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1
Computational sub-structure analysis of multidimensional detonation waves Ralf Deiterding Computer Science and Mathematics Division, Oak Ridge National Laboratory Governing equations We solve the inhomogeneous Euler equations for multiple species that read q t + ∇· f (q)= s(q) (1) with vector of state q =(ρ 1 ,...,ρ K , ρu 1 ,...,ρu d , ρE ) T and flux functions f n (q)=(ρ 1 u n ,...,ρ K u n , ρu 1 u n + δ 1n p, . . . , ρu d u n + δ dn p, u n (ρE + p)) T . The equation of state follows Dalton’s law and the ideal gas law p = K i=1 p i = RT K i=1 ρ i W i with the caloric equation h = K i=1 Y i h i (T ) , h i (T )= h 0 i + T T 0 c pi (σ )dσ which requires computation of T = T (ρ,e) from the implicit equation K i=1 ρ i h i (T ) - ρe -RT K i=1 ρ i W i =0 . For chemistry, the source term is s(q)= ( W 1 ˙ ω 1 ,...,W K ˙ ω K , 0,..., 0, 0 ) T with the reaction rates for detailed kinetics ˙ ω i = J j =1 (ν r ji - ν f ji ) k f j K l=1 ρ l W l ν f jl - k r j K l=1 ρ l W l ν r jl . In here, all results were obtained with a mechanism for H 2 - O 2 - Ar combustion with 34 elementary reactions and 9 species. Numerical methods Numerical source term incorporation and extension to multiple dimensions are done with the method of fractional steps. -→ Numerical decoupling of hydrodynamic and chemical time steps. Time-explicit 2nd order TVD shock-capturing method for thermally perfect gases: In most cells an approximate Riemann solver with numerical flux function F(Q l , Q r )= 1 2 f (Q l )+ f (Q r ) - 3 ι=1 |s ι |W ι with s 1 =ˆ u 1 - ˆ c, s 2 =ˆ u 1 , s 3 =ˆ u 1 +ˆ c is used. The waves W ι are defined as W 1 = a 1 ˆ r 1 , W 2 = K +2 m=2 a m ˆ r m , W 3 = a K +3 ˆ r K +3 , where a 1 = Δp - ˆ ρˆ cΔu 1 2ˆ c 2 ,a i+1 =Δρ i - ˆ Y i Δp ˆ c 2 ,a K +2 =ˆ ρΔu n ,a K +3 = Δp +ˆ ρˆ cΔu 1 2ˆ c 2 , and ˆ r 1 =( ˆ Y 1 ,..., ˆ Y K , ˆ u 1 - ˆ c, ˆ u 2 , ˆ H - ˆ u 1 ) T , ˆ r i+1 =(δ 1i ,...,δ Ki , ˆ u 1 , ˆ u 2 , ˆ u 2 1 +ˆ u 2 2 - ˆ φ i /(ˆ γ - 1)) T , ˆ r K +2 = (0,..., 0, 1, ˆ u 2 ) T , ˆ r K +3 =( ˆ Y 1 ,..., ˆ Y K , ˆ u 1 +ˆ c, ˆ u 2 , ˆ H +ˆ u 1 ) T . with the usual Roe average given by ˆ v := √ ρ l v l + √ ρ r v r √ ρ l + √ ρ r for u n ,Y i ,T,h i ,H := E + p/ρ and ˆ ρ := √ ρ l ρ r and the specific averages ˆ γ := ˆ c p ˆ c v with ˆ c {p/v } = K i=1 ˆ Y i ˆ c {p/v }i , ˆ c {p/v }i = 1 T r - T l T r T l c {p,v }i (τ ) dτ and ˆ φ i := (ˆ γ - 1) ˆ u 2 1 +ˆ u 2 2 2 - ˆ h i +ˆ γ R W i ˆ T, ˆ c := K i=1 ˆ Y i ˆ φ i - (ˆ γ - 1)(ˆ u 2 1 +ˆ u 2 2 - ˆ H ) 1/2 . The entropy correction reads |¯ s ι | = |s ι | , |s ι |≥ 2η, |s 2 ι |/(4η )+ η, |s ι | < 2η and is applied in 1d to ι =1, 3 with η = 1 2 (|u 1,r -u 1,l |+|c r -c l |), where in 2d and 3d a multidimensional evaluation of η for all fields is used as carbuncle fix. Example: ˜ η j,k+ 1 2 = max η j + 1 2 ,k ,η j - 1 2 ,k ,η j, k+ 1 2 ,η j - 1 2 ,k+1 ,η j + 1 2 ,k+1 The correction F i = F ρ · Y l i , F ρ ≥ 0 , Y r i , F ρ < 0 ensures the positivity of the mass fractions Y i . If internal energy or pressure in the intermediate states Q l = Q l + W 1 and Q r = Q r - W 3 are unphysical, i.e. ρ l/r ≤ 0 or e l/r ≤ 0, the HLL flux F(Q l , Q r )= f (Q l ) , 0 <s 1 , s 3 f (Q l ) - s 1 f (Q R )+ s 1 s 3 (Q r - Q l ) s 3 - s 1 , s 1 ≤ 0 ≤ s 3 , f (Q r ) , 0 >s 3 , with the wave speed estimation s 1 = min(u 1,l - c l ,u 1,r - c r ), s 3 = max(u 1,l + c l ,u 1,r + c r ) is used. Source term integration: • Standard solver for stiff ODE’s, e.g., semi-implicit Rosenbrock-Wanner method • Automatic stepsize adjustment to allow for an efficient treatment of chemical time scales smaller than the global time-step Blockstructured AMR Locally high resolution, which is essential for the accurate computation of deto- nation waves, is achieved by blockstructured Adaptive Mesh Refinement (AMR). • Discretization necessary only for single rectangular grid • Spatial and temporal refinement, no global time step restriction • Blockstructured data guarantees high computational performance • Non-conformal finite volumes unavoid- able and require special treatment Our framework AMROC provides a generic object-oriented implementation of the blockstructured AMR method that is applicable to any explicit FV scheme for (1): • Parallel hierarchical data structures employ MPI library • Data follows “floor plan” of a single Grid Hierarchy. • Data of all levels resides on same node → most AMR operations are local • Neighboring grids are synchronized transparently even over processor borders when boundary conditions are applied • Distribution algorithm: Generalization of Hilbert’s space-filling curve Refinement indicators Physically motivated refinement is achieved through scaled gradients |w (Q j +1,k ) - w (Q jk )| > w , |w (Q j,k +1 ) - w (Q jk )| > w , |w (Q j +1,k +1 ) - w (Q jk )| > w or estimating the leading-order term of the local error of quantity w by Richardson extrapolation τ w jk := |w ( ¯ Q jk (t +Δt)) - w (Q jk (t +Δt))| 2 o+1 - 2 . In practice, we use the criterion τ w jk max(|w (Q jk (t +Δt))|,S w ) >η r w that combines relative and absolute error. Verification: Detonation ignition in a shock tube Shock-induced detonation ignition of a hydrogen- oxygen-argon mixture at molar ratios 2:1:7 in a 1d shock tube closed at the left end. • Insufficient resolution leads to inaccurate results • Reflected shock is captured by the FV scheme correctly at all resolutions, but the detonation approximation is resolution dependent as can be seen in time t m , when both waves merge • Fine mesh necessary in the induction zone at the head of the detonation and AMR is highly suitable to reduce the compute time 1000 1500 2000 2500 3000 6.4 6.36 6.32 6.28 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Temperature [K] Mass fraction [-] x 1 [cm] T Y H Y O Y H 2 1000 1500 2000 2500 3000 6.02 5.98 5.94 5.9 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Temperature [K] Mass fraction [-] x 1 [cm] T Y H Y O Y H 2 Approximation of the detonation wave at t = 170 μs for the uniform resolutions Δx 1 = 100 μm and Δx 1 =6.25 μm(∼ 24 Pts/l ig ). 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 4.5 5 5.5 6 6.5 7 Pressure [MPa] x 1 [cm] ∆x 1 =6.25 μm ∆x 1 =100 μm ∆x 1 =200 μm 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 4 4.5 5 5.5 6 6.5 7 7.5 8 Pressure [MPa] x 1 [cm] Pressure Level 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 6.05 6.03 6.01 5.99 5.97 5.95 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Pressure [MPa] Y O [-] x 1 [cm] Uniform Adaptive Left: Comparison of pressure distribution at t = 170 μs for different mesh sizes. Middle: Domains of refinement levels. Right: Direct comparison of the distributions of p and Y O in the detonation wave with a uniform simulation. Uniform Adaptive Δx 1 [μm] Cells t m [μs] Time [s] l max r l t m [μs] Time [s] 400 300 166.1 31 200 600 172.6 90 2 2 172.6 99 100 1200 175.5 277 3 2,2 175.8 167 50 2400 176.9 858 4 2,2,2 177.3 287 25 4800 177.8 2713 4 2,2,4 177.9 393 12.5 9600 178.3 9472 5 2,2,2,4 178.3 696 6.25 19200 178.6 35712 5 2,2,4,4 178.6 1370 Comparison of uniformly refined and dynamic adaptive simulations run on a single CPU of a Intel Xeon 3.4 GHz processor. Y i S Y i · 10 -4 η r Y i · 10 -3 O 2 10.0 2.0 H 2 O 7.8 8.0 H 0.16 5.0 O 1.0 5.0 OH 1.8 5.0 H 2 1.3 2.0 ρ =0.07 kg m -3 , p = 50 kPa Refinement indicator values for the one- dimensional ignition problem. Embedding of complex domains Usage of a signed distance function ϕ for efficient implicit boundary representation on the hierarchical Cartesian mesh. • ϕ = 0 marks the boundary, boundary normal n = ∇ϕ/|∇ϕ| • Our current implementation evaluates ϕ in the mid- points only and uses mesh cells directly to prescribe boundary conditions -→ proper AMR refinement of the boundary necessary • Non-oscillating, 1st order accurate interpola- tion/extrapolation to construct ghost cell values • Mirroring of the primitive values ρ i , u, p and inversion of normal velocity component by u = (2w · n - u · n)n +(u · t)t =2 ( (w - u) · n ) n + u Interpolation from interior cells to construct mirrored values in ghost cells (gray). Verification: Shock-induced combustion around a sphere A spherical projectile of radius 1.5 mm travels with constant velocity v I = 2170.6m/s through a hydrogen-oxygen-argon mixture (molar ratios 2:1:7) at 6.67 kPa and T = 298 K and exhibits shock-induced combustion ignition. • Cylindrical symmetric simulation on AMR base mesh of 70 × 40 cells • Comparison of 3-level computation with refinement factors 2,2 (∼ 5 Pts/l ig ) and a 4-level computation with refinement factors 2,2,4 (∼ 19 Pts/l ig ) at t = 350 μs • Higher resolved computation captures combustion zone visibly better and at slightly different position (see below) Iso-contours of p (black) and Y H 2 (white) on domains of different refinement levels (gray) at t = 350 μs for a 3-level computation with r 1,2 = 2 (left) and a 4-level computation with r 1,2 =2,r 3 = 4 (right). Left: Active refinement indicators on l = 2 at t = 350 μs. Blue: ρ , light blue: p , green shades: η r Y i , red: embedded boundary. Right: Domain decomposition to 8 processors for the 4-level hierarchy depicted in the graphic above. 4 Procs 8 Procs 16 Procs Task Time [s] % Time [s] % Time [s] % Fluid dynamics 22374 34.6 11261 32.3 5742 26.7 Chemical kinetics 25280 39.1 12455 35.8 6291 29.2 Boundary setting 7758 12.0 5888 16.9 5761 26.8 Embedded boundary 5429 8.4 2839 8.2 1703 7.9 Recomposition 3210 5.0 1919 5.5 1623 7.5 Misc 638 0.9 456 1.3 416 1.9 Total / Speed-up 64689 34818 1.86 21536 1.62 Breakdown of computational costs for parallel 4-level simulations with r 1,2,3 = 2 running from t = 150 μs to t e = 400 μs. Y i S Y i · 10 -4 η r Y i · 10 -4 O 2 10.0 4.0 H 2 O 5.8 3.0 H 0.2 10.0 O 1.4 10.0 OH 2.3 10.0 H 2 1.3 4.0 ρ =0.02 kg m -3 , p = 16 kPa Refinement indicator values for simulat- ing shock-induced combustion around the projectile. Cellular detonation and triple point structures Self-sustaining detonations are inherently unstable and exhibit transverse pres- sure waves that propagate perpendicular to the detonation front. The transverse Schlieren and PLIF images of a regularly oscillating detonation front in hydrogen-oxygen-argon (image courtesy J. E. Shepherd, Graduate Aeronautical Laboratory, California Institute of Technology). waves form evolving triple point patterns at the det- onation front with severely enhanced pressure, temper- ature and thereby reaction. Despite considerable exper- imental efforts, a general model for the instability and the detailed flow field in triple points in detonations is not available yet. Objective of this work is to decipher the de- tailed hydrodynamic flow conditions in triple point sub-structures under periodic and tran- sient conditions. In experiments, basically two different configurations have been qualitatively distinguished. Experimentally known triple point structures. Regular detonation structures in 2d • Mixture: H 2 :O 2 : Ar at molar ratios 2:1:7 at initially 298 K and 10 kPA • Initialization with Chapman-Jouguet ZND solution and an irregular pocket to quickly trigger symmetry breaking. Tube width 3.2 cm • Simulation until a single perturbation has developed into two regularly oscillating cells with width λ =1.6 cm • 67.6 Pts/l ig . 4 additional refinement levels (2,2,2,4) on base mesh 2000 × 128 • Computation of triple point trajectories by tracking the magnitude of the vor- ticity vector on a uniform mesh at level 1 Left: Oscillating detonation front on computed triple points tracks. Right: Schlieren plot on refinement levels. Left: Front on triple point tracks. 10 μs between snapshots (marked with open squares in (M s , Φ 1 )-plane, cf. right column). Middle and right: Clearly established Double-Mach reflection pattern shortly before triple point collision. Only the strong structure appears in the simulations and can clearly be identi- fied as a double-Mach reflection pattern. p/p 0 ρ/ρ 0 T [K] q [m/s] α γ M A 1.00 1.00 298 1775 -39.5 1.557 5.078 B 31.45 4.17 2248 447 -64.4 1.487 0.477 C 31.69 5.32 1775 965 -64.4 1.500 1.153 D 19.17 3.84 1487 1178 -73.2 1.509 1.533 E 35.61 5.72 1856 901 -67.9 1.497 1.053 F 40.61 6.09 1987 777 -70.0 1.494 0.880 10 20 30 40 50 0 5 10 15 20 25 30 35 40 p/p 0 θ Incident Reflected Values shortly before triple point collision in frame of reference of primary triple point. Inflow (A), Mach stem (B), reflected wave (C), incident shock (D), 2nd reflected (E), 2nd Mach stem (F). (q , α) are the polar coordinates of the velocity vector in the Galilean frame of reference. Right: shock polars for first triple point in left table. Cellular structures in 3d • H 2 :O 2 : Ar /2:1:7at6.67 kPA with cell width λ = 3 cm in 3d and 2d (not shown) • High-resolution simulation of the quarter of a rectangular detonation cell • Simulation under Galilean transformation -→ inflow of unreacted gas with CJ velocity • 44.8 Pts/l ig . 2 additional refinement levels (2,4) on base mesh 400 × 24 × 24, adaptive Simulation uses ∼ 18 · 10 6 cells instead of 118 · 10 6 (uniform) •∼ 55, 000 h CPU on 128 Compaq Alpha processors Regularly oscillating detonations that exhibited a strong structure in 2d, show only the weak structure in 3d as the transverse wave is weaker as in 2d. The resulting structure in 3d can be identified as a transitional Mach reflection. Left, middle: Schlieren plots of ρ (left) and Y OH in the first half of a detonation cell. Data displayed is for 5.0 cm < x 1 < 7.0 cm and mirrored at x 2 = 0 and x 3 = 0. The blue of ρ visualizes the induction length. Right: Planes with schlieren plots of ρ through the origin normal to x 2 and x 3 -axis unfolded exhibit the transitional Mach structure behind the triple points, which is characterized by a kink instead of secondary triple point on the transverse wave. Detonation structure in smooth pipe bends • Initialization with 5 regularly oscillating detonation cells in H 2 :O 2 : Ar /2:1:7 at initially 298 K and 10 kPa, tube with 8 cm • Pipe bend with radius 8 cm. Angle θ = {15 o , 30 o , 45 o , 60 o } • 67.6 Pts/l ig . 4 additional refinement levels with factors (2,2,2,4) • Adaptive computations use ∼ 7 · 10 6 cells (∼ 5 · 10 6 on highest level) instead of ∼ 1.2 · 10 9 cells (uniform grid) •∼ 70, 000 h CPU each on 128 CPUs Pentium-4 2.2 GHz Color plot of T and schlieren plot of ρ on triple point tracks and on refinement regions (middle, right) for θ = 45 o after t = 150 μs simulated time. Triple point tracks for θ = 15 o (left, top), θ = 30 o (left, bottom), and θ = 60 o (right) display the fundamental flow features. Top: Front on triple point tracks for θ = 15 and 110 μs, 140 μs, 170 μs, 200 μs simulated time. Right, top: Strengthening of double-Mach reflection pattern (solid squares in (M s , Φ 1 )-plane) under expansion at 130 μs and 140 μs. Right, bottom: Weakening transitional Mach reflection in over-driven region (open circles in (M s , Φ 1 )-plane) at 130 μs and 140 μs. Triple point re-initiation with change from transitional to double-Mach reflection (solid circles in (M s , Φ 1 )-plane) when the oscillation becomes regular again after the bend, θ = 15 at 200 μs, 210 μs, and 220 μs. 35 40 45 50 55 60 65 70 1 2 3 4 5 6 7 8 Φ 1 M s NR SMR TMR/DMR RR Domains and transition boundaries of different shock wave reflection phenomena. Frame of reference of first triple point. Φ 1 is angle between incident shock and vector (q,α), M s = sin(Φ 1 )M . NR: no reflection, SMR: single Mach reflection, TMR/DMR: transitional/double- Mach reflection, RR: regular reflection. Under transient boundary conditions, the regularly oscillating detonation exhibits both weak and strong triple point structures. Detailed quantitative triple point analysis for the thermally perfect, but non-reactive, H 2 - O 2 - Ar mixture (using the oblique shock relations to transform the Eulerian data into the frame work reference of the primary triple point) indicates strongly: • Triple point structures in self-sustained detonations exist only in the transitional and double-Mach, but not in the single-Mach reflection regime (cf. above (M s , Φ 1 )-plane) • The relative strength S = p C -p D p D of the transverse pressure wave primarily de- termines the reflection type • A change of the reflection type happens especially in triple point collisions Publications R. D.. A parallel adaptive method for simulating shock-induced combustion with detailed chemical kinetics in complex domains, Computers & Structures, submitted, 2008. R. D. Dynamically adaptive simulation of regular detonation structures using the Cartesian mesh refinement framework AMROC. Int. Journal Computational Science and Engineering, accepted and in press, 2008.
Transcript of aves - rdeiterding.website · aves ng ry ns ead q t + f (q = s (q (1) e q ( 1 K u 1 u d E) T ions f...
Com
puta
tionalsub-s
tructu
reanalysis
ofm
ultid
imensio
naldeto
nation
waves
Ralf
Deiterd
ing
Com
pute
rScie
nce
and
Math
em
atics
Div
isio
n,O
ak
Rid
ge
NationalLabora
tory
Govern
ing
equations
We
solv
eth
ein
hom
ogeneous
Eule
requations
for
multip
lesp
ecie
sth
at
read
qt+∇·f
(q)=
s(q)
(1)
with
vecto
rofst
ate
q=
(ρ1,.
..,ρ
K,ρ
u1,.
..,ρ
ud,ρ
E)T
and
flux
functions
f n(q
)=
(ρ1u
n,.
..,ρ
Ku
n,ρ
u1u
n+
δ 1np,.
..,ρ
udu
n+
δ dnp,u
n(ρ
E+
p))
T.
The
equation
ofst
ate
follow
sD
alton’s
law
and
the
idealgas
law
p=
K ∑ i=1
pi=R
TK ∑ i=
1
ρi
Wi
with
the
calo
ric
equation
h=
K ∑ i=1
Yih
i(T),
hi(
T)=
h0 i+
∫ T T0c p
i(σ)d
σ
whic
hre
quires
com
puta
tion
of
T=
T(ρ
,e)
from
the
implicit
equation
K ∑ i=1
ρih
i(T)−
ρe−R
TK ∑ i=
1
ρi
Wi=
0.
For
chem
istr
y,th
eso
urc
ete
rmis
s(q)
=(W
1ω1,.
..,W
Kω
K,0
,...
,0,0
)Tw
ith
the
reaction
rate
sfo
rdeta
iled
kin
etics
ωi=
J ∑j=
1(ν
r ji−
νf ji) kf j
K ∏ l=1
ρl
Wl ν
f jl
−k
r j
K ∏ l=1
ρl
Wl ν
r jl .
Inhere
,all
resu
lts
were
obta
ined
with
am
echanism
for
H2−
O2−
Ar
com
bust
ion
with
34
ele
menta
ryre
actions
and
9sp
ecie
s.
Num
ericalm
eth
ods
Num
ericalso
urc
ete
rmin
corp
ora
tion
and
exte
nsion
tom
ultip
ledim
ensionsare
done
with
the
meth
od
of
fractionalst
eps.
−→Num
ericaldecoupling
of
hydro
dynam
ic
and
chem
icaltim
est
eps.
Tim
e-e
xplicit
2nd
ord
erT
VD
shock-c
aptu
ring
meth
od
forth
erm
ally
perfectgase
s:
Inm
ost
cells
an
appro
xim
ate
Rie
mann
solv
er
with
num
ericalflux
function
F(Q
l,Q
r)=
1 2
f(Q
l)+
f(Q
r)−
3 ∑ ι=1
|sι|W
ι w
ith
s 1=
u1−
c,s 2
=u1,
s 3=
u1+
cis
use
d.
The
waves
Wιare
defined
as
W1
=a1r 1
,W
2=
K+
2 ∑m
=2
amr m
,W
3=
aK
+3r K
+3,
where
a1=
∆p−
ρc∆
u1
2c2
,a
i+1=
∆ρi−
Yi∆
p
c2,
aK
+2=
ρ∆
un
,a
K+
3=
∆p+
ρc∆
u1
2c2
,
and
r 1=
(Y1,.
..,Y
K,u
1−
c,u2,H
−u1)T
,
r i+
1=
(δ1i,
...,
δ Ki,
u1,u
2,u
2 1+
u2 2−
φi/
(γ−
1))
T,
r K+
2=
(0,.
..,0
,1,u
2)T
,
r K+
3=
(Y1,.
..,Y
K,u
1+
c,u2,H
+u1)T
.
with
the
usu
alRoe
avera
ge
giv
en
by
v:=
√ρlv
l+√
ρrvr
√ρl+√
ρr
for
un,Y
i,T
,hi,
H:=
E+
p/ρ
and
ρ:=
√ρlρ
rand
the
specifi
c
avera
ges
γ:=
c p c vw
ith
c {p/v}=
K ∑ i=1
Yic{p
/v}i
,c {
p/v}i
=1
Tr−
Tl
∫ T r Tl
c {p,v}i(τ
)dτ
and
φi:=
(γ−
1)
( u2 1+
u2 2
2−
hi) +
γR W
iT
,c:=
K ∑ i=1
Yiφ
i−
(γ−
1)(
u2 1+
u2 2−
H) 1
/2
.
The
entr
opy
corr
ection
reads
|sι|
=
{ |sι|
,|s
ι|≥
2η
,
|s2 ι|/(4
η)+
η,|s
ι|<
2η
and
isapplied
in1d
toι=
1,3
with
η=
1 2(|
u1,r−
u1,l|+|c
r−
c l|),w
here
in2d
and
3d
am
ultid
imensional
evalu
ation
of
ηfo
rall
field
sis
use
das
carb
uncle
fix.
Exam
ple
:
ηj,
k+
1 2=
max{ η
j+1 2,k
,ηj−
1 2,k
,ηj,
k+
1 2,η
j−1 2,k
+1,η
j+1 2,k
+1
}T
he
corr
ection
F? i=
Fρ·{ Y
l i,
Fρ≥
0,
Yr i
,F
ρ<
0
ensu
res
the
positivity
ofth
em
ass
fractions
Yi.
Ifin
tern
alenerg
yorpre
ssure
inth
ein
term
edia
test
ate
sQ
? l=
Ql+
W1
and
Q? r=
Qr−
W3
are
unphysical,
i.e.
ρ? l/
r≤
0or
e? l/r≤
0,th
eHLL
flux
F(Q
l,Q
r)=
f(Q
l),
0<
s 1,
s 3f(
Ql)−
s 1f(
QR)+
s 1s 3
(Qr−
Ql)
s 3−
s 1,
s 1≤
0≤
s 3,
f(Q
r)
,0
>s 3
,
with
the
wave
speed
est
imation
s 1=
min
(u1,l−
c l,u
1,r−
c r),
s 3=
max(u
1,l+
c l,u
1,r
+c r
)is
use
d.
Sourc
ete
rmin
tegra
tion:
•Sta
ndard
solv
erfo
rst
iffO
DE’s,e.g
.,se
mi-im
plicit
Rose
nbro
ck-W
annerm
eth
od
•Auto
matic
stepsize
adju
stm
ent
toallow
for
an
effi
cie
nt
treatm
ent
of
chem
ical
tim
esc
ale
ssm
aller
than
the
glo
baltim
e-s
tep
Blo
ckstr
uctu
red
AM
R
Locally
hig
hre
solu
tion,w
hic
his
ess
entialfo
rth
eaccura
tecom
puta
tion
of
deto
-
nation
waves,
isachie
ved
by
blo
ckst
ructu
red
Adaptive
Mesh
Refinem
ent
(AM
R).
•D
iscre
tization
necess
ary
only
for
single
recta
ngula
rgrid
•Spatial
and
tem
pora
lre
finem
ent,
no
glo
baltim
est
ep
rest
riction
•Blo
ckst
ructu
red
data
guara
nte
es
hig
h
com
puta
tionalperform
ance
•Non-c
onfo
rmal
finite
volu
mes
unavoid
-
able
and
require
specia
ltr
eatm
ent
Ourfram
ework
AM
RO
Cpro
vid
es
ageneric
obje
ct-
oriente
dim
ple
menta
tion
ofth
e
blo
ckst
ructu
red
AM
Rm
eth
od
thatis
applicable
toany
explicit
FV
schem
efo
r(1
):
•Para
llelhie
rarc
hic
aldata
stru
ctu
res
em
plo
yM
PIlibra
ry
•D
ata
follow
s“floor
pla
n”
ofa
single
Grid
Hie
rarc
hy.
•D
ata
ofall
levels
resides
on
sam
enode→
most
AM
Ropera
tions
are
local
•Neig
hboring
grids
are
synchro
nized
transp
are
ntly
even
over
pro
cess
or
bord
ers
when
boundary
conditio
ns
are
applied
•D
istr
ibution
alg
orith
m:
Genera
lization
ofHilbert
’ssp
ace-fi
llin
gcurv
e
Refinem
ent
indicato
rs
Physically
motivate
dre
finem
ent
isachie
ved
thro
ugh
scale
dgra
die
nts
|w(Q
j+1,k)−
w(Q
jk)|
>ε w
,|w
(Qj,
k+
1)−
w(Q
jk)|
>ε w
,|w
(Qj+
1,k
+1)−
w(Q
jk)|
>ε w
orest
imating
the
leadin
g-o
rderte
rmofth
elo
calerr
orofquantity
wby
Ric
hard
son
extr
apola
tion
τw jk
:=|w
(Qjk
(t+
∆t)
)−
w(Q
jk(t
+∆
t))|
2o+
1−
2.
Inpra
ctice,we
use
the
crite
rion
τw jk
max(|
w(Q
jk(t
+∆
t))|
,Sw)
>η
r w
that
com
bin
es
rela
tive
and
abso
lute
err
or.
Verification:
Deto
nation
ignitio
nin
ashock
tube
Shock-induced
deto
nation
ignitio
nof
ahydro
gen-
oxygen-a
rgon
mix
ture
at
mola
rra
tios
2:1
:7in
a1d
shock
tube
clo
sed
at
the
left
end.
•In
suffi
cie
ntre
solu
tion
leads
toin
accura
tere
sults
•Reflecte
dsh
ock
iscaptu
red
by
the
FV
schem
e
corr
ectly
at
all
reso
lutions,
but
the
deto
nation
appro
xim
ation
isre
solu
tion
dependentas
can
be
seen
intim
et m
,w
hen
both
waves
merg
e
•Fin
em
esh
necess
ary
inth
ein
duction
zone
at
the
head
of
the
deto
nation
and
AM
Ris
hig
hly
suitable
tore
duce
the
com
pute
tim
e
10
00
15
00
20
00
25
00
30
00
6.4
6.3
6 6
.32
6.2
8 0 0
.002
0.0
04
0.0
06
0.0
08
0.0
1
0.0
12
0.0
14
Temperature [K]
Mass fraction [-]
x1 [cm
]
TY
HY
OY
H2
10
00
15
00
20
00
25
00
30
00
6.0
2 5
.98
5.9
4 5
.9 0 0
.002
0.0
04
0.0
06
0.0
08
0.0
1
0.0
12
0.0
14
Temperature [K]
Mass fraction [-]
x1 [cm
]
TY
HY
OY
H2
Appro
xim
ation
of
the
deto
nation
wave
at
t=
170
µsfo
rth
euniform
reso
lutions
∆x1
=100
µm
and
∆x1
=6.2
5µm
(∼24Pts
/l i
g).
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
4.5
5 5
.5 6
6.5
7
Pressure [MPa]
x1 [cm
]
∆x1=
6.2
5 µ
m∆x
1=
100 µ
m∆x
1=
200 µ
m
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
4 4
.5 5
5.5
6 6
.5 7
7.5
8
Pressure [MPa]
x1 [cm
]
Pre
ssure
Level
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
6.0
5 6
.03
6.0
1 5
.99
5.9
7 5
.95
0 0.0
02
0.0
04
0.0
06
0.0
08
0.0
1
0.0
12
0.0
14
Pressure [MPa]
YO [-]
x1 [
cm
]
Un
ifo
rmA
da
ptive
Left
:Com
pariso
nofpre
ssure
distr
ibution
at
t=
170
µs
fordiff
ere
ntm
esh
sizes.
Mid
dle
:D
om
ain
sofre
finem
entle
vels.
Rig
ht:
Direct
com
pariso
nofth
edistr
ibutions
of
pand
YO
inth
edeto
nation
wave
with
auniform
sim
ula
tion.
Uniform
Adaptive
∆x1[µ
m]
Cells
t m[µ
s]T
ime
[s]
l max
r lt m
[µs]
Tim
e[s]
400
300
166.1
31
200
600
172.6
90
22
172.6
99
100
1200
175.5
277
32,2
175.8
167
50
2400
176.9
858
42,2
,2177.3
287
25
4800
177.8
2713
42,2
,4177.9
393
12.5
9600
178.3
9472
52,2
,2,4
178.3
696
6.2
519200
178.6
35712
52,2
,4,4
178.6
1370
Com
pariso
nofuniform
lyre
fined
and
dynam
icadaptive
sim
ula
tionsru
non
asingle
CPU
ofa
Inte
lXeon
3.4
GHz
pro
cess
or.
Yi
SY
i·1
0−4
ηr Y
i·1
0−3
O2
10.0
2.0
H2O
7.8
8.0
H0.1
65.0
O1.0
5.0
OH
1.8
5.0
H2
1.3
2.0
ε ρ=
0.0
7kgm−3,
ε p=
50kPa
Refinem
entin
dic
ato
rvalu
esfo
rth
eone-
dim
ensionalig
nitio
npro
ble
m.
Em
beddin
gofcom
ple
xdom
ain
s
Usa
ge
ofa
signed
dista
nce
function
ϕfo
reffi
cie
ntim
plicit
boundary
repre
senta
tion
on
the
hie
rarc
hic
alCart
esian
mesh
.
•ϕ
=0
mark
sth
eboundary
,boundary
norm
al
n=∇
ϕ/|∇
ϕ|
•O
urcurr
ent
imple
menta
tion
evalu
ate
sϕ
inth
em
id-
poin
tsonly
and
use
sm
esh
cells
directly
topre
scribe
boundary
conditio
ns−→
pro
per
AM
Rre
finem
ent
of
the
boundary
necess
ary
•Non-o
scilla
ting,
1st
ord
er
accura
tein
terp
ola
-
tion/extr
apola
tion
toconst
ruct
ghost
cell
valu
es
•M
irro
ring
ofth
eprim
itiv
evalu
es
ρi,
u,
pand
invers
ion
ofnorm
alvelo
city
com
ponent
by
u′ =
(2w·n−
u·n
)n+
(u·t
)t=
2((
w−
u)·n
)n+
u
Inte
rpola
tion
from
inte
riorcells
to
const
ructm
irro
red
valu
esin
ghost
cells
(gra
y).
Verification:
Shock-induced
com
bustion
aro
und
asphere
Asp
herical
pro
jectile
of
radiu
s1.5
mm
travels
with
const
ant
velo
city
vI
=
2170.6
m/s
thro
ugh
ahydro
gen-o
xygen-a
rgon
mix
ture
(mola
rra
tios
2:1
:7)
at
6.6
7kPa
and
T=
298K
and
exhib
its
shock-induced
com
bust
ion
ignitio
n.
•Cylindricalsy
mm
etr
icsim
ula
tion
on
AM
Rbase
mesh
of70×
40
cells
•Com
pariso
nof3-levelcom
puta
tion
with
refinem
entfa
cto
rs2,2
(∼5Pts
/l i
g)and
a4-levelcom
puta
tion
with
refinem
ent
facto
rs2,2
,4(∼
19Pts
/l i
g)
at
t=
350
µs
•Hig
her
reso
lved
com
puta
tion
captu
res
com
bust
ion
zone
visib
lybett
er
and
at
slig
htly
diff
ere
nt
position
(see
belo
w)
Iso-c
onto
urs
of
p(b
lack)
and
YH
2(w
hite)
on
dom
ain
sofdiff
ere
nt
refinem
ent
levels
(gra
y)
at
t=
350
µs
for
a3-level
com
puta
tion
with
r 1,2
=2
(left
)and
a4-levelcom
puta
tion
with
r 1,2
=2,r
3=
4(r
ight)
.
Left
:Active
refinem
ent
indic
ato
rson
l=
2at
t=
350
µs.
Blu
e:
ε ρ,light
blu
e:
ε p,gre
en
shades:
ηr Y
i,re
d:
em
bedded
boundary
.Rig
ht:
Dom
ain
decom
position
to8
pro
cess
ors
for
the
4-levelhie
rarc
hy
depic
ted
inth
egra
phic
above.
4Pro
cs
8Pro
cs
16
Pro
cs
Task
Tim
e[s
]%
Tim
e[s]
%T
ime
[s]
%Flu
iddynam
ics
22374
34.6
11261
32.3
5742
26.7
Chem
icalkin
etics
25280
39.1
12455
35.8
6291
29.2
Boundary
sett
ing
7758
12.0
5888
16.9
5761
26.8
Em
bedded
boundary
5429
8.4
2839
8.2
1703
7.9
Recom
position
3210
5.0
1919
5.5
1623
7.5
Misc
638
0.9
456
1.3
416
1.9
Tota
l/
Speed-u
p64689
34818
1.8
621536
1.6
2
Bre
akdow
nof
com
puta
tional
cost
sfo
rpara
llel
4-level
sim
ula
tions
with
r 1,2
,3=
2ru
nnin
gfrom
t=
150
µs
tot e
=400
µs.
Yi
SY
i·1
0−4
ηr Y
i·1
0−4
O2
10.0
4.0
H2O
5.8
3.0
H0.2
10.0
O1.4
10.0
OH
2.3
10.0
H2
1.3
4.0
ε ρ=
0.0
2kgm−3,
ε p=
16kPa
Refinem
entin
dic
ato
rvalu
esfo
rsim
ula
t-
ing
shock-induced
com
bust
ion
aro
und
the
pro
jectile
.
Cellulardeto
nation
and
trip
lepoin
tstr
uctu
res
Self-s
ust
ain
ing
deto
nations
are
inhere
ntly
unst
able
and
exhib
ittr
ansv
ers
epre
s-
sure
waves
that
pro
pagate
perp
endic
ula
rto
the
deto
nation
front.
The
transv
ers
e
Schliere
nand
PLIF
images
of
are
gula
rly
osc
illa
ting
deto
nation
front
inhydro
gen-o
xygen-a
rgon
(im
age
court
esy
J.
E.
Shepherd
,G
raduate
Aero
nauticalLabora
tory
,Califo
rnia
Inst
itute
ofTechnolo
gy).
waves
form
evolv
ing
trip
le
poin
tpatt
ern
sat
the
det-
onation
front
with
severe
ly
enhanced
pre
ssure
,te
mper-
atu
reand
there
by
reaction.
Desp
ite
considera
ble
exper-
imenta
leffort
s,a
genera
l
modelfo
rth
ein
stability
and
the
deta
iled
flow
field
in
trip
lepoin
tsin
deto
nationsis
not
available
yet.
Obje
ctive
of
this
work
isto
decip
her
the
de-
tailed
hydro
dynam
icflow
conditio
ns
intr
iple
poin
tsu
b-s
tructu
res
under
periodic
and
tran-
sient
conditio
ns.
Inexperim
ents
,basically
two
diff
ere
nt
configura
tions
have
been
qualita
tively
distinguished.
Experim
enta
lly
know
ntr
iple
poin
tst
ructu
res.
Regulardeto
nation
str
uctu
res
in2d
•M
ixtu
re:
H2
:O
2:Ar
at
mola
rra
tios
2:1
:7at
initia
lly
298K
and
10kPA
•In
itia
lization
with
Chapm
an-J
ouguet
ZND
solu
tion
and
an
irre
gula
rpocket
to
quic
kly
trig
ger
sym
metr
ybre
akin
g.
Tube
wid
th3.2
cm
•Sim
ula
tion
untila
single
pert
urb
ation
hasdevelo
ped
into
two
regula
rly
osc
illa
ting
cells
with
wid
thλ
=1.6
cm
•67.6
Pts
/l i
g.
4additio
nalre
finem
ent
levels
(2,2
,2,4
)on
base
mesh
2000×
128
•Com
puta
tion
oftr
iple
poin
ttr
aje
cto
ries
by
trackin
gth
em
agnitude
ofth
evor-
ticity
vecto
ron
auniform
mesh
at
level1
Left
:O
scilla
ting
deto
nation
front
on
com
pute
dtr
iple
poin
tstr
acks.
Rig
ht:
Schliere
nplo
ton
refinem
ent
levels.
Left
:Fro
nt
on
trip
lepoin
ttr
acks.
10
µs
betw
een
snapsh
ots
(mark
ed
with
open
square
sin
(Ms,Φ
1)-
pla
ne,cf.
right
colu
mn).
Mid
dle
and
right:
Cle
arly
est
ablish
ed
Double
-Mach
reflection
patt
ern
short
lybefo
retr
iple
poin
tcollisio
n.
Only
the
stro
ng
stru
ctu
reappears
in
the
sim
ula
tions
and
can
cle
arly
be
identi-
fied
asa
double
-Mach
reflection
patt
ern
.
p/p0
ρ/ρ0
T[K
]q[m
/s]
αγ
MA
1.0
01.0
0298
1775
-39.5
1.5
57
5.0
78
B31.4
54.1
72248
447
-64.4
1.4
87
0.4
77
C31.6
95.3
21775
965
-64.4
1.5
00
1.1
53
D19.1
73.8
41487
1178
-73.2
1.5
09
1.5
33
E35.6
15.7
21856
901
-67.9
1.4
97
1.0
53
F40.6
16.0
91987
777
-70.0
1.4
94
0.8
80
10
20
30
40
50 0
5 1
0 1
5 2
0 2
5 3
0 3
5 4
0
p/p0
θ
Incid
ent
Refle
cted
Valu
essh
ort
lybefo
retr
iple
poin
tcollisio
nin
fram
eofre
fere
nce
ofprim
ary
trip
lepoin
t.
Inflow
(A),
Mach
stem
(B),
reflecte
dwave
(C),
incid
ent
shock
(D),
2nd
reflecte
d
(E),
2nd
Mach
stem
(F).
(q,
α)
are
the
pola
rcoord
inate
sof
the
velo
city
vecto
rin
the
Galile
an
fram
eofre
fere
nce.
Rig
ht:
shock
pola
rsfo
rfirs
ttr
iple
poin
tin
left
table
.
Cellularstr
uctu
res
in3d
•H
2:O
2:Ar/
2:1
:7
at
6.6
7kPA
with
cell
wid
thλ
=3cm
in3d
and
2d
(not
show
n)
•Hig
h-r
eso
lution
sim
ula
tion
ofth
equart
er
ofa
recta
ngula
rdeto
nation
cell
•Sim
ula
tion
under
Galile
an
transf
orm
ation−→
inflow
of
unre
acte
dgas
with
CJ
velo
city
•44.8
Pts
/l i
g.
2additio
nalre
finem
ent
levels
(2,4
)on
base
mesh
400×
24×
24,
adaptive
Sim
ula
tion
use
s∼
18·1
06
cells
inst
ead
of118·1
06
(uniform
)
•∼
55,0
00h
CPU
on
128
Com
paq
Alp
ha
pro
cess
ors
Regula
rly
osc
illa
ting
deto
nationsth
atexhib
ited
ast
rong
stru
ctu
rein
2d,sh
ow
only
the
weak
stru
ctu
rein
3d
as
the
transv
ers
ewave
isweakeras
in2d.
The
resu
ltin
g
stru
ctu
rein
3d
can
be
identified
as
atr
ansitionalM
ach
reflection.
Left
,m
iddle
:Schliere
nplo
tsof
ρ(left
)and
YO
Hin
the
firs
thalf
ofa
deto
nation
cell.
Data
displa
yed
isfo
r5.0
cm
<
x1
<7.0
cm
and
mirro
red
at
x2
=0
and
x3
=0.
The
blu
eof
ρvisualizes
the
induction
length
.Rig
ht:
Pla
nes
with
schliere
nplo
tsof
ρth
rough
the
origin
norm
alto
x2
and
x3-a
xis
unfo
lded
exhib
itth
etr
ansitionalM
ach
stru
ctu
rebehin
d
the
trip
lepoin
ts,w
hic
his
chara
cte
rized
by
akin
kin
stead
ofse
condary
trip
lepoin
ton
the
transv
ers
ewave.
Deto
nation
str
uctu
rein
sm
ooth
pip
ebends
•In
itia
lization
with
5re
gula
rly
osc
illa
ting
deto
nation
cells
inH
2:O
2:Ar/
2:1
:7
at
initia
lly
298K
and
10kPa,tu
be
with
8cm
•Pip
ebend
with
radiu
s8cm
.Angle
θ={1
5o,3
0o,4
5o,6
0o}
•67.6
Pts
/l i
g.
4additio
nalre
finem
ent
levels
with
facto
rs(2
,2,2
,4)
•Adaptive
com
puta
tions
use
∼7·1
06
cells
(∼5·1
06
on
hig
hest
level)
inst
ead
of
∼1.2·1
09
cells
(uniform
grid)
•∼
70,0
00h
CPU
each
on
128
CPUs
Pentium
-42.2
GHz
Colo
rplo
tof
Tand
schliere
nplo
tof
ρon
trip
lepoin
ttr
acks
and
on
refinem
ent
regio
ns
(mid
dle
,right)
for
θ=
45o
aft
er
t=
150
µs
sim
ula
ted
tim
e.
Triple
poin
ttr
acks
for
θ=
15o
(left
,to
p),
θ=
30o
(left
,bott
om
),and
θ=
60o
(rig
ht)
displa
yth
efu
ndam
enta
lflow
featu
res.
Top:
Fro
nt
on
trip
lepoin
ttr
acks
for
θ=
15
and
110
µs,
140
µs,
170
µs,
200
µs
sim
ula
ted
tim
e.
Rig
ht,
top:
Str
ength
enin
gof
double
-Mach
reflection
patt
ern
(solid
square
sin
(Ms,Φ
1)-
pla
ne)
underexpansion
at
130
µs
and
140
µs.
Rig
ht,
bott
om
:W
eakenin
gtr
ansitionalM
ach
reflection
inover-
driven
regio
n(o
pen
circle
sin
(Ms,Φ
1)-
pla
ne)
at
130
µs
and
140
µs.
Triple
poin
tre
-initia
tion
with
change
from
transitionalto
double
-Mach
reflection
(solid
circle
sin
(Ms,Φ
1)-
pla
ne)
when
the
osc
illa
tion
becom
es
regula
ragain
aft
er
the
bend,
θ=
15
at
200
µs,
210
µs,
and
220
µs.
35
40
45
50
55
60
65
70 1
2 3
4 5
6 7
8
Φ1
Ms
NR
SMR
TMR/
DMR
RR
Dom
ain
sand
transition
boundaries
of
diff
ere
nt
shock
wave
reflection
phenom
ena.
Fra
me
ofre
fere
nce
offirs
t
trip
lepoin
t.Φ
1is
angle
betw
een
incid
ent
shock
and
vecto
r(q
,α),
Ms=
sin(Φ
1)M
.NR:no
reflection,SM
R:
single
Mach
reflection,T
MR/D
MR:tr
ansitional/
double
-
Mach
reflection,RR:re
gula
rre
flection.
Under
transient
boundary
conditio
ns,
the
regula
rly
osc
illa
ting
deto
nation
exhib
its
both
weak
and
stro
ng
trip
lepoin
tst
ructu
res.
Deta
iled
quantita
tive
trip
lepoin
t
analy
sis
for
the
therm
ally
perfect,
but
non-r
eactive,H
2−
O2−
Ar
mix
ture
(using
the
oblique
shock
rela
tions
totr
ansf
orm
the
Eule
rian
data
into
the
fram
ework
refe
rence
ofth
eprim
ary
trip
lepoin
t)in
dic
ate
sst
rongly
:
•Triple
poin
tst
ructu
resin
self-s
ust
ain
ed
deto
nationsexistonly
inth
etr
ansitional
and
double
-Mach,
but
not
inth
esingle
-Mach
reflection
regim
e(c
f.above
(Ms,Φ
1)-
pla
ne)
•T
he
rela
tive
stre
ngth
S=
pC−
pD
pD
of
the
transv
ers
epre
ssure
wave
prim
arily
de-
term
ines
the
reflection
type
•A
change
ofth
ere
flection
type
happens
esp
ecia
lly
intr
iple
poin
tcollisio
ns
Publications
R.D
..A
para
lleladaptive
meth
od
forsim
ula
ting
shock-induced
com
bust
ion
with
deta
iled
chem
icalkin
etics
incom
ple
x
dom
ain
s,Com
pute
rs&
Str
uctu
res,
subm
itte
d,2008.
R.D
.D
ynam
ically
adaptive
sim
ula
tion
ofre
gula
rdeto
nation
stru
ctu
resusing
the
Cart
esian
mesh
refinem
entfram
ework
AM
RO
C.In
t.Journ
alCom
puta
tionalScie
nce
and
Engin
eering,accepte
dand
inpre
ss,2008.
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