + N + O 1.318E-8 -1.5 1.043E-18 2.0976E-46 0.27 0.0 -1.043E-18

15 NO + NO <--> NO + N + O 1.318E-8 -1.5 1.043E-18 2.0976E-46 0.27 0.0 -1.043E-18

16 N2 + O <--> NO + N 1.12E-16 0.0 5.175E-19 2.490E-17 0.0 0.0 -5.175E-19

17 NO + O <--> O2 + N 5.279E-21 1.0 2.719E-19 1.598E-18 0.5 4.968E-20 -2.719E-19

1D Shock Simulation

X

Pressure

StreamwiseVelocity

Inle

tB

oundary

Specula

rW

all

Simulation is initialized with a bulk velocitydirected towards the specular wall at the rightboundary of the domain, with pre-shock density,temperature, and chemical composition.

Vbulk

X

Pressure

StreamwiseVelocity

Inle

tB

oundary

Specula

rW

all

ShockPropagatesUpstream

X

Pressure

StreamwiseVelocity

Inle

tB

oundary

Specula

rW

all

The shock is allowedto move a reasonabledistance away from thewall before any form ofsampling begins.

X

Upstream PressureSampling Region

Inle

tBoundary

Specula

rW

allPressure

Downstream Pressure Sampling Region

X0.0

0.5

1.0

1.5 Normalized PressureBoxcar Averaged Normalized Pressure

Inle

tBoundary

Specula

rW

all

Shock Position

X0.0

0.5

1.0

1.5Boxcar Averaged Normalized Pressure (Time = t)Boxcar Averaged Normalized Pressure (Time = t + t)

Inle

tBoundary

Specula

rW

all

Shock Position(time = t)

Shock Position(time = t + t)

s

Shock PropagationSpeed = VP = s/t

X0.0

0.5

1.0

1.5 Boxcar Averaged Normalized Pressure (Time = t + t)

Inle

tBoundary

Specula

rW

all

Shock Position(time = t + t)

Shock SamplingRegion

VSampling Region = VP

• We require the ability to simulate a 1-D shock without knowing the post-shock conditions a priori.• To this end, we simulate an unsteady 1-D shock, and make use of a sampling region which moves with the shock.

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.

Sensitivity Analysis Parameters𝑘 (𝑇 )=𝚲𝑇𝜼𝑒−𝑬𝒂 /𝑘𝑇 10𝛼=𝚲

Throughout the sensitivity analysis, the ratio of forward to backward rate for a given reaction is kept constant, since these ratios should be fixed by the equilibrium constant.

# Reaction αmin αnom αmax Nominal Forward Rate Constants

Λ η EA

1 N2 + N2 <--> N2 + N + N -13.099 -12.099 -11.099 7.968E-13 -0.5 1.561E-18

2 N + N2 <--> N + N + N -8.161 -7.161 -6.161 6.9E-8 -1.5 1.561E-18

3 O2 + N2 <--> O2 + N + N -13.497 -12.497 -11.497 3.187E-13 -0.5 1.561E-18

4 O + N2 <--> O + N + N -13.497 -12.497 -11.497 3.187E-13 -0.5 1.561E-18

5 NO + N2 <--> NO + N + N -13.497 -12.497 -11.497 3.187E-13 -0.5 1.561E-18

6 N2 + O2 <--> N2 + O + O -11.922 -10.922 -9.922 1.198E-11 -1.0 8.197E-19

7 N + O2 <--> N + O + O -12.222 -11.222 -10.222 5.993E-12 -1.0 8.197E-19

8 O2 + O2 <--> O2 + O + O -11.268 -10.268 -9.268 5.393E-11 -1.0 8.197E-19

9 O + O2 <--> O + O + O -10.824 -9.824 -8.824 1.498E-10 -1.0 8.197E-19

10 NO + O2 <--> NO + O + O -12.222 -11.222 -10.222 5.993E-12 -1.0 8.197E-19

11 N2 + NO <--> N2 + N + O -10.181 -9.181 -8.181 6.59E-10 -1.5 1.043E-18

12 N + NO <--> N + N + O -8.880 -7.880 -6.880 1.318E-8 -1.5 1.043E-18

13 O2 + NO <--> O2 + N + O -10.181 -9.181 -8.181 6.59E-10 -1.5 1.043E-18

14 O + NO <--> O + N + O -8.880 -7.880 -6.880 1.318E-8 -1.5 1.043E-18

15 NO + NO <--> NO + N + O -8.880 -7.880 -6.880 1.318E-8 -1.5 1.043E-18

16 N2 + O <--> NO + N -16.951 -15.951 -14.951 1.12E-16 0.0 5.175E-19

17 NO + O <--> O2 + N -21.277 -20.277 -19.277 5.279E-21 1.0 2.719E-19

Scenario: 1-D Shock

• Shock speed is ~8000 m/s, M∞ ≈ 23.• Upstream number density = 3.22×1021 #/m3.• Upstream composition by volume: 79% N2, 21% O2.• Upstream temperature = 300 K.

Results: Nominal Parameter Values

X (m)

Density

(kg/m

3 )

-0.05 0 0.05 0.1 0.15 0.20.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030 BulkN2NO2ONO

Bulk

O2, NO

N

O

N2

X (m)

Tem

pera

ture

(K)

0 0.05 0.1 0.15 0.20

5000

10000

15000

20000

25000

TtransTrotTvibT

Quantity of Interest (QoI)

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

X

QoI

We cannot yet simulate EASTresults, so we must choose a temporary, surrogate QoI.

?

Sensitivity Analysis - Methods

Two measures for sensitivity were used in this work.• Pearson correlation coefficients:

• The mutual information:

Both measures involve global sensitivity analysis based on a Monte Carlo sampling of the parameter space, and thus the same datasets can beused to obtain both measures.

{𝑸𝒐𝑰 }={𝑸𝒐𝑰𝟏𝑸𝒐𝑰𝟐𝑸𝒐𝑰𝟑

⋮𝑸𝒐𝑰𝒏

}X (m)

Ttrans,b

ulk

(K)

0 0.05 0.1 0.15 0.20

5000

10000

15000

20000

25000

For our analysis, each blue dotrepresents a single, scalar QoI.

Scalar QoI used in later schematicsdemonstrating calculation of correlationcoefficients and the mutual information.

Sensitivity Analysis - Scalar vs. Vector QoI

We use two QoI’s in this work, the bulk translational temperature and the density of NO. Each of these is a vector QoI, with values over a range of points in space.

log10(O2 + NO --> O + O + NO)

Ttr

ans,b

ulk

(K),

atx

=0.0

45

m

-12 -11.5 -11 -10.56000

6500

7000

7500

8000

8500

9000

r2 = 0.00005

Sensitivity Analysis: Correlation Coefficient

log10(NO + N --> N + O + N)

Ttr

ans,b

ulk

(K),

atx

=0.0

45

m

-8.5 -8 -7.5 -76000

6500

7000

7500

8000

8500

9000

r2 = 0.0984

log10(N2 + N --> N + N + N)

Ttr

ans,b

ulk

(K),

atx

=0.0

45

m

-8 -7.5 -7 -6.56000

6500

7000

7500

8000

8500

9000

r2 = 0.7208

Sensitivity Analysis: Mutual Information

1

QoI

1

QoI

Sensitivity Analysis: Mutual Information

0.6560.4920.3280.1640.000

p(1,QoI)

1

p(

1)

QoI

p(QoI)

Sensitivity Analysis: Mutual Information

1

p(

1)

QoI

p(QoI)

0.1310.0980.0660.0330.000

p(1)p(QoI)

Sensitivity Analysis: Mutual Information

𝑰 (𝜽𝟏 ,𝑸𝒐𝑰 )=∫𝜽𝟏

∫𝑸𝒐𝑰

𝒑 (𝜽𝟏 ,𝑸𝒐𝑰 )[𝒍𝒏( 𝒑 (𝜽𝟏 ,𝑸𝒐𝑰 )𝒑 (𝜽𝟏)𝒑 (𝑸𝒐𝑰 )) ]𝒅𝑸𝒐𝑰 𝒅𝜽𝟏

0.6560.4920.3280.1640.000

p(1,QoI)

0.1310.0980.0660.0330.000

p(1)p(QoI)Hypothetical joint PDF for case where the QoI is indepenent of θ1.

Actual joint PDF of θ1 and the QoI, from a Monte Carlo sampling of theparameter space.

0.0410.0310.0210.0100.000

𝐩 (𝛉𝟏 ,𝐐𝐨𝐈 )[𝐥𝐧 ( 𝐩 (𝛉𝟏 ,𝐐𝐨𝐈)𝐩 (𝛉𝟏 )𝐩(𝐐𝐨𝐈)) ]

QoI

θ1

Kullback-Leibler divergence

X (m)

Ttrans,b

ulk

(K)

0 0.05 0.1 0.15 0.20

5000

10000

15000

20000

25000

For our analysis, each blue dotrepresents a single, scalar QoI.

Scalar QoI used in earlier schematicsdemonstrating calculation of correlationcoefficients and the mutual information.

Sensitivities vs. X for Ttrans,bulk as QoI

X (m)

r2

Mutu

alI

nfo

rmation

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1

N2 + N --> N + N + N (r2)NO + N --> N + O + N (r2)NO + O --> N + O + N (r2)N2 + O --> NO + N (r2)N2 + N --> N + N + N (MI)NO + N --> N + O + N (MI)NO + O --> N + O + N (MI)N2 + O --> NO + N (MI)

Sensitivities vs. X for ρNO as QoI

X (m)

r2

Mutu

alI

nfo

rmation

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1N2 + N --> N + N + N (r2)NO + N --> N + O + N (r2)NO + O --> N + O + N (r2)N2 + O --> NO + N (r2)N2 + N --> N + N + N (MI)NO + N --> N + O + N (MI)NO + O --> N + O + N (MI)N2 + O --> NO + N (MI)

X (m)

r2

Mutu

alI

nfo

rmation

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1N2 + O --> NO + N (r2)N2 + O --> NO + N (MI)

At this x-location, r2 is zerowhile the mutual informationis greater than zero.

r2 vs. Mutual Information

log10(N2 + O --> NO + N)

NO

(kg/m

3 ),atx

=0.

021

m

-16.5 -16.0 -15.5 -15.0

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05r2 = 0.0000002MI = 0.0668

X (m)

NO

(kg/m

3 )

-0.01 0 0.01 0.02 0.03 0.04 0.050

2E-05

4E-05

6E-05

8E-05

Lowest for N2 + O <--> NO + NNominal for N2 + O <--> NO + NHighest for N2 + O <--> NO + N

Sensitivities vs. X for ρNO as QoI

r2

Mutu

alI

nfo

rmation

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1N2 + N --> N + N + N (r2)NO + N --> N + O + N (r2)N2 + N --> N + N + N (MI)NO + N --> N + O + N (MI)

X (m)

log10(N2 + N <--> N + N + N)

NO

(kg/m

3),

atx

=0.1

41

m

-8 -7.5 -7 -6.50.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04 r2 = 0.3053

log10(NO + N <--> N + O + N)

NO

(kg/m

3),

atx

=0.0

03

m

-8.5 -8 -7.5 -70.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04r2 = 0.2902

Variance of ρNO vs. X

Variance Weighted Sensitivities for ρNO as QoI

X

var(

NO)x

r2(N

orm

aliz

ed)

var(

NO)x

Mutu

alI

nfo

rmation

(Norm

aliz

ed)

0 0.05 0.1 0.150.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

N2 + N <--> N + N + N (r2)O2 + O <--> O + O + O (r2)NO + N <--> N + O + N (r2)NO + O <--> N + O + O (r2)N2 + O <--> NO + N (r2)NO + O <--> O2 + N (r2)N2 + N <--> N + N + N (MI)O2 + O <--> O + O + O (MI)NO + N <--> N + O + N (MI)NO + O <--> N + O + O (MI)N2 + O <--> NO + N (MI)NO + O <--> O2 + N (MI)

X

var(

NO)x

r2(N

orm

aliz

ed)

var(

NO)x

Mutu

alI

nfo

rmation

(Norm

aliz

ed)

0 0.005 0.01 0.0150.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0N2 + N <--> N + N + N (r2)O2 + O <--> O + O + O (r2)NO + N <--> N + O + N (r2)NO + O <--> N + O + O (r2)N2 + O <--> NO + N (r2)NO + O <--> O2 + N (r2)N2 + N <--> N + N + N (MI)O2 + O <--> O + O + O (MI)NO + N <--> N + O + N (MI)NO + O <--> N + O + O (MI)N2 + O <--> NO + N (MI)NO + O <--> O2 + N (MI)

Overall Sensitivities

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Ove

rall

Sens

itivi

ty (N

orm

aliz

ed)

Parameter #

r^2 (QoI_1)MI (QoI_1)r^2 (QoI_2)MI (QoI_2)

N2 + N <--> N + N + N

r2 (QoI = Ttrans,bulk)Mutual Information (QoI = Ttrans,bulk)r2 (QoI = ρNO)Mutual Information (QoI = ρNO)

NO + N <--> N + O + N

NO + O <--> N + O + O

N2 + O <--> NO + N

NO + O <--> O2 + N

Conclusions

• Global, Monte Carlo based sensitivity analysis can provide a great deal of insight into how various parameters affect a given QoI.• Sensitivities based on r2 are mostly similar to those based on the mutual information, but there are notable differences for some parameters.• Which parameters have the highest sensitivities depends strongly on which QoI is chosen.• If our goal is to calibrate as many parameters as possible, and we had available data, we would use ρNO as our QoI, since it is sensitive to several more of our parameters than Ttrans,bulk.

Future Work

• Synthetic data calibrations for a 1-D shock with the current code.• Upgrade the code to allow modelling of ionization and electronic excitation.• Couple the code with a radiation solver.• Sensitivity analysis for a 1-D shock with the additional physics included.• Synthetic data calibrations with the upgraded code.• Calibrations with real data from EAST or similar facility.