High Accuracy Shock-Fitted Computation of Unsteady ...

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High Accuracy Shock-Fitted Computation of Unsteady Detonation with Detailed Kinetics Tariq D. Aslam ([email protected]) Los Alamos National Laboratory; Los Alamos, NM Joseph M. Powers ([email protected]) University of Notre Dame; Notre Dame, IN 10 th US National Congress on Computational Mechanics Columbus, Ohio 18 July 2009

Transcript of High Accuracy Shock-Fitted Computation of Unsteady ...

Page 1: High Accuracy Shock-Fitted Computation of Unsteady ...

High Accuracy Shock-Fitted Computation of

Unsteady Detonation with Detailed Kinetics

Tariq D. Aslam ([email protected])

Los Alamos National Laboratory; Los Alamos, NM

Joseph M. Powers ([email protected])

University of Notre Dame; Notre Dame, IN

10th US National Congress on Computational Mechanics

Columbus, Ohio

18 July 2009

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Outline

• Gas phase detonation introduction

• Length scale requirements from steady traveling wave

solutions for H2-air (review)

• Unsteady dynamics of ozone detonation

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Fundamentals of Detonation

-0.010 -0.005 0.000HmL x`

500 000

1.´106

1.5´106

2.´106

2.5´106

P HPaL

• Detonation: shock-

induced combustion

process

• applications:

aerospace propulsion,

safety, military, internal

combustion engines,

etc.

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Length and Time Scale Discussion

Simplistic linear advection-reaction model:

∂ψ

∂t︸︷︷︸

evolution

+ uo

∂ψ

∂x︸ ︷︷ ︸

advection

= −kψ︸︷︷︸

reaction

dt= −kψ : time scale τ =

1

k

uo

dx= −kψ : length scale ℓ =

uo

kFast reaction (large k) induces small length and time scales.

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Motivation

• Detailed kinetics models are widely used in detonation simula-

tions.

• The finest length scale predicted by such models is usually not

clarified and often not resolved.

• Tuning computational results to match experiments without first

harmonizing with underlying mathematics renders predictions

unreliable.

• See Powers and Paolucci, AIAA Journal, 2005.

• We explore the transient behavior of detonations with fully re-

solved detailed kinetics.

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Verification and Validation

• verification: solving the equations right (math).

• validation: solving the right equations (physics).

• Main focus here on verification

• Some limited validation possible, but detailed valida-

tion awaits more robust measurement techniques.

• Verification and validation always necessary but never

sufficient: finite uncertainty must be tolerated.

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Model: Steady 1D Reactive Euler Equations

ρu = ρoD,

ρu2 + p = ρoD

2 + po,

e +u2

2+

p

ρ= eo +

D2

2+

po

ρo

,

p = ρℜT

NX

i=1

Yi

Mi

,

e =

NX

i=1

Yi

hoi,f +

Z T

To

cpi(T̂ ) dT̂ −ℜT

Mi

«

,

dYi

dx=

Mi

ρoD

JX

j=1

νijAjTβj e

−EjℜT

«

0

B

B

B

B

@

NY

k=1

ρYk

Mk

«ν′kj

| {z }

forward

−1

Kcj

NY

k=1

ρYk

Mk

«ν′′kj

| {z }

reverse

1

C

C

C

C

A

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Eigenvalue Analysis of Local Length Scales

Algebraic reduction yields

dY

dx= f(Y).

Local behavior is modeled by

dY

dx= J · (Y − Y

∗) + b, Y(x∗) = Y∗.

whose solution is

Y(x) = Y∗ +

(

P · eΛ(x−x∗) · P−1 − I

)

· J−1 · b.

Here, Λ has eigenvalues λi of Jacobian J in its diagonal. Lengthscales given by

ℓi(x) =1

|λi(x)| .

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Computational Methods: Steady Detonation

• A standard ODE solver (DLSODE) was used to inte-

grate the equations.

• Standard IMSL subroutines were used to evaluate the

local Jacobians and eigenvalues at every step.

• The Chemkin software package was used to evaluate

kinetic rates and thermodynamic properties.

• Computation time was typically one minute on a 1GHz

HP Linux machine.

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Physical System

• Hydrogen-air detonation: 2H2 +O2 + 3.76N2.

• N = 9 molecular species, L = 3 atomic elements,

J = 19 reversible reactions.

• po = 1 atm.

• To = 298K .

• Identical to system studied by both Shepherd (1986)

and Mikolaitis (1987).

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Detailed Kinetics Modelj Reaction Aj βj Ej

1 H2 + O2 ⇋ OH + OH 1.70 × 1013 0.00 47780

2 OH + H2 ⇋ H2O + H 1.17 × 109 1.30 3626

3 H + O2 ⇋ OH + O 5.13 × 1016 −0.82 16507

4 O + H2 ⇋ OH + H 1.80 × 1010 1.00 8826

5 H + O2 + M ⇋ HO2 + M 2.10 × 1018 −1.00 0

6 H + O2 + O2 ⇋ HO2 + O2 6.70 × 1019 −1.42 0

7 H + O2 + N2 ⇋ HO2 + N2 6.70 × 1019 −1.42 0

8 OH + HO2 ⇋ H2O + O2 5.00 × 1013 0.00 1000

9 H + HO2 ⇋ OH + OH 2.50 × 1014 0.00 1900

10 O + HO2 ⇋ O2 + OH 4.80 × 1013 0.00 1000

11 OH + OH ⇋ O + H2O 6.00 × 108 1.30 0

12 H2 + M ⇋ H + H + M 2.23 × 1012 0.50 92600

13 O2 + M ⇋ O + O + M 1.85 × 1011 0.50 95560

14 H + OH + M ⇋ H2O + M 7.50 × 1023 −2.60 0

15 H + HO2 ⇋ H2 + O2 2.50 × 1013 0.00 700

16 HO2 + HO2 ⇋ H2O2 + O2 2.00 × 1012 0.00 0

17 H2O2 + M ⇋ OH + OH + M 1.30 × 1017 0.00 45500

18 H2O2 + H ⇋ HO2 + H2 1.60 × 1012 0.00 3800

19 H2O2 + OH ⇋ H2O + HO2 1.00 × 1013 0.00 1800

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Temperature Profile

10−5

10−4

10−3

10−2

10−1

100

101

1500

2000

2500

3000

3500

x (cm)

T (

K)

• Temperature flat in the

post-shock induction

zone 0 < x <

2.6 × 10−2 cm.

• Thermal explosion

followed by relaxation

to equilibrium at

x ∼ 100 cm.

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Mole Fractions versus Distance

10−4

10−3

10−2

10−1

100

101

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

x (cm)10

−5

H O2 2

OH

H

O

H O2H 2

HO2

O2

N 2

H O2 2

O O2

H 2

OH

Xi

• significant evolution at

fine length scales x <

10−3 cm.

• results agree with

those of Shepherd.

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Eigenvalue Analysis: Length Scale Evolution

10−5

10−4

10−3

10−2

10−1

100

101

10−5

10−4

10−3

10−2

10−1

100

101

102

x (cm)

(cm

)i

• Finest length scale:

2.3 × 10−5 cm.

• Coarsest length scale

3.0 × 101 cm.

• Finest length scale

similar to that

necessary for

numerical stability of

ODE solver.

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Verification: Comparison with Mikolaitis

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−13

10−11

10−9

10−7

10−5

10−3

10−1

t (s)

H O2 2

OH

H

OH O2

HO2Xi

• Lagrangian calculation

allows direct

comparison with

Mikolaitis’ results.

• agreement very good.

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Grid Convergence

10−10

10−8

10−6

10−4

10−10

10−8

10−6

10−4

10−2

100

x (cm)

1

2.008

1

1.006

First OrderExplicit Euler

Second OrderRunge-Kutta

ε OH

• Finest length scale

must be resolved to

converge at proper

order.

• Results are

converging at proper

order for first and

second order

discretizations.

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Numerical Stability

10−4

10−3

10−2

10−7

10−6

10−5

x (cm)

∆x = 1.00 x 10 cm (stable)-5

∆x = 2.00 x 10 cm (stable)-4

∆x = 2.38 x 10 cm (unstable)-4

XH

• Discretizations finer than

finest physical length

scale are numerically

stable.

• Discretizations coarser

than finest physical

length scale are

numerically unstable.

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Unsteady Model: Reactive Euler Equations

• one-dimensional,

• inviscid,

• detailed mass action kinetics with Arrhenius tempera-

ture dependency,

• ideal mixture of calorically imperfect ideal gases

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Model: Unsteady Reactive Euler PDEs

∂ρ

∂t+

∂x(ρu) = 0,

∂t(ρu) +

∂x

(ρu2 + p

)= 0,

∂t

(

ρ

(

e +u2

2

))

+∂

∂x

(

ρu

(

e +u2

2+

p

ρ

))

= 0,

∂t(ρYi) +

∂x(ρuYi) = Miω̇i,

p = ρℜTN∑

i=1

Yi

Mi

,

e = e(T, Yi),

ω̇i = ω̇i(T, Yi).

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Computational Method

• Shock fitting coupled with a fifth order method for

continuous regions

– Fifth order WENO5M for spatial discretization

– Fifth order Runge-Kutta for temporal discretization

• see Henrick, Aslam, Powers, J. Comp. Phys., 2006, for

full details on shock fitting

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Outline of Shock Fitting Method

• Transform from lab frame to shock-attached frame,

(x, t) → (ξ, τ)

– example mass equation becomes

∂ρ

∂τ+

∂ξ(ρ (u−D)) = 0

• In interior

– fifth order WENO5M for spatial discretization

– fifth order Runge-Kutta for temporal discretization

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Outline of Shock Fitting Method, cont.

• At shock boundary, one-sided high order differences

are utilized

• Note that some form of an approximate Riemann

solver must be used to determine the shock speed,

D, and thus set a valid shock state

• At downstream boundary, a zero gradient (constant

extrapolation) approximation is utilized

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Summary of Shock-Fitting Method

−17.5 −12.5 −10 −7.5 −5 −2.5

10

20

30

40

ξ

boundary interior

near

shock

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Difficulties in Unsteady Calculations

• Note that H2-air steady detonation had length scales

spanning six orders of magnitude

• This is feasible for steady calculations but extremely

challenging in a transient calculation.

• To cleanly illustrate the challenges of coupled length

and time scales, we choose a realistic problem with

less stiffness that we can verify and validate: ozone

detonation.

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Ozone Reaction Kinetics

Reaction Afj , Ar

j βfj , βr

j Efj , Er

j

O3 + M ⇆ O2 + O + M 6.76 × 106 2.50 1.01 × 1012

1.18 × 102 3.50 0.00

O + O3 ⇆ 2O2 4.58 × 106 2.50 2.51 × 1011

1.18 × 106 2.50 4.15 × 1012

O2 + M ⇆ 2O + M 5.71 × 106 2.50 4.91 × 1012

2.47 × 102 3.50 0.00

see Margolis, J. Comp. Phys., 1978, or Hirschfelder, et al.,

J. Chem. Phys., 1953.

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Validation: Comparison with Observation

• Streng, et al., J. Chem. Phys., 1958.

• po = 1.01325 × 106 dyne/cm2, To = 298.15 K ,

YO3= 1, YO2

= 0, YO = 0.

Value Streng, et al. this study

DCJ 1.863 × 105 cm/s 1.936555 × 105 cm/s

TCJ 3340 K 3571.4 K

pCJ 3.1188 × 107 dyne/cm2 3.4111 × 107 dyne/cm2

Slight overdrive to preclude interior sonic points.

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Stable Strongly Overdriven Case: Length Scales

Do = 2.5 × 105 cm/s.

10−10

10−8

10−6

10−4

10−2

100

10−8

10−7

10−6

10−5

10−4

10−3

x (cm)

length scale (cm)

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Mean-Free-Path Estimate

• The mixture mean-free-path scale is the cutoff mini-

mum length scale associated with continuum theories.

• A simple estimate for this scale is given by Vincenti

and Kruger, ’65:

ℓmfp =M√

2Nπd2ρ∼ 10−7 cm.

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Stable Strongly Overdriven Case: Mass Fractions

Do = 2.5 × 105 cm/s.

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−5

10−4

10−3

10−2

10−1

100

101

x (cm)

Yi

O

O

O

2

3

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Stable Strongly Overdriven Case: Temperature

Do = 2.5 × 105 cm/s.

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

2800

3000

3200

3400

3600

3800

4000

4200

4400

x (cm)

T (K)

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Stable Strongly Overdriven Case: Pressure

Do = 2.5 × 105 cm/s.

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

0.9 x 10

1.0 x 10

1.1 x 10

1.2 x 10

x (cm)

P (dyne/cm2) 8

8

8

8

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Stable Strongly Overdriven Case: Transient

Behavior for Various Resolutions

Initialize with steady structure ofDo = 2.5×105 cm/s.

2.480 x 105

2.485 x 105

2.490 x 105

2.495 x 105

2.500 x 105

2.505 x 105

0 2 x 10-10

4 x 10-10

6 x 10-10

8 x 10-10

1 x 10-9

∆x=2.5x10 -8 cm

∆x=5x10 -8 cm

∆x=1x10 -7 cm

D (

cm

/s)

t (s)

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3 Cases Near Neutral Stability: Transient Behavior

x

x

x

x

x

x

x x x x

o

o

o

5

5

5

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Slightly Unstable Case: Transient Behavior

Initialized with steady structure,Do = 2.4 × 105 cm/s.

x

x

x

x

x

x

x x x x x

o5

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Case After Bifurcation: Transient Behavior

Initialized with steady structure ofDo = 2.1×105 cm/s.

150000

200000

250000

300000

350000

400000

450000

0

t (s)

D (

cm /

s)

2 x 105 x 10 -9

1.5 x 10-8

1 x 10-8 -8

D = 2.1 x 10 cm/s5

o

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Long Time Dmax/D0 versus DCJ/D0

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Effect of Resolution on Unstable Moderately

Overdriven Case

∆x Numerical Result

1 × 10−7 cm Unstable Pulsation

2 × 10−7 cm Unstable Pulsation

4 × 10−7 cm Unstable Pulsation

8 × 10−7 cm O2 mass fraction > 1

1.6 × 10−6 cm O2 mass fraction > 1

• Algorithm failure for insufficient resolution

• At low resolution, one misses critical dynamics

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Examination of H2-Air Results

Reference ℓind (cm) ℓf (cm) ∆x (cm) Under-resolution

Oran, et al., 1998 2 × 10−1 2 × 10−4 4 × 10−3 2 × 101

Jameson, et al., 1998 2 × 10−2 5 × 10−5 3 × 10−3 6 × 101

Hayashi, et al., 2002 2 × 10−2 1 × 10−5 5 × 10−4 5 × 101

Hu, et al., 2004 2 × 10−1 2 × 10−4 3 × 10−3 2 × 101

Powers, et al., 2001 2 × 10−2 3 × 10−5 8 × 10−5 3 × 100

Osher, et al., 1997 2 × 10−2 3 × 10−5 3 × 10−2 1 × 103

Merkle, et al., 2002 5 × 10−3 8 × 10−6 1 × 10−2 1 × 103

Sislian, et al., 1998 1 × 10−1 2 × 10−4 1 × 100 5 × 103

Jeung, et al., 1998 2 × 10−2 6 × 10−7 6 × 10−2 1 × 105

All are under-resolved, some severely.

Page 39: High Accuracy Shock-Fitted Computation of Unsteady ...

Conclusions

• Unsteady detonation dynamics can be accurately sim-

ulated when sub-micron scale structures admitted by

detailed kinetics are captured with ultra-fine grids.

• Shock fitting coupled with high order spatial discretiza-

tion assures numerical corruption is minimal.

• Predicted detonation dynamics consistent with results

from one-step kinetic models.

• At these length scales, diffusion will play a role and

should be included in future work.

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Moral

You either do detailed kinetics with the

proper resolution,

or

you are fooling yourself and others, in

which case you should stick with

reduced kinetics!