Notes Lecture 7 - Princeton University · Lecture 7 Detonation Waves Moshe&Matalon& & & & & & & &...

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


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


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


⇢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


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


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


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


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


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


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


�

!

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


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


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

Notes Lecture 7 - Princeton University · Lecture 7 Detonation Waves Moshe&Matalon& & & & & & & &...

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