effects of diffusion on the dynamics of detonation tariq d...
TRANSCRIPT
![Page 1: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/1.jpg)
Effects of Diffusion on the Dynamics of Detonation
Tariq D. AslamLos Alamos National Laboratory; Los Alamos, NM
Joseph M. Powers∗University of Notre Dame; Notre Dame, IN
Gordon Research Conference - Energetic MaterialsTilton, New Hampshire
June 13-18, 2010
∗and contributions from many others
![Page 2: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/2.jpg)
Motivation
• Computational tools are critical in modeling of high
speed reactive flow.
• Steady wave calculations reveal sub-micron scale struc-
tures in detonations with detailed kinetics (Powers and
Paolucci, AIAA J., 2005).
• Small structures are continuum manifestation of molec-
ular collisions.
• We explore the transient behavior of detonations with
fully resolved detailed kinetics.
![Page 3: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/3.jpg)
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.
![Page 4: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/4.jpg)
Some Length Scales inherent in PBXs
Micrograph of PBX 9501 (from C. Skidmore)
![Page 5: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/5.jpg)
Some Length Scales Due to Diffusion
Shock Rise in Aluminum (from V. Whitley)
10 ps rise time at 10 km/s yields scale of 10−7 m.
![Page 6: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/6.jpg)
Modeling Issues for PBXs:
• Inherently 3D, multi-component mixture,
• Massive disparity in scales,
• Many parameters are needed and many are unknown∗:
– elastic constants
– equation of states
– thermal conductivities, viscosities for constituents
– heat capacities
– reaction rates
– species diffusion
∗see Menikoff and Sewell, CTM 2002
![Page 7: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/7.jpg)
Before climbing Everest, we need to step back a bit...
Let’s examine detonation dynamics of gases...
1. Inviscid, one-step Arrhenius chemistry
2. Inviscid, detailed chemistry
3. Diffusive, one-step Arrhenius chemistry
4. Diffusive, detailed chemistry
![Page 8: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/8.jpg)
Model: Reactive Euler Equations
• one-dimensional,
• unsteady,
• inviscid,
• detailed mass action kinetics with Arrhenius tempera-
ture dependency,
• ideal mixture of calorically imperfect ideal gases
![Page 9: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/9.jpg)
General Review of Pulsating Detonations
• Erpenbeck, Phys. Fluids, 1962,
• Fickett and Wood, Phys. Fluids, 1966,
• Lee and Stewart, JFM, 1990,
• Bourlioux, et al., SIAM J. Appl. Math., 1991,
• He and Lee, Phys. Fluids, 1995,
• Short, SIAM J. Appl. Math., 1997,
• Sharpe, Proc. R. Soc., 1997.
![Page 10: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/10.jpg)
Review of Recent Work of Special Relevance
• Kasimov and Stewart, Phys. Fluids, 2004: published
detailed discussion of limit cycle behavior with shock-
fitting; error ∼ O(∆x).
• Ng, Higgins, Kiyanda, Radulescu, Lee, Bates, and
Nikiforakis, CTM, in press, 2005: in addition, consid-
ered transition to chaos; error ∼ O(∆x).
• Present study similar to above, but error ∼ O(∆x5).
![Page 11: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/11.jpg)
Model: Reactive Euler Equations
• one-dimensional,
• unsteady,
• inviscid,
• one step kinetics with finite activation energy,
• calorically perfect ideal gases with identical molecular
masses and specific heats.
![Page 12: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/12.jpg)
Model: Reactive Euler Equations
∂ρ
∂t+
∂
∂ξ(ρu) = 0,
∂
∂t(ρu) +
∂
∂ξ
`
ρu2 + p´
= 0,
∂
∂t
„
ρ
„
e +1
2u2
««
+∂
∂ξ
„
ρu
„
e +1
2u2 +
p
ρ
««
= 0,
∂
∂t(ρλ) +
∂
∂ξ(ρuλ) = αρ(1 − λ) exp
„
−
ρE
p
«
,
e =1
γ − 1
p
ρ− λq.
![Page 13: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/13.jpg)
Unsteady Shock Jump Equations
ρs(D(t) − us) = ρo(D(t) − uo),
ps − po = (ρo(D(t) − uo))2
(
1
ρo− 1
ρs
)
,
es − eo =1
2(ps + po)
(
1
ρo− 1
ρs
)
,
λs = λo.
![Page 14: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/14.jpg)
Model Refinement
• Transform to shock attached frame via
x = ξ −∫ t
0
D(τ)dτ,
• Use jump conditions to develop shock-change equa-
tion for shock acceleration:
dD
dt= −
(
d(ρsus)
dD
)−1 (
∂
∂x(ρu(u − D) + p)
)
.
![Page 15: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/15.jpg)
Numerical Method
• point-wise method of lines,
• uniform spatial grid,
• fifth order spatial discretization (WENO5M) takes PDEs
into ODEs in time only,
• fifth order explicit Runge-Kutta temporal discretization
to solve ODEs.
• details in Henrick, Aslam, Powers, JCP, 2006.
![Page 16: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/16.jpg)
Numerical Simulations
• ρo = 1, po = 1, L1/2 = 1, q = 50, γ = 1.2,
• Activation energy, E, a variable bifurcation parameter,
25 ≤ E ≤ 28.4,
• CJ velocity: DCJ =√
11 +√
615 ≈ 6.80947463,
• from 10 to 200 points in L1/2,
• initial steady CJ state perturbed by truncation error,
• integrated in time until limit cycle behavior realized.
![Page 17: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/17.jpg)
Stable Case, E = 25: Kasimov’s Shock-Fitting
6.81
6.82
6.83
6.84
0 50 100 150 200 250 300t
D
N = 1001/2
N = 2001/2
exact
• N1/2 = 100, 200,
• minimum error in D:
∼ 9.40 × 10−3,
• Error in D converges
at O(∆x1.01).
![Page 18: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/18.jpg)
Stable Case, E = 25: Improved Shock-Fitting
6.809465
6.809470
6.809475
6.809480
6.809485
0 50 100 150 200 250 300
t
D exact
N = 201/2
• N1/2 = 20, 40,
• minimum error in D:
∼ 6.00 × 10−8, for
N1/2 = 40.
• Error in D converges
at O(∆x5.01).
![Page 19: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/19.jpg)
Linearly Unstable, Non-linearly Stable Case: E = 26
6.2
6.6
7.0
7.4
0 100 200 300 400 500t
D
• One linearly unstable
mode, stabilized by
non-linear effects,
• Growth rate and fre-
quency match linear
theory to five decimal
places.
![Page 20: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/20.jpg)
D, dDdt Phase Plane: E = 26
-0.2
0.0
0.2
0.4
6.2 6.6 7.0 7.4
D
dDdt
stationarylimitcycle
• Unstable spiral at early
time, stable period-1
limit cycle at late time,
• Bifurcation point of
E = 25.265 ± 0.005
agrees with linear
stability theory.
![Page 21: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/21.jpg)
Period Doubling: E = 27.35
6
7
8
0 100 200 300 400t
D
• N1/2 = 20,
• Bifurcation to period-
2 oscillation at E =
27.1875 ± 0.0025.
![Page 22: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/22.jpg)
D, dDdt Phase Plane: E = 27.35
-1
0
1
6 7 8D
dDdt
stationarylimit cycle • Long time period-2
limit cycle,
• Similar to independent
results of Sharpe and
Ng.
![Page 23: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/23.jpg)
Transition to Chaos and Feigenbaum’s Number
limn→∞
δn =En − En−1
En+1 − En
= 4.669201 . . . .
n En En+1 − Ei δn
0 25.265 ± 0.005 - -
1 27.1875 ± 0.0025 1.9225 ± 0.0075 3.86 ± 0.05
2 27.6850 ± 0.001 0.4975 ± 0.0325 4.26 ± 0.08
3 27.8017 ± 0.0002 0.1167 ± 0.0012 4.66 ± 0.09
4 27.82675 ± 0.00005 0.02505 ± 0.00025 -
......
......
∞ 4.669201 . . .
![Page 24: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/24.jpg)
Bifurcation Diagram
7
8
9
10
25 26 27 28
E
Dmax
E <
25.
265,
line
arly
sta
ble
25.265 < E < 27.1875,Period 2 (single period mode)0
27.1
875
< E
< 2
7.68
50P
erio
d 21
StablePeriod 6
StablePeriod 3
Sta
ble
Per
iod
5
Sta
ble
Per
iod
2n
![Page 25: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/25.jpg)
D versus t for Increasing E
6
7
8
9
3700 3800 3900 4000t
D
6
7
8
9
3700 3800 3900 4000t
D
a b
d
e f
c
6
7
8
9
3700 3800 3900 4000t
D
6
7
8
9
3700 3800 3900 4000t
D
6
7
8
9
3700 3800 3900 4000t
D
6
7
8
9
10
3700 3800 3900 4000
t
D
![Page 26: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/26.jpg)
Model: Reactive Euler PDEs with Detailed Kinetics
∂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).
![Page 27: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/27.jpg)
Computational Methods
• Steady wave structure
– LSODE solver with IMSL DNEQNF for root finding
– Ten second run time on single processor machine.
– see Powers and Paolucci, AIAA J., 2005.
• Unsteady wave structure
– Shock fitting coupled with a high order method for
continuous regions
– see Henrick, Aslam, Powers, J. Comp. Phys., 2006,
for full details on shock fitting
![Page 28: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/28.jpg)
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.
![Page 29: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/29.jpg)
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.
![Page 30: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/30.jpg)
Stable Strongly Overdriven Case: Length Scales
D = 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)
![Page 31: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/31.jpg)
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.
![Page 32: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/32.jpg)
Stable Strongly Overdriven Case: Mass Fractions
D = 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
![Page 33: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/33.jpg)
Stable Strongly Overdriven Case: Temperature
D = 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)
![Page 34: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/34.jpg)
Stable Strongly Overdriven Case: Pressure
D = 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
![Page 35: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/35.jpg)
Stable Strongly Overdriven Case: Transient
Behavior for various resolutions
Initialize with steady structure of D = 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)
![Page 36: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/36.jpg)
Unstable Moderately Overdriven Case: Transient
Behavior
Initialize with steady structure of D = 2 × 105 cm/s.
2 x 105
3 x 105
4 x 105
0 1 x 10-9
2 x 10-9
3 x 10-9
∆x=2x10 -7 cm
D (
cm
/s)
t (s)
![Page 37: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/37.jpg)
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
![Page 38: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/38.jpg)
Long Time Relative Maxima in D/Do versus Inverse
Overdrive
![Page 39: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/39.jpg)
Diffusive Modeling in Gaseous Detonation
∂ρ
∂t+
∂
∂x(ρu) = 0,
∂
∂t(ρu) +
∂
∂x
(
ρu2 + p − τ)
= 0,
∂
∂t
(
ρ
(
e +u2
2
))
+∂
∂x
(
ρu
(
e +u2
2
)
+ jq + (p − τ) u
)
= 0,
∂
∂t(ρYB) +
∂
∂x(ρuYB + jm
B ) = ρr,
![Page 40: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/40.jpg)
Diffusive Modeling in Gaseous Detonation
p = ρRT,
e = cvT − qYB =p
ρ (γ − 1)− qYB ,
r = H(p − ps)a (1 − YB) e−E
p/ρ ,
jmB = −ρD∂YB
∂x,
τ =4
3µ
∂u
∂x,
jq = −k∂T
∂x+ ρDq
∂YB
∂x.
![Page 41: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/41.jpg)
To compare with previous one-step work...
• Need to choose scale ratios between diffusion and
reaction
• Choose half-reaction length scale to be 1µm.
• Choose diffusive length scale to be 100nm
D = 10−4 m2/s,
k = 10−1 W/m/K,
µ = 10−4 Ns/m2,
For ρo = 1 kg/m3, Le = Sc = Pr = 1.
![Page 42: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/42.jpg)
Numerical Methods
• 5th Order WENO Schemes for Hyperbolic Compo-
nents
• 4th Order Central Difference Scheme for Parabolic
Components
• 3rd Order Explicit Runge-Kutta Time Integration
• Expect Fully 4th Order Convergence Rates Under
Resolution
![Page 43: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/43.jpg)
Density for a stable detonation E = 25
0 1 2
x 10−4
2
4
6
8ρ
(kg/
m3 )
x (m)
t = 5.0× 10−8 s
t = 0.0 s
![Page 44: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/44.jpg)
Density for a stable detonation E = 25 - zoom
1.5 1.52 1.54 1.56 1.58 1.6
x 10−4
2
4
6
8ρ
(kg/
m3 )
x (m)
t = 5.0× 10−8 s
![Page 45: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/45.jpg)
Pressure for a stable detonation E = 25 - zoom
0 1 2
x 10−4
1000
2000
3000
4000
x (m)
Pre
ssur
e (k
Pa)
t = 0.0 s
t = 5.0× 10−8 s
![Page 46: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/46.jpg)
Pressure vs. time for a unstable detonation E = 28
0 0.5 13000
3500
4000
4500
5000
5500
6000P
ress
ure
(kP
a)
t (µs)
![Page 47: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/47.jpg)
Pressure vs. time for a unstable detonation
E = 29.5
Period Doubling
![Page 48: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/48.jpg)
Pressure vs. time for a unstable detonation E = 32
Chaotic Dynamics
![Page 49: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/49.jpg)
Diffusive Bifurcation Diagram
With a relatively small amount of diffusion, a substantial
stabilization occurs.
![Page 50: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/50.jpg)
Where we are headed with all this...
Multi-D WAMR simulation of 2H2 : O2 : 7Ar
![Page 51: Effects of Diffusion on the Dynamics of Detonation Tariq D ...powers/paper.list/gordon.slides.2010.pdf · 1×10−7 cm Unstable Pulsation 2×10−7 cm Unstable Pulsation 4×10−7](https://reader034.vdocuments.us/reader034/viewer/2022050323/5f7c5342fd8a9c25761326c6/html5/thumbnails/51.jpg)
Conclusions
• Unsteady detonation dynamics can be accurately simulated when
sub-micron scale structures admitted by detailed kinetics are
captured with ultra-fine grids.
• Shock fitting coupled with high order spatial discretization assures
numerical corruption is minimal.
• For resolved diffusive effects, relatively simple numerical methods
work fine.
• Predicted detonation dynamics consistent with results from invis-
cid models...
• At these sub-micron length scales, diffusion plays a substantial
role.