Similarity solutions of the wood-kirkwood model in detonation theory

OIX?O-7225/91 $3.00 + 0.00 Copyright @ 1991 Pergamon Press plc

Int. 1. Engng Sci. Vol. 29, No. 4, pp. 523-532, 1991 Printed in Great Britain. All rights reserved

SIMILARITY SOLUTIONS OF THE WOOD-KIRKWOOD MODEL IN DETONATION THEORY M. TORRISI Dipartimento

and S. LOMBARD0

di Matematica, Universita di Catania, Viale A. Doria, 6-95125 Catania, Italy (Communicated

by E. S. SUHUBi)

Abstmct-We

find some classes of exact similarity solutions in detonation theory by using group analysis methods and we apply a procedure to determine the time and width of reaction.

1. INTRODUCTION We consider the Wood-Kirkwood model [l, 21 for a reactive gas into which a detonation wave is propagating. This model is obtained reducing to the axis a two dimensional flow in a cylindrical charge, so that the boundary effects caused by the walls of the cylinder enter only indirectly through a divergence term in the mass conservation equation. This model has been studied extensively in literature and a wide list of reference can be found in the book of Fickett and Davis [3]. Recently Logan [4], assuming as boundary conditions the Rankine-Hugoniot relations, performed a standard group analysis [5-71 and discussed the reduced system of ordinary differential equations in a limit case. In this work, starting from group analysis performed by Logan [4], we transform the system of partial differential equations in a new system of partial differential equations and after having found the associated group [8,9], by an opportune transformation of variables we get a new system in which the new independent variables do not appear. Thus we are able to find a class of solutions of the last system compatible with the boundary conditions. Once these solutions have been obtained, by going back to the original variables, we obtain similarity solutions. A detailed discussion of these results is carried out and a procedure to calculate the time of reaction tR and the width of reaction zR is suggested. Finally by observing that, in a special case, we can reduce the Wood-Kirkwood system to the case of planar Z.N.D. model [3], we also obtain similarity solutions in this case.

2. GOVERNING

EQUATIONS

FOR THE

WOOD-KIRKWOOD

MODEL

We consider a binary reacting gas mixture produced by shock wave propagation in a gas in a quiescent state and initiating an irreversible exothermic chemical reaction which takes place behind the shock. We assume that the reaction is like A- B while we consider as reaction progress variable the mass fraction of product B which we denote with A. As well known, moreover we assume that the flow in reacting zone is compressible, adiabatic and inviscid. To be precise by adiabatic we mean that in reaction zone the heat flow between fluid elements is negligible; in fact, generally, in detonation phenomena the hydrodynamic timescale is much faster than the timescale for heat conduction. In this zone the governing equations are given by the usual conservation laws of gasdynamic to which we add the chemical reaction equation, for which we write: pt + v ’ (pv) = 0 v,+v.vv+p-‘vp=o

e,+v.Ve+pp-‘V.v=O At + v . VA = Q(& p, p).

(2.1)

We denote with p, p, e, v the density, the pressure, the specific internal energy, the particle velocity respectively; Q is the rate function which is a constitutive relation. 523

524

M. TORRISI and S. LOMBARD0

Moreover as we assume the reactant and product polytropic state relation for internal energy is

have same adiabatic

W - qA

e =P]~(Y -

constant

y the

(2.2)

where q is the specific heat of reaction. The medium considered in this paper is a semiinfinitely long cylinder of finite radius in which we introduce a system of eulerian coordinates denoting with the z-axis the axis of cylinder, with r the radial distance from the axis and assuming as origin of the z-axis the position at which a piston, generating the shock wave, shocks the gas at initial time [3,4] (t = 0). Then the governing equations in cylindrical symmetry, taking (2.2) into account, become: pt+up,+op,+pu,+po,+polr=O pu,

+ puu,

po,

+ puo,

+ pcill&

+ pz = 0

+ pow,

+ pr = 0

($ > +

PI + UP, + WPr -

(Pt

UP,

+WP,) =(Y - l>qPQ@>

P, P)

4 + uk, + 04 = Q(& P, P)

(2.3)

where u and w are particle velocity along the z-axis and particle velocity along the radius r. As in Wood and Kirkwood [2] and following [4] we specialize the system (2.3) to axis of the cylinder (r+ 0): limo=0

lim pr = 0.

r-r0

r-+0

(2.4)

While from the definition of derivative lim o = 0,. r-+0 r

(2.5)

Then the (2.3) assume the following form

u, + uu, +

PI + UP,

-

(

$

>

0f

pz=o

(PI + UP=)

= (Y -

l)wQ&

P, P)

A + uA, = Q@, P, P)

(2.6)

In (2.6.1) the function w, is treated as parameter function containing information about the effect of cylinder wall confinement and it is assumed depending on u, A and p [4]. We assume for (2.6) as boundary condition the Rankine-Hugoniot relations for the fluid dynamic shock. By considering the strong shock approximation and taking into account that the shock progresses in a quiescent non reacting gas with initial state p. = u. = 0 and p = p. we can write the R-H relations as u,=-D

2 Y-+1

Pl =

Y+l

y_lPo

2PoDZ

pl=y+l

Where D is the shock speed and the subscripts 0 and 1 refer respectively evaluated ahead and behind the shock front. We complete (2.7) by adding the following condition for A [lo]: il, = 0; by this condition we assume there is no chemical reaction at the shock front.

to the quantities

(2.8)

525

The Wood-Kirkwood model 3. SIMILARITY

ANALYSIS

As

shown by Logan [4] the system (2.6) with auxiliary conditions (2.7) and (2.8) is invariant with respect to the infinitesim~ tr~sfo~ations [5-7f t*=t+&t

z “=Z+&f;

U*=U+&qJ

p*=p+&n

p*=p+&x

A*=A+Ep

c=bz+d x=0

Q,=(b-a)u p=2(b--a)A

(3.1)

where E is the parameter and t=at+c

n=2(b-a)p

(3.2)

with a, b, c, d constant. In this work, for simplicity’s sake, we suppose a, b, c, d # 0.

bfa

(3.3)

The invariance with respect to (3.1) lead to the identification of the following classes to which the functions Q and W, must belong: Q=P!f(j*

P)

0, = u x/3-1)if L ( u29P

(3.4) (3.5)

)

where f and g are arbitrary functions and 2b-3a

s=2(b -

(3.6)

a) *

Furthermore, the invariance allows us to apply the following transformation of variables [93: T = ln(qt + 1) 1 o = (cgt + 1)‘2 c4z

+

p(tt z) = R(K o) u(t, z) = (Cjt + 1)“~‘U(T, a) p(t, 2) = (cg + 1)*(=2-l)P(T,u) qt, z) = (c,t -f- 1)2’“2-1’A(1;

II)

(3.7)

where we put c2 = bfu, c3 = u/c, c4 = bfd and hence

Then in the new variables the system (2.6) becomes: R,+(~U-c20)R,+~RU*=-~R~2~~-1~g UT + %I-c2a ( c3 6

E

+

u

-

U,+:R-‘P,=(l-q)U >

c2u

(

fh-+

w

-

c,o)A,

P, + yz U,P = 2(1- c2)P _ 2r p~m-“g

>

+ (Y-1) cj

MpBf

c3

=

2(1- $)A+-$

P@f

(3.8)

M. TORRISI and S. LOMBARD0

526

Since the shock front must be a similarity curve passing through t = 0, z = 0, its equation is: c&z + 1 = (cg + 1)‘2 from which it follows D(t) = D&t

+ l)EZ--1

where

corresponds

to a noted initial shock velocity and the R-H conditions become:

NT, 1)

=~PO

2L),

U(T, 1) =-

Y+l

(3.9) The system (3.8) is again invariant

with respect

to the transformation

group (associated

group) [8,91:

p* = /&2+Ap u* = pu CT*=/&7 R’=jiAR T’=T f * = p2-B(2+9 g+ = ru2-28g A.* = c12~ A E R - (0, -2) I provided that

f

(3.10)

and g have the following functional forms:

(3.11) where 8 and f are arbitrary functions of their arguments. Then, by assuming f and g like (3.11) by applying the following change of variables

(3.12) to the system (3.8) and to the auxiliary conditions (3.9), we obtain

1 v,,

+

YZ

v,v,,

+

(2

vz

-

c2 >

v,,

-- 2Y v~(l-~)/Av~~-l)v~~

=-(2+A)(~v2-c2)V,-Y~V,V,+2(1-c2)h +Y-1 cj

~v~l-~)(l+~A)v~ .f(y)

c3

V,(T, 0) = y+l y-pJ*

WT 0)

=f$,.

200 V,( T, 0) = y+l’ V,(T, 0) = 0.

(3.14)

527

The Wood-Kirkwood model

It is easy to observe that if we look for “steady” solutions we obtain the so-called reduced system of ODE’s given by the classical procedure of similarity, Here it is interesting to note, instead, that, if we tind some solutions of (3.13) with (3.14) we can study the propagation of the weak discontinuities of VI, V,, V,, Vi across a curve rp(T, q) = 0 in a state characterized by this solution. This fact is very useful, as observed in [9], when the solutions found are “steady” or “constant”; in fact in the original independent variables these solutions are in general non constant and non steady.

4. SOME

CLASSES

OF EXACT

SOLUTIONS

Now we look for solutions of system (3.13), with auxiliary conditions (3.14) of the form [ll]:

v, = v&J =

Y+l -

V+V,=200

y-lPO

2 v, = v, = Y+l

y+I POG

v, = v,G-J 11)

(4.1)

this implies

v,v, v~(~--BYA+~v;W)~ ---c,(a+1)+e,av,-z C3

=

0

-~V:+V+Z+A);=O 1 -(2 + A + y)

2 V,V,

+ (c2A + 2)V, -

z V;(1--B)‘AV;(p-1)V3g+ @$) ~v$1-fi)(1+2’A)v~ =

0

(4.2) so that we must require: jj = go = const.

j! = f. = const.

(4.3)

Therefore we can solve the system (4.2) for some of the constants which appear there. We observe that we can consider fo, c2 known from the functional form of the rate law and q known from the chemical characteristics of the reactive medium, while we must determine c3, c4 and A from (4.2). Moreover, we must add to system (4.2) the relation

~o~czc3

(4.4)

c4

as compatibility condition if go is known or as a fourth equation go[2, 31. Then if go # 0, taking into account (4.1) and

v,

Y-1

v,v”,

2

-=-

that allows us to determine

’

assuming l-2cz#O,

(4.4’)

after some calculations we obtain A=

-2c,y + y + 1 C*(Y- 1)

(4.5) (4.6)

and, ifA+y+2#0, (4.7)

528

M. TORRISI and S. LOMBARD0

or,ifA+y+2=Oand

y#2, (4.7’)

In this case the compatibility

of A + y + 2 = 0 and (4.5) implies 1

c2=2 #j=-3y-4

and hence

2(Y - 1) * The (4.4) yields the following constraint go:

for the constitutive

y, q, $,, c2 and for

parameters

(4.8) whenA+

y+2#0,

or (y - l)(y - 2)

w--4)mY--1) (4.8’)

Y

whenA+y+2=Oand yf2. We could obtain from (4.8) or (4.8’) go so that w, is identified as classically requested [2,31. Finally, by going back to the original variables t, z we obtain the following solution:

in

(4.9) (4.10) (4.11) (4.12) where c3 and c4 are given by (4.6) and (4.7) or (4.7’) with constraints (4.8) or (4.8’), while V4(T, 11) is the solution of the fourth equation of system (3.13) with the condition (3.14.IV); i.e. solution of the problem [ll]:

~v,(W-V=O

(4.12’)

Concerning the solution of (4.12)‘, taking into account (4.6), (4.7) or (4.7’) and (4.8) or (4.8’), we obtain: V

(T

4

a)

J

=

(% - 3, + y(l - *“) Dz,[d(y+l-k&(1-+ (1+

y)(l+

Y-

2c2)

_

11

(4.13)

4

forA+y+2#Oandc2#(y+1)/2; 4(Y - 1) 0; W7’9 0) -- (y + l)z 4 In 0

(4.13’)

forA+y+2#Oandc2=(y+1)/2;

4

v,U’, 4 = yl;_:l) (a” - 1) forA+y+2=Oand

yf2.

(4.13”)

The Wood-Kirkwood model

529

Hence, going back to the original variables we obtain respectively:

forA+y+2#0andc,#(y+1)/2; 4(yn(tJ r)=(y+

1) 0;

1 2 c4z + 1 ( cg + 1 > In (cg + l)(y+l)n c4z

+

(4.14’)

forA+y+2#Oandc2=(y+1)/2; A(t, 2) =

y;y;21) 4 (SY[ ((c3;:;L)h - l]

(4.14”)

forA+y+2=Oand y#2. Before the analysis of particular cases we observe that we are able to specify the signs of c3, c4 and g, once the value c2 is fIxed from the constitutive relation (3.4) or (3.5). In fact from (4.8) and (4.8’) it follows respectively sign g, = sign (y - 2) - & [ ifA+y+2#0,

2

I

or sign go = sign(y - 2)

ifA+y+2=0, Moreover,

(4.15) (4.15’)

y#2. as v$i-B’~Av;$B-” > 0,

(4.6) yields sign c3 = sign 1 _g02c2 . Then taking into account the assumptions

(4.16)

(3.3), (3.10), (4.4’) we obtain the following cases:

CaseA:A+y+2#Owithfivesubcases: (A,) l 0 (Ai2) 0 0, c3 > 0, c4 < 0 (A14) c2 > (y - 3)/2(y - 2); then go < 0, c3 > 0, c4 > 0. We do not consider the subcase c2 = (y - 3)/2(y - 2) because go goes to infinite. (A,) y =2: (A,,) c2 c 0; then go < 0, c3 < 0, c4 > 0 (A22) OO, c~ 0, c3 > 0, c4 < 0 (A32) (y - 3)/2(y - 2) < ~2 < 0 then go < 0, ~3 < 0, ~4 > 0 (A33) 0 4 then go > 0, cg < 0, c4 < 0. (A4) y=3 (A4i) c2<0 thengo>O, c3>0, c4C0 (A42) 0~~2~4 theng, 12then go > 0, c3 < 0, c4 < 0. (A,) Y ’ 37 (ASI) c2<0 theng,>O, c3>0, c4<0 (AS2) 0 < c2 < (y - 3)/2( y - 2) then go > 0, c3 > 0, c4 > 0 (A53) (y - 3)/2(y - 2) < c2 < ? then go < 0, c3 < 0, c4 < 0 (AS4) c2 > $ then go > 0, c3 < 0, c4 < 0.

530

M. TORRISI and S. LOMBARD0

CaseB: A+ y+2=Owithtwosubcases: (B,) 1< y<2. Then c2>Ogo0 cd>0 (B,) y>2. Then c,COgO>O c,>O c,
(Case

-3c,+2

A, Subcase

A4).

(4.17)

C2

&To 22/(3cz-2)

c~I(l-c~)(3c~-Z)~~/(l-c~)

PO

(4.18) (4.19) (4.20)

With (4.20) c3 and cq become: (4.21) C4 =

-4h2

W(3q-2)

P

c~I(:-c~)(3~~-2)~~/(1-c~)

(4.22)

and the solution assumes the following form: p(t, z) = 2P,[

(~~t~~l~~2](2-3c*~‘c* (4.23) 1 c,t + 1

c4z

u(t, z) =go-

+

(4.24)

1 (S,‘1

(c4z + 1)(2--c&2 P(t,r)

= f POG 0;

qt, 2) = ‘i(C,cz

for A +5#0

and c2#2,

2) 4

(c4z [

(C,t

[ (c3t + 1)4-3=2 + +

1)4(%-l)h 1)2(cZ-1)

-

(4.25) (4.26)

or (4.26’)

for A+520 and c2=2. It must be stressed that from the previous analysis, in the Case A.,, as regards the constants go, c3, c4 for the previous solution we obtain when: (9 cp < 0 a decelerating shock with decreasing pressure at shock front while we have a break of solution at 2 = -l/c,; (ii) O 213 we have a break also of p; (iv) c2> 1 a decelerating shock with decreasing pressure while at t = -l/c3 there is a break of u, L and if c2 < 4/3 there is a break also of p. At z = -l/c4 there is again a break of p and if c2 > 2, this also occurs for p. If 4/3 < c2 < 2 only p has no break.

The Wood-Kirkwood model

531

We now consider a special value of c2 of the Subcase (i) in order to illustrate determination of the reaction time, CR,and the determination of the reaction width zR. So we take c2 = -2 and use for J. the (4.26). Because the reaction finish when P = Pp (density of product) and A.= 1, this implies Pp = 2Po

c4.+ (c3fR

+ +

1 (%)‘I 1

-4

1)-2

-

the

(4.27) (4.28)

from which we obtain: (4.29) where m:=-

2Po

and

<:=

PIJ while c3, c4 are given by (4.21) and (4.22) for c2= -2. Bearing in mind that the solution falls down at z = -1/c4, in order that the reaction finishes before this happens, we must require that 0 < 5‘< 1 then we obtain:

l
(4.30)

PP

where m* is the real solution of the following equation m 114 =$(m-1)

(4.31)

from which we get the limit value of m giving 5 = 1.

5. THE In this case g = 0 and consequently

SPECIAL

CASE g,= 0

from (3.11) and (3.5) w, = 0

(5.1)

so that the system (2.6) coincides with the ZND model in the planar case [3]. Then specializing the (4.2), by following the same procedure, we obtain

1 c2 = 2 so that is 2 + A + y # 0 and as a consequence

(5.3)

of which

c3 = -2(~)1+yp~-YLI~o

(5.4)

while from the initial shock velocity we can obtain c4=

-

(5 >

l+yppDoqfo.

(5.5)

V,, instead, is given by the solution of the problem (4.12’) where A and c2 are given from (5.2) and (5.3).

532

M. TORRISI

and S. LOMBARD0

Then:

It is worthwhile noticing that, taking into account (5.4) and (5.5) we may obtain some comments about the feature of these solutions. Finally specializing the previous results for y = 3 we obtain c4z + 1 P(C 2) = 2PO (C$ + l)‘n

u(t, z) +()--

PO, z) =; PO@

+ 1 c,t + 1

(5.11)

(c,z + 1)3 1)5’2

(5.12)

c4z

(C$ +

(~~:,,'I Q'-gg'& c4=-$-3& (c,t + 1) 12 q [ (C4Z + 1)4 -

A@,2) = --where

(5.10)

1 0;

(5.13) (5.14)

Ac~~~dg~~-~is work was supported by C.N.R. (S.L. and M.T.) through G.N.F.M. and by M.U.R.S.T. (48% and 68%) (M.T.). One of the authors (S.L.) wishes to thank the Dipa~mento di Matematica del~u~ve~it~ di Catania for the support received during the present work.

REFERENCES [l] W. W. WOOD and J. G. KJRKWOOD, J. C&=ur. Phyr. 22, 1915 (1954). [Z] W. W. WOOD and 1. G. KIRKWOOD, 1. Chem. Phys. 22,1920 (1954). [3] W. FiCKEZT and W. C. DAVIS, Detonation. University of California Press (1979). [4] J. D. LOGAN, 1. Phys. A: Math. Gen. 21, 643 (1988). [J] G. W. BLUMAN and J. D. COLE, Similarity Methods for Differential Equations. Springer, New York (1374). (61 L. V. OVSIANNIKOV, Group Analysis of Difierential Equations (Translation Editor W. F. AMES). Academic Press, New York (1982). Russian edition: Nauka, Moscow (1978). [‘7) W. F. AMES, Nonlinear Partial Differential Equations in Engineering, Vol. II, Chap. 5. Academic Press, New York (1972). f8] L. DRESNBR, Similarity Solutions of Non-Lineur Partial Diffe~ntia~ Eqwtions, RNM88. Pitman, London (1987). [9] W. F. AMES and A. DONATO, ht. J. Non-Lin. Me&. 23, 167 (1988). [lo] R. R. ROSALBS, Stability theory for shocks in reacting gases: Mach stems in detonation waves. In Reacting Flows: Combustion on Chemical Reactors (Edited by G. S. S. LUDFORD), LAM, Vol. 24. AMS, Providence, RI (1986). [ll] M. TORRISI, Int. 1. Non-L&z. Mech. 24,441 (1989). (Received 27 July 1990; accepted 21 August 1990)