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