High Accuracy Shock-Fitted Computation of

Unsteady Detonation with Detailed Kinetics

Tariq D. Aslam (aslam@lanl.gov)

Los Alamos National Laboratory; Los Alamos, NM

Joseph M. Powers (powers@nd.edu)

University of Notre Dame; Notre Dame, IN

10th US National Congress on Computational Mechanics

Columbus, Ohio

18 July 2009

Outline

• Gas phase detonation introduction

• Length scale requirements from steady traveling wave

solutions for H2-air (review)

• Unsteady dynamics of ozone detonation

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.

Length and Time Scale Discussion

Simplistic linear advection-reaction model:

∂ψ

∂t︸︷︷︸

evolution

+ uo

∂ψ

∂x︸ ︷︷ ︸

advection

= −kψ︸︷︷︸

reaction

dψ

dt= −kψ : time scale τ =

1

k

uo

dψ

dx= −kψ : length scale ℓ =

uo

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

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.

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.

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

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)| .

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.

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).

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

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.

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.

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.

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.

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.

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.

Unsteady Model: Reactive Euler Equations

• one-dimensional,

• inviscid,

• detailed mass action kinetics with Arrhenius tempera-

ture dependency,

• ideal mixture of calorically imperfect ideal gases

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).

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

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

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

Summary of Shock-Fitting Method

−17.5 −12.5 −10 −7.5 −5 −2.5

10

20

30

40

ξ

boundary interior

near

shock

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.

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.

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.

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)

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.

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

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)

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

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)

3 Cases Near Neutral Stability: Transient Behavior

x

x

x

x

x

x

x x x x

o

o

o

5

5

5

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

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

Long Time Dmax/D0 versus DCJ/D0

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

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.

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.

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!