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Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 1
Lecture 7
Detonation Waves
Moshe Matalon
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CJ
CJ
1/⇢0
p0
p1
v1 = 1/⇢1
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 2
Deflagrations
Thermal di↵usivity Dth ⇠ 10�5 m2/s
Chemical time tc ⇠ 10�3 � 10�5 s
Speed SL ⇠pDth/tc SL ⇡ 10�1�1 m/s
Wave thickness lf ⇠pDthtc lf ⇡ 10�1cm
p1/p0 ⇠ 1, T1/T0 ⇠ 6
SL
lf
cold
fresh mixture
hot
products
propagation by di↵usion
produce heat
Moshe Matalon
Vwave = SL
in the laboratory frame
u0 =0p1 , ⇢1
T1 , Yi1
p0 , ⇢0
T0 , Yi0
cold, fresh gas
SLu1=�(��1)SL
in the frame attached to the wave
p1 , ⇢1
T1 , Yi1
p0 , ⇢0
T0 , Yi0
cold, fresh gas
u1 = ��SL u0 = �SL
fluid particle velocity u|wave = u|lab � Vwave
lf
burned gas
burned gas
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 3
Detonations
fresh mixture
propagation by shock compression
rapid, violent, spectacular
lR
detonated
products
D
shock pressures up to 500,000 atm
temperatures up to 5,500K
power density ⇠ 20·109 Watt/cm
2
shock followed by a fast flame
Moshe Matalon
Vwave = D
in the laboratory frame
u0 =0p1 , ⇢1
T1 , Yi1
p0 , ⇢0
T0 , Yi0
cold, fresh gas
Du
fluid particle velocity u|wave = u|lab � Vwave
in the frame attached to the wave
p1 , ⇢1
T1 , Yi1
p0 , ⇢0
T0 , Yi0
cold, fresh gas
u0 = �D�(D � u)
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 4
⇢Dh
Dt=
Dp
Dt
D⇢
Dt+ ⇢r · u = 0
⇢Du
Dt= �rp
Di↵usion e↵ects, all important in flames, are negligible in detonations becauseof their extremely high propagation velocity. Here u is the gas particle velocity,and v = 1/⇢ will be used to denote the specific volume (volume per unit mass).
Governing Equations - Euler equations
These equations are supplemented with an equation of state, which for ideal
gases is p = ⇢ ¯RT , and a caloric equation of state h =
PN
i=1 Yi
ho
i
+
RT
T
o
cp
dT
where cp
=
PN
i=1 Yi
cp
i
is the mixture specific heat. Assuming equal specific
heats, with cp
independent of temperature, the enthalpy of the mixture is
h =
NX
i=1
Yi
ho
i
+ cp
(T � T o
)
⇢DYi
Dt= !i
Moshe Matalon
The chemical reaction is assumed to be a one-step irreversible reaction R ! P,
and the reaction rate is described in terms of a single progress variable � (for
example, the mass fraction of the products P) such that � = 0 corresponds to
the unreacted material and � = 1 to completed reaction.
D�
Dt= ! ! = k(1� �)⌫e�E/RT
The reaction rate is assumed of the form
Caloric equation of state
pv =� � 1
�cpTEquation of state
c =p�pv
Speed of sound
h =�
� � 1pv � �Q
h = YR
ho
R
+ YP
ho
P
+ cp
(T � T o
) = (1� �)ho
R
+ �ho
P
+
� ¯R(� � 1)
T + const.
= ��(ho
R
� ho
P
) +
� ¯R� � 1
T + const. = ��Q+
�
� � 1
pv + const.
where use has been made of cp =
���1
¯R
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 5
wave-attached frame
For a steadily propagating detonation the shock velocity is constant and the
flow in the reaction zone steady in a frame attached to the shock (the following
rarefaction is necessarily unsteady).
State within the reaction zone
State ahead of the shock
p0, v0(=⇢�10 ), u0, �0(= 0)
p, v(=⇢�1), u, �
laboratory frame
u0 =0
D
u
following flow
The following flow
is determined by
the rear boundary
condition, and usually
a rarefaction.
u0 =0
cold, fresh gas
Du
cold, fresh gas
u0 = �D�(D � u)
u0 =�D
�(D � u)
Moshe Matalon
Steady, one dimensional conservation laws
)
d
dx
(⇢u) = 0
⇢u
du
dx
= �dp
dx
⇢u
dh
dx
= u
dp
dx
d
dx
(⇢u) = 0
d
dx
�p+ ⇢u
2�= 0
d
dx
�h+ 1
2u2�= 0
in a frame attached to the wave
⇢0D = ⇢(D � u)
p0 + ⇢0D2 = p+ ⇢(D � u)2
h0 +12D
2 = h+ 12 (D � u)2
Rayleigh line and Hugoniot
) ✓p
p0+
� � 1
� + 1
◆✓v
v0� � � 1
� + 1
◆= 1�
✓� � 1
� + 1
◆2
+ 2� � 1
� + 1
�Q
p0v0
p� p0 = �m2(v � v0), m = ⇢0D
these relations connect the state at any point within the reaction zone to the state ahead of the shock
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 6
p0
v
p
v0
S
� = 0
� = 1
CJ
W
The Hugoniot is parametrized by �,with � = 0 corresponding to an inert shockand � = 1 to a completely reacted state.
Moshe Matalon
Zel’dovich, von Neueman, Doring
The ZND structure
Ahead of the wave, the gas is quiescent and there is insignificant reaction. Pas-
sage through the lead shock the gas is compressed, the pressure increases tremen-
dously and its temperature rises thousands of degrees. The ensuing chemical
reaction goes to completion very rapidly in a relatively thin reaction zone (or
fire) behind the shock.
shock followed by a fast flame
p
⇢
T
induction lead shock
D
fire
reaction zone
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 7
The two extreme Hugoniot curves correspond to � = 0 and � = 1.For a given shock velocity D, all states within the reaction zone must lie on thecorresponding Rayleigh line
p� p0 = �⇢20D2(v � v0)
Since the Hugoniot reaction must also be satisfied, the portion of the Rayleighline relevant to the ZND structure is that bounded by the two extreme Hugoniotcurves (i.e., the solid portion NS).
The lowest possible Rayleigh line is the one tangent to the complete-reactionHugoniot (i.e., corresponding to � = 1). The final state in this case is theChapman-Jouguet (CJ) state. The corresponding detonation speed, DCJ, is theminimum speed consistent with the conservation laws. The flow at the pointCJ is sonic.
Starting with an initial state (p0, v0), the state of a gas particle jumps to thepoint N along the shock-Hugoniot (i.e., corresponding to �= 0) upon passagethrough the lead shock. As the particle reacts, � increases and the state of theparticle slides down along the Rayleigh line towards the end point S crossingHugoniot curves of increasing � (i.e., corresponding to partial reaction Hugoniotcurves). At the end of the reaction zone, the particle reaches the �=1 Hugoniotat the final state S. The flow at the point S is subsonic.
Moshe Matalon
p0
v
p
v0
S
� = �1
� = 0
� = 1
N
CJ
W
N - von Neumann point
S - strong detonation
W - weak detonation
CJ - Chapman-Jouguet detonation
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 8
p0
v
p
v0
S
� = �1
� = 0
� = 1
N
N
CJ
W
N - von Neumann point
S - strong detonation
W - weak detonation
CJ - Chapman-Jouguet detonation
Moshe Matalon
The ZND structure is not possible for weak detonations.
The ZND structure does not restrict the propagation speed D for strong deto-
nations. Therefore, wave speeds depend on the experimental configuration, or
on the rear BCs. Strong detonations are therefore overdriven detonations,
namely forced to run at velocity D > DCJ by being pushed from behind by a
piston, say. The question remains on how to determine D.
The CJ detonation, is an unsupported detonation, namely one that is not
pushed from behind, and travels at a speed DCJ determined by the conservation
laws.
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 9
up > u⇤CJ
The rear BC can be thought to be a hypothetical piston following the wave. The
question is how to determine D for a given piston velocity up (in a laboratory-
fixed frame).
We denote by u⇤the particle speed at the end of the reaction zone and u⇤
CJthe
corresponding value for the CJ detonation.
The detonation is overdriven.
The detonation speed D is chosen
such that up = u⇤(D), and the
following flow is uniform.
D
x
reaction zone
up
u⇤steady following flow
u = up
As up is reduced towards u⇤CJ
, the same qualitative picture remains, with theNeuman state N on the shock-Hugoniot dropping lower and the final state Sapproaching the CJ point.
Moshe Matalon
D
x
reaction zone
up
constant state
u⇤CJ
rarefaction
As up is reduced towards u⇤CJ
, the final state S approaches the CJ point and
D = DCJ . What happens when up is reduced further?
up < u⇤CJ
A further reduction in up belowu⇤
CJleaves the detonation speed and
the reaction zone unchanged, since thefinal state has reached its lowestvalue on the fully-reacted Hugoniot.
The detonation wave (including
the reaction zone) is now unsupported
and continue to propagate at speed DCJ ,
una↵ected by the following flow.
The following flow, however, must now be reduced to match the BC. And, unlike
the previously uniform state, it is replaced by a (time-dependent) rarefaction
wave, which could be followed by a constant state as necessary. The smaller up,
the larger the amplitude of the rarefaction.
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 10
The state of the gas immediately behind the shock is easily obtained from
p� p0 = �m2(v � v0)
Using the first relation to eliminate v/v0, one gets a quadratic equation for p,with two solutions
p =1
� + 1(p0 + ⇢0D
2)±(✓
�p0� + 1
� ⇢D2
� + 1
◆2
� 2�Q� � 1
� + 1⇢20D
2
)1/2
vs = v0 �2(M2
0 � 1)
(� + 1)M20
v0ps = p0 +2�p0� + 1
(M20 � 1)
D � us =⇢
⇢sD
where M0 = D/c0 and c20 = �p0/⇢0
The inert shock solution (denoted by subscript s) is found by setting � = 0.One of the two solutions is the undisturbed state p = p0, v = v0. The other isthe state of an inert shock:
where m = ⇢0D.
✓p
p0+
� � 1
� + 1
◆✓v
v0� � � 1
� + 1
◆= 1�
✓� � 1
� + 1
◆2
+ 2� � 1
� + 1
�Q
p0v0
Moshe Matalon
which in a frame attached to the shock is given by
The spatial distribution behind the shock is determined from
D�
Dt= !
which can be integrated to give
The end state is found when � = 1.
(D � u)d�
dx
= k(1� �)⌫e�R/RT
! = k(1� �)⌫e�E/RT
x =
Z �
0
[D � u(�)]
k(1� �)⌫e
E/RTd�
A natural length scale is the “half-reaction” length scale, obtained by setting
� = 1/2, namely
`1/2 =
Z 1/2
0
[D � u(�)]
k(1� �)⌫eE/RT d�
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 11
�
!
Distance from the shock
Reaction zone structure of an unsupported
detonation, with all variables plotted
as a function of the distance from the
lead shock, scaled with `1/2.
A prominent feature is the appearance
of an induction zone, where there is only
a small amount of reaction, followed by
a rapid reaction zone that is well-separated
from the shock.
Moshe Matalon
⇢ D�u
pT
Distance from the shock Distance from the shock
Ficke& & Davis, 1979 Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 12
Curved detonations
ns
h
o
c
k
unreacted
mixture
reacted
products
D
⇢@h
@n� @p
@n= 0
For weak curvature, in a frame attached to the shock
@
@n(⇢(u�D)) + ⇢u = 0
⇢(u�D)@u
@n+
@p
@n= 0
(u�D)@�
@n= !
These equations are quasi-steady (and therefore independent of initial data) and
quasi-planar (requiring only knowledge of the state of immediately behind the
shock). A solution exists only if D = D(); i.e., D and satisfy an eigenvaluerelation that depends on the kinetics.
Moshe Matalon
ANRV294-FL39-12 ARI 12 December 2006 6:4
-0.1 0 0.1 0.2 0.3 0.4 0.5
5
6
7
8
9
Theoretical Dn(!)DNSDSD
!(mm)-1!(mm)
-1
Dn(
mm
/µse
c)
Dn(
mm
/µse
c)
b
High velocity branch
(Ignition)
Low velocity branch
(Extinction)
9
8
7
6
5
4
3
20.5 1.0 1.5
!1 !2
DCJ
c0
(Dn)1
(Dn)0x
0
a
Figure 6Detonation velocity vs curvature curves. (a) A Z-shaped Dn(!) curve showing critical curvatureand scenarios for ignition and extinction. (b) A comparison of the Dn(!) relation, thedetonation shock dynamics (DSD) simulation, and the direct numerical simulation (DNS) fora spherical detonation in plastic-bonded explosive (PBX) 9501 from Lambert et al. (2005).
stationary (see Figure 4b); (b) from a shock polar analysis, compute the angle "c forthe interaction of the explosive with the inert (see Figure 4c); (c) when the angle "
of the incoming detonation satisfies " < "s , apply an extrapolation condition at thedetonation-confinement boundary, !nedge · !"" = 0, where !nedge is the normal to theboundary; and (d) when " > "s , either set " = "c or " = "s , depending on whichgives the lower shock polar match pressure. Specifics on the boundary conditions forDSD can be found in Aslam & Bdzil (2002), Aslam et al. (2004), Bdzil et al. (1995),and Sharpe & Bdzil (2006).
Use of the level-set methods for DSD was set forth in an article by Aslam et al.(1996). It is now the most widely used detonation front propagation method. Earliermethods were either front centric (Bdzil & Fickett 1992) or ray-based (Lambourn& Swift 1990, Swift & Lambourn 1993), and required detailed logic to resolve frontmerging and bifurcation. The level-set method (Osher & Sethian 1988) embeds thepropagating detonation front in a surface of one higher dimension, and then evolvesthat surface according to
#$
#t+ Dn(!)| !"$| = 0. (12)
Bifurcation of the front (such as around obstacles) and merging of two fronts intoone are captured automatically when the $ = 0 contour is extracted from $(!r). The
278 Bdzil · Stewart
Annu. R
ev. F
luid
Mec
h. 2007.3
9:2
63-2
92. D
ow
nlo
aded
fro
m a
rjourn
als.
annual
revie
ws.
org
by P
rofe
ssor
D. S
cott
Ste
war
t on 0
1/1
3/0
7. F
or
per
sonal
use
only
.
Detonation velocity vs curvature
Bdzil & Stewart (2007)
For weak curvature ⌧ 1, and for a rate law of the form ! = k(1 � �)⌫ the
following relations are obtained
D ⇠ DCJ � ↵ for 0 < ⌫ < 1
D ⇠ DCJ � � ln� ↵ for ⌫ = 1
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-‐Champaign 13
The planar structure is highly unstable and is prone to result in transient three-
dimensional structures.
Strehlow, 1968
Cellular structure of a hydrogen-oxygen mixture
Moshe Matalon