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CopyNo. :X,^ CONFIDENTIAL ;^^;.Xnc.?27

NEW YORK UNIVCRSITY

INSTITUTE OV MATHEMAT;CAL SCIENCES
LlUi^: • '.

25 Waverly PI ce, ; , y



REMARKS ON THE MATHEMATICAL THEORY OF
DETONATION AND DEFLAGRATION WAVES IN GASES

(Supplement to the Manual on Supersonic Flow and Shock Waves)

SS.P I 7 ,369



Prepared for the

APPLIED MATHEMATICS PANEL

NATIONAL DEFENSE RESEARCH CGMMIHEE

By the

Applied Mathematics Group

New York University



affecting fhe National Defense of the
United States within the meaning of the
Espionage Act. U. S. C. 50; 31 and 32.
Its t-ansmisslon or the revelation of its
contents In any manner to an unauthorized
person Is prohibited by law.



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PREFACE

In the present supplement to the Manual on Supersonic
Plow and Shock Waves the gas dynamical phenomena of one-
dlmenslonal flow Involving detonation and combustion are
analyzed from the mathematician's viewpoint. As In the
Manual, content and emphasis In the present supplement are
conditioned by the background from which the writers happened
to approach the subject J Important points, such as the
finite width of the reaction zone, are touched upon only In
an appendix and In the bibliography.

Jouguet, In his classical work, was concerned mainly
with the discussion of the discontinuous reaction front;
but only the consideration of the flow as a whole can supply
the Information necessary to determine the dynamical phenomena
Involving detonation or combustion. G. I. Taylor has studied
detonation processes under such aspects. In the recent
research program of the Applied Mathematics Group of New York
University a somewhat more systematic analysis of the
mathematical possibilities of flows Involving reaction procesaei
became desirable, and the present report gives an account of
such investigations.

R. Courant
Director, Contract OEMsr-945



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REMARKS ON THE MATHEMATICAL THEORY OF DETONATION
AND DEFLAGRATION WAVES IN GASES

(Supplement to the Manual on Supersonic
Flow and Shock Waves )

AMP Report 38. 3R



In a process of detonation as well as of combustion
(or deflagration) a "reaction front" sweeps over the com-
bustible substance, here assumed as a gas, and separates
the explosive from the bxirnt gases. As seen In the "Manual
on Supersonic Flow and Shock Waves" (AMP Report 38. 2R) there
Is a close analogy between "shock fronts" and "reaction
fronts," both are discontinuity stirfaces across which the
state of the substance undergoes sudden changes by the three
laws of mass, of momentum, and of energy; the difference Is
merely that across reaction fronts the energy -balance con-
tains as one term the energy liberated by the reaction, while
no such term appears In the energy condition for shocks.*

Together with the differential equations In the zones
of continuity and with Initial and boundary conditions, the
transition conditions should determine the flow altogether.
However, while such a determinancy exists for gas motions
Involving shocks only, a different situation presents itself
as soon as "reaction fronts" occur (we shall use this term
Indiscriminately for the transition front due to detonation
or deflagration); then the conditions as usually envisaged
In non-reactive gas flows are no longer svifflclent to determine
the flow, and additional conditions must be found; In the case
of deflagration or combustion the degree of indeterminacy is
even higher than for detonation fronts.

» See Manual [l]. Chapter III, Section 50.

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Thus the following general problem arises: To what
extent are reaction waves left undetermined by the laws of
conservation? ?rhat are appropriate additional conditions
to determine the reaction front? These questions are of a
strictly mathematical character. They refer to the determinacy
of the solutions of the underlying differential equations satis-
fying well-defined transition, initial, and boTindary conditions.

Iii5)licitly the assumption seems to have always been made
that for detonation waves the well known Chapman- Jouguet rule
as an additional condition suffices to determine the process
while for combustion fronts other conditions have to be supplied
from a specific theory of the mechanism of combustion. Our
following mathematical analysis has the objective of clarifying
this statement and of specifying which reaction fronts are com-
patible with conditions as they actually may occur. Moreover,
it will be seen in what manner reaction fronts can and must be
accompanied by rarefaction waves, compression waves or shocks.
The mathematical analysis, restricted here to the case of one-
dimensional motion in gases , will be based on the theory of
characteristics; in particular, on the concept of domain of
dependence*

While the following discussion stresses the mathematical
aspect of the theory, it may have physical significance in parti-
cular for the understanding of reaction processes taking place
in gases which are in non -uniform motion.

Remarks on the mathematical theory of detonation and deflagraticn
waves in gases *

1. Basic notions concerning reaction fronts .

We first recall a number of well-known facts concerning
reaction fronts in gases. The gases , before and after reaction
are assumed ideal but not necessarily polytropic. In other words,

♦ See Courant-Hllbert [2], Chapter 5, and the Manual [1], Chapt II.

** For the physico-chemical notions and facts in the present re-
port Lewis and v. Elbe [3]. For the theory of reaction fronts
see Jouguet [4] ,[5], Becker [6], v. Neumann [7], Taylor [14].

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the internal energy Is a function of the temperature but not
necessarily proportional to lt» By p, />, r = p , c we
denote pressure, density, specific volume, and sound speed of
the burnt gas quantities referring to the unburnt gas or
"explosive" are characterized by a subscript (o). We Intro-
duce the quantity

9 =rp,

which Is proportional to the temperature T. By e, 1 = e + 0,
g ne denote respectively the Internal energy, enthalpy, and
energy of formation per unit mass. The quantities

E = e + g, I=l + g

may conveniently be called "chemical energy" and "chemical
enthalpy" p.u. mass respectively. We ass\ime that these quanti-
ties depend only on the temperature and hence " on 9.

2
The sound speed c is defined by the relation dp -c dyo=

if dE + pdr = 0; the latter relation expresses that p and t
are coupled adlabatlcally. We introduce a function i= ^(9)
of 9 by the relation



(1)



dE _ 1
35 - 7^



» 9 = m'T^T, R being the gas constant, M the molecular weight.
Since the composition of the burnt gas and hence its molecu-
lar weight depends on the temperature, the dependence of 9
on T is not strictly linear.

** With reference to absolute zero temperature. By liberated
energy f we shall denote the difference go-g of the energies
of formation at absolute zero temperature (not, as more
customary, at a standard temperature such as 298°K. )

*** Strictly speaking this is not correct for the burnt gas, which
at different ten^jeratures has a different chemical composi-
tion, since the state of equilibrium depends not only on the
temperature but also to a certain degree on the pressure. In
the following we ignore this dependence.



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Then (see App. II) the sound speed is given by^

(2) c^ =^© .

We assume the gas flow to be one -dimensional, taking
place in a long cylindrical tube. By u, ^ , and

(3) V = u - ^

we denote gas velocity, reaction front velocity, and relative
velocity respectively.

The laws of conservation in the regions where the flow
is continuous and adiabatic are expressed by the differential
equations



E^. + uE^ + p{T^ + ur^)



Across a discontinuity front the laws of conservation are

I P-" =/'o% *

II P ^p^^ = Po ^ Po^% >
m |v2 -.I=|v2^I^ .



* For we have identically

dE + pdr = -i— (rdp + )?pdr) = :T~-(dp - ^©d^),
3r - 1 J - 1

hence vanishing of dE + pdr Inplies dp/d/o =^Q.
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The mechanical relations I and II Imply relations



(4) '^^-A'-fK

'o



p-p



(5) = vv



(6) (p-Po)( ^o-^^ = oo .



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A transition to a state represented by a point on the
upper branch corresponds to a detonation, while the lower
branch corresponds to a deflagration. If the reaction front
Is "facing to the right," l»e. If the explosive gas (0) Is
to the right. and the burnt gas to the left, (the positive x-axls
defining the direction to the right), the mass flux Is from the
right to the left, v < 0, and consequently



P-P,



_o

u-u.



Therefore, the right upper and the left lower, branch correspond
to forward facing reactions. It Is then clear that for a for-
ward facing detonation u > u^, while for a forward facing
deflagration u < u^. In other words:

A detonation accelerates the burnt gas toward the explosive ,
a deflagration accelerates the bxirnt gas away from the explosive .

Among all conceivable reaction processes which lead from
a fixed state (p^, u^, r^) to states (p, u, r,), there are two
of particular significance, one a deflagration and one a detona-
tlon. They are characterized by the " Chapman- Jouguet law
which can be stated in various equivalent forms; for most re-
action processes it becomes plausible by many considerations (see
e.g. subsequent discussions in this supplement), but must be con-
sidered as a hypothesis rather than a law of general validity.
The "Chapman- Jouguet reactions" can be defined by any of the
following conditions P, Q, R, S, T:

(P) ^ = ^^

^^' .dr r-r



(That there are Just two solutions of this equation, one with
p > p and one wlthT>r , follows from the assumptions
d^p/d-r^ > 0, ^ < 0, and p^ < p^ or r^ < Tg, see p. 10. ) On
the graph of p as a function of r (Pig. 1) the point corresponding

» See Manual [1].

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to this transition is characterized by the condition that
the tangent at this point passes through the point (t ,p ).
Similarly, on the graph In the (u,p) -plane (Fig. 2) this
point Is characterized by the condition that the tangent
at this point passes through (u^,p^)* . Equivalent properties
are:** (See Manual p. 131)

(Q) The propagation speed |v^| of the front observed from the
explosive is an extremum for fixed values of ^ fPojT ) (Chapman
1899), or

dv = .

(R) The entropy of the burnt gas Is an extremum.

(S) The propagation speed Iv| of the front observed from the
burnt gas Is sonic (Jouguet 1905)



(T) ^ = -i'fi

dr r



The two Chap man- Jouguet reaction processes, corresponding
to the points D and C, differ In that for detonations the speed
(▼qI and the entropy are relative minima , while for deflagrations
the speed |v^| and the entropy are relative maxima , the state
(0) ahead of the wave always assTimed to be fixed (see App.I^).***

♦ Prom (4) we have by (P)

2(u-u^)du = ('r^-r)dp-cp-pQ)dT = 2(r^-T)dp , whence
du/dp = (u-u^)/(p-p^).

♦♦ (For the equivalence of the conditions (P) to (T) (see App.I.]0<
A procedure for computing the Chapman -Jouguet reaction process
when the state In front of it is given is outlined In Appendix

♦** Emphasis is laid on the relative character of these oxtrema.
The absolute value of the maximum for deflagrations may be
less than that of the minimum for detonations.



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the extremal properties of the speed |v | can be tmderstood*



from the relation (4),

2 2 _ P-Po

P-Po
and the fact that Is the slope of the segment connecting

the points ('^q,Pq) anS ('r',p) together with d p/dt^ > 0.

According to whether or not a discontinuity effects a
smaller or larger change, a detonation implying a transition
of Pq to p < p or p > p_j will be called weak or strong
respectively; a deflagration implying a transition of p to p > p«
or p < Pj^ will also be called weak or strong respectively. Both
a constant volvime detonation and a constant pressure deflagration
are weak.

The answer to our main question, the problem of deter-
minacy of reaction fronts, will depend decisively on whether
the flow, observed from the reaction front, is sonic or sub-
sonic or supersonic. This question is answered by a statement
(Jouguet [5] (which we shall prove in Appendix 1.3), and which
will serve as the basis of the subsequent analysis:
The gas flow relative to the reaction front is

supersonic ahead of a detonation front,
supersonic behind a weak detonation front,
subsonic behind a strong detonation front,
subsonic ahead of a deflagration front,
subsonic behind a weak deflagration front,
supersonic behind a strong deflagration front.
We shall refer to and make use of these properties as Jouguet 's
rule in the following discussion.

* One might also use the relation (7)

V - P'Pq
o = " '^o u-u^, '

p-p
and the fact that is the slope of the segment connacting

the points (u^jp^) anS (u,p). This fact leads to an obvious

geometric construction of v in the (u,p) -plane, see Courant-
Priedrichs [8], p. 51. °

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11 CONFIDENTIAL



2. general remarks on the determinacy of a flow Involving
a discontinuity front .

As was said before, we restrict our discussion to the
case of gases filling a long tube and asstime that all processes
are one -dimensional, (an assumption which will be actually
satisfied for combustion processes only If the tube has a con-
siderable width).

Assuming that the state of the gas at some Initial time
is given and that at this initial time a reaction front occxxrs
at some place, we ask how far the flow is then determined by
the three laws of conservation and the boundary conditions Im-
posed at the ends of the tube. (These conservation laws are,
as said before, nothing but the flow differential equations In
the continuous regions of the flow and the transition relations
I, II, III at a reaction front. )

We specify our initial conditions by assvuning that at
time t = 0, the explosive is at rest in the half infinite tube
X ^ 0, that a reaction begins at t = at the closed end x = 0,
and that the closed end is operated as a piston moving with the
prescribed velocity TJ.

Usually one would be concerned with the case U = 0, which
means that the end of the tube is held fixed; however, consider-
ing the general case of an arbitrary value TJ not only contributes
much to the clarification of the problem of determinacy, but al-
so is helpful for the understanding of phenomena such as occur
when an explosive gas moves against a wall and is ignited there;
likewise for an understanding of the phenomena of boosting, the
study of the case U ^ may be useful.

The mathematical determinacy of flows involving reaction
fronts depends essentially on the geometrical relationship be-
tween the characteristics of the system of differential equations
I, II, III, and the line W representing the reaction front in
the (x,t) plane as well as the line ir representing the moving
piston at the closed end of the tube. As usual, the one dimensional

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COIlFIDENriAL



12




/f'et^ct/f?} fr



Figure 3
Possible relation between characteristic
and reaction front* (For other
possibilities see Flg.7.)

motion is represented in the (x,t) -plane. For Isentroplc flow

the system reduces to the system of the differential equation

I and II only (see Manual [l]. Chapter III) and for any flow

there exists two sets of characteristics C_, C^ In the (x,t)-plane,

defined (see Manual, Chapter IV ) by

C^ : dx = (u+c)dt; along C_|_ we have dp +/?c du =
C_ : dx = (u-c)dt; along C_ we have dp -/3c du =

If the flow Is not Isentroplc the third family of characteristics
C , the streamlines, have to be considered



dx



udt; along C we have d E + pdt



For the following discussion we envisage the case of isentroplc
flow, keeping in mind that the situation is analogous In the
general case. Furthermore, if nothing is said to the contrary,
we shall visualize the flow in front and behind the reaction
front as consisting of zones of constant flow and siniple waves,
so that either both sets C_ and C^ or at least one is straight.
(The following statements, however, remain valid also for flows



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13



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In which the characteristics are all curved). We Imagine
the characteristics described In the direction of Increasing
values of t. Then a line element X through a point P In the
(x,t) -plane Is called space-like If the two characteristics
leading Into P are both on the same side of ?i ; otherwise
Is called time-like (see Fig. 3).





Flgtire 4



space-like



time -like



A line or curve through a zone of flow In the {x,t) - plane Is
called space-like if all Its line elements are space-like.
(E.g. t = 0, the x-axls Is space-like.)

If Initial values of u and /O are arbitrarily prescribed
along a space-like curve L (see Pig. 5), then the initial value



M-f




Figure 5
Domain of dependence,
AB, of a point P.



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probleia Is uniquely solvable, and the values of u and p at
a point P, as In Pig. 5, depend only on the Initial values of
u and (o on that part AB of L which Is cut out hy the two
characteristics drawn backwards from L domain of dependence).

If L were time -like, then only one of the two char a ot er-
istics would Intersect L; along a time -like line not both
quantities u and ^ can be prescribed arbitrarily, but only
one of them. The same Is true If L Is characteristic.

If u and (^ are prescribed on the space-like Initial
curve L: AB then the flow Is determined between L and the


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