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Shock-Fitted Numerical Solutions forTwo-Dimensional Detonationswith Periodic Boundary Conditions

Andrew Henrick, Tariq D. Aslam, Joseph M. Powers11

th SIAM International Conference on Numerical Combustion

Granada, Spain

Los Alamos National LaboratoryUniversity of Notre Dame

April 23, 2006

Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.1/20

Presentation Outline

IntroductionMotivation & BackgroundGeneral Formulation

Shock-FittedTransformationNumerical Method

1-D Limiting CaseComparison withLinear Stability TheoryPulsating Detonation

2-D Results


Background

2-D shock geometry2-D Euler equations withreaction2 species chemicalkineticsCalorically perfect idealgas mixtureHigh-order convergence

Confiner

Unshocked HE

ConfinerShock Wave

Sonic Locus

Deflection of Confiner

Shocked HE


Background


!

"

Shock Wave

Perio

dic

B.C.

Perio

dic

B.C.

Neumann Condition


Background


!"

!t+

!

!x("u) +

!

!y("v) = 0,

!

!t("u) +

!

!x("u2 + p) +

!

!y("uv) = 0,

!

!t("v) +

!

!x("vu) +

!

!y("v2 + p) = 0,

!

!t

„

"

„

e +1

2(u2 + v2)

««

+

!

!x

„

"u

„

e +1

2(u2 + v2) +

p

"

««

+

!

!y

„

"v

„

e +1

2(u2 + v2) +

p

"

««

= 0.


Background


A ! B

Let # denote mass fraction of B

!

!t("#) +

!

!x("u#) = a"(1 " #) exp

„

"E"

p

«


Background


e =1

! ! 1

p

"! q#


Background


!x

Erro

r

O(!x5)

O(!x)


Motivation

What are the state-of-the-art numerical techniques used fordiscontinuous problems?

Shock capturingRobustNumerical viscosity reduces convergence to O(!x)

Shock trackingRobustDescription of discontinuous motion variesConverges at O(!x)


Motivation

Accuracy loss due to differentiation across discontinuities.High order convergence can be achieved throughshock-fitting

Governing equations are posed in fitted coordinatesSolution is smooth within each domainAnalytic jump conditions used to compute shock speedRestricted to embedded shocks


Motivation


Governing equations are posed in fitted coordinatesShock location is fixed

Solution is smooth within each domainAnalytic jump conditions used to compute shock speedRestricted to embedded shocks


Motivation


Governing equations are posed in fitted coordinatesShock location is fixed

Solution is smooth within each domainRegular finite differencing is adequate

Analytic jump conditions used to compute shock speedRestricted to embedded shocks


Shock-Fit Transformation

Consider transform

" = "(x, y, t), ! = !(x, y, t), # = t

applied to

$F

$t+

$f i

$yi = B Dn =

!f i

"

#F $%i

where only derivatives aretransformed.

! = 0

%i

F(1)

F(2)

f i(1)

f i(2)


Shock-Fit Transformation

Consider transform

" = "(x, y, t), ! = !(x, y, t), # = t

applied to

$F

$t+

$f i

$yi = B Dn =

!f i

"

#F $%i

where only derivatives aretransformed.

x and y momentum are stillsolved

! = 0

%i

F(1)

F(2)

f i(1)

f i(2)


Transformed Equations

The resulting fitted equations are

$

$#("

gF ) +$

$!j

!

"gF

$!j

$t+"

gf i$!j

$yi

"

="

gB

Conservation form with proper shock speed

Dn =

%"gF !"j

!t +"

gf i !"j

!yi

&

!"gF

" %j

"g =

#

#

#

#

#

#

!y!"

#

#

#

#

#

#is the determinant of the metric tensor

!!"i

!t = !"i

!yj!yj

!# # U i = !"i

!yj U j


Formulation: Conserved Quantities

!"

!t+

!

!x("u) +

!

!y("v) = 0,

!

!t("u) +

!

!x("u2 + p) +

!

!y("uv) = 0,

!

!t("v) +

!

!x("vu) +

!

!y("v2 + p) = 0,

!

!t

„

"

„

e +1

2(u2 + v2)

««

+

!

!x

„

"u

„

e +1

2(u2 + v2) +

p

"

««

+

!

!y

„

"v

„

e +1

2(u2 + v2) +

p

"

««

= 0,

!

!t("#) +

!

!x("u#) = a"(1 " #) exp

„

"E"

p

«

,

e =1

$ " 1

p

"" q#.

$

%

%

%

%

%

%

%

&

F1

F2

F3

F4

F5

'

(

(

(

(

(

(

(

)

=

$

%

%

%

%

%

%

%

&

&

&u

&v

&*

e + 12(u2 + v2)

+

&'

'

(

(

(

(

(

(

(

)


Formulation: Shock-Fitted Eqns.

!

!%

2

6

6

6

6

6

6

6

4

F !1

F !2

F !3

F !4

F !5

3

7

7

7

7

7

7

7

5

+!

!&

0

B

B

B

B

B

B

B

B

B

@

!&

!t

2

6

6

6

6

6

6

6

4

F !1

F !2

F !3

F !4

F !5

3

7

7

7

7

7

7

7

5

+1#

g

!y

!'

2

6

6

6

6

6

6

6

6

6

4

F !2

F !22

F !1

+#

gp

F !2F !3

F !1

F !2F !4

F !1

+#

gF !2

F !1

p

F !2F !5

F !1

3

7

7

7

7

7

7

7

7

7

5

"1#

g

!x

!'

2

6

6

6

6

6

6

6

6

6

4

F !3

F !2F !3

F !1

F !23

F !1

+#

gp

F !3F !4

F !1

+#

gF !3

F !1

p

F !3F !5

F !1

3

7

7

7

7

7

7

7

7

7

5

1

C

C

C

C

C

C

C

C

C

A

+!

!'

0

B

B

B

B

B

B

B

B

B

@

!'

!t

2

6

6

6

6

6

6

6

4

F !1

F !2

F !3

F !4

F !5

3

7

7

7

7

7

7

7

5

"1#

g

!y

!&

2

6

6

6

6

6

6

6

6

6

4

F !2

F !22

F !1

+#

gp

F !2F !3

F !1

F !2F !4

F !1

+#

gF !2

F !1

p

F !2F !5

F !1

3

7

7

7

7

7

7

7

7

7

5

+1#

g

!x

!&

2

6

6

6

6

6

6

6

6

6

4

F !3

F !2F !3

F !1

F !23

F !1

+#

gp

F !3F !4

F !1

+#

gF !3

F !1

p

F !3F !5

F !1

3

7

7

7

7

7

7

7

7

7

5

1

C

C

C

C

C

C

C

C

C

A

=

2

6

6

6

6

6

6

6

4

0

0

0

0

(

3

7

7

7

7

7

7

7

5

p =($ " 1)#

g

„

F !4 "

(F !2)

2 + (F !3)

2

2F !1

+ qF !5

«

, ( =a#

g(F !

1 " F !5) exp

„

"EF !1

#gp

«


Formulation: Shock Change Eqn.

At the shock, E = &(e + 12(u2 + v2)) = f(Dn).

$Dn

$#=

!

$E

$Dn

#

#

#

#

S

"!1 $E

$#

#

#

#

#

S


Formulation: Shock Change Eqn.

At the shock, E = &(e + 12(u2 + v2)) = f(Dn).

|v| = vivj = u2 + v2 is invariant.

$Dn

$#=

!

$E

$Dn

#

#

#

#

S

"!1 $E

$#

#

#

#

#

S

!E!#

#

#

Sis already calculated in the flow field.

Thus system is closed.


Shock-Fitted Geometry

(g(2)

g(1)

g(1)

g(2)!

Dn

)(1)

)(2)S

contravariant basiscovariant basis

g(i) lie along fitted coords.

g(i) are reciprocal basis! = U|S is the shockvelocity

Since S : !2 = 0

g(1) is embedded in theshockg(2)||"

Note that !1 in general con-tributes artificial tangentialshock velocity.

Thus, Dn = cos(())(2) $ |!|,in general.


Shock-Fitted Geometry: x % "

*

Dn

Shocked HE

Quiescent HE

g(1)

g(2)g(1)

g(2)

!yS

!t

=$(2

)=

Do

!1=

y1=

cons

tant

S : !2 = y2 ! yS(x, t) = 0

g(1) =

0

@

1!yS

!"1

1

A , g(2) =

0

@

0

1

1

A ,

g(1) =

0

@

1

0

1

A , g(2) =

0

@

" !yS

!y1

1

1

A ,

and the grid velocity components are

!&

!t= 0

!'

!t= "

!yS

!t


Shock-Fitted Geometry: x % "

*

Dn

Shocked HE

Quiescent HE

g(1)

g(2)g(1)

g(2)

!yS

!t

=$(2

)=

Do

!1=

y1=

cons

tant

S : !2 = y2 ! yS(x, t) = 0

In this case"

g = 1

( = *, the shock angle

Do =!y

S

!t = Dn

,

1 +-

!yS

!y1

.2

relating the phase speed, theshock surface, the normalshock speed, and the shockslope.


Numerical Method

The method of lines is used to separate spatial and temporalintegration

O(!t5) Runge-Kutta integration in timeO(!x5) or O(!y5) WENO5M with Lax-Friedrich’s fluxsplitting to differentiate in spaceAt the shock

Rankine-Hugoniot jump conditions used directlyOne-sided finite differencing usedShock-change equation gives evolution of Dn


1-D Results: Pulsating Detonations


2-D Results: E = 0, q = 15

4.3125

4.313

4.3135

4.314

4.3145

4.315

4.3155

4.316

0 10 20 30 40 50 60 70

Dn

t" = tk

!x = 0.05f(x)

f(x) = D + a1ea2x sin(a3x + a4)

N1/2 !x Range Growth Rate (a2) Frequency (a3)

5 0.2 [20:40] 0.024362 1.1053010 0.1 [30:45] 0.024348 1.1048720 0.05 [40:60] 0.024210 1.10437


2-D Results: Detonation Cells

"

t


2-D Results: Detonation Cells

"

t

"

!


2-D Results: Linear Stability

2.8

2.802

2.804

2.806

2.808

2.81

2.812

2.814

2.816

2.818

2.82

0 10 20 30 40 50 60 70 80 90 100

t

Dn

!x = 0.1


Conclusions

Need for highly resolved solutions to sandwich testShock-fitting is a viable high order solution technique forproblems involving a single embedded shockConservative 2-D shock-fitted equations derived andimplementedWENO5M combined with LLF splitting

allows for high order convergencegives discrete conservation away from the shockcorrectly captures shocks (degrade to & O(!x))

Areas of current researchValidation from linear stability theory in progressExtension to more complicated geometries


WENO5M

Weighted Essentially Non-Oscillatory (WENO) Schemes

Stencil 0

Stencil 1

Stencil 2

!xj j + 1

fj+1/2 Fifth order scheme overallfk = h + O(!x3)

fj±1/2 =/2

k=0 )(M)k fk

j±1/2

Schemes differ throughformulation of )

(M)k

Ideal weights:

)0 = 1/10, )1 = 6/10, )2 = 3/10.


WENO5M

WENO5 Modified (Mapped)

)(M)k =

*"k

P2i=0 *"

i

where *"k = gk()

(JS)k )

gk()) =)()k + )2

k " 3)k) + )2)

)2k + (1 " 2)k))

, ) $ [0, 1]

where )(JS)k are those used by Jiang and

Shu (1996)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k = 0

k = 2

k = 1

!

"* k

Identity mapping


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