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.