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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
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.2/20
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
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.3/20
Background
2-D shock geometry2-D Euler equations withreaction2 species chemicalkineticsCalorically perfect idealgas mixtureHigh-order convergence
!
"
Shock Wave
Perio
dic
B.C.
Perio
dic
B.C.
Neumann Condition
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.3/20
Background
2-D shock geometry2-D Euler equations withreaction2 species chemicalkineticsCalorically perfect idealgas mixtureHigh-order convergence
!"
!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.
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.3/20
Background
2-D shock geometry2-D Euler equations withreaction2 species chemicalkineticsCalorically perfect idealgas mixtureHigh-order convergence
A ! B
Let # denote mass fraction of B
!
!t("#) +
!
!x("u#) = a"(1 " #) exp
„
"E"
p
«
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.3/20
Background
2-D shock geometry2-D Euler equations withreaction2 species chemicalkineticsCalorically perfect idealgas mixtureHigh-order convergence
e =1
! ! 1
p
"! q#
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.3/20
Background
2-D shock geometry2-D Euler equations withreaction2 species chemicalkineticsCalorically perfect idealgas mixtureHigh-order convergence
!x
Erro
r
O(!x5)
O(!x)
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.3/20
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)
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.4/20
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
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.5/20
Motivation
Accuracy loss due to differentiation across discontinuities.High order convergence can be achieved throughshock-fitting
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
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.5/20
Motivation
Accuracy loss due to differentiation across discontinuities.High order convergence can be achieved throughshock-fitting
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-Fitted Numerical Solutions for Two-Dimensional Detonations – p.5/20
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-Fitted Numerical Solutions for Two-Dimensional Detonations – p.6/20
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-Fitted Numerical Solutions for Two-Dimensional Detonations – p.6/20
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-Fitted Numerical Solutions for Two-Dimensional Detonations – p.6/20
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)
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.6/20
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
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.7/20
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)
+
&'
'
(
(
(
(
(
(
(
)
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.8/20
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
«
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.9/20
Formulation: Shock Change Eqn.
At the shock, E = &(e + 12(u2 + v2)) = f(Dn).
$Dn
$#=
!
$E
$Dn
#
#
#
#
S
"!1 $E
$#
#
#
#
#
S
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.10/20
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 Numerical Solutions for Two-Dimensional Detonations – p.10/20
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 Numerical Solutions for Two-Dimensional Detonations – p.11/20
Shock-Fitted Geometry: x % "
*
Dn
Shocked HE
Quiescent HE
g(1)
g(2)g(1)
g(2)
!yS
!t
=$(2
)=
Do
!1=
y1=
cons
tant
S : !2 = y2 ! yS(x, t) = 0
g(1) =
0
@
1!yS
!"1
1
A , g(2) =
0
@
0
1
1
A ,
g(1) =
0
@
1
0
1
A , g(2) =
0
@
" !yS
!y1
1
1
A ,
and the grid velocity components are
!&
!t= 0
!'
!t= "
!yS
!t
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.12/20
Shock-Fitted Geometry: x % "
*
Dn
Shocked HE
Quiescent HE
g(1)
g(2)g(1)
g(2)
!yS
!t
=$(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.
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.12/20
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
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.13/20
1-D Results: Pulsating Detonations
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.14/20
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
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.15/20
2-D Results: Detonation Cells
"
t
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.16/20
2-D Results: Detonation Cells
"
t
"
!
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.16/20
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
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.17/20
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
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.18/20
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.
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.19/20
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
Shock-Fitted Numerical Solutions for Two-Dimensional Detonations – p.20/20