TECHNISCHE HOGESCHOOL DELFT LUCHTVAART- EN F.!;iMfFVAARTTECHNJE«{

B!8L«0THEE!C

### Kluyverweg 1 - DFirr

### Dynamic Behaviour of Combustible Gases

### Between a Shock Wave and a Following Flame

**Cranfield **

### College of Aeronautics Report No. 8616

March 19S6

### by J F Clarke and Z W Wang*

### College of Aeronautics

### Cranfield Institute of Technology

### Cranfield, Bedford MK43 OAL, England

Present address: Department of Engineering Mechanics, Xian Jiaotang University, Xian, China.

**T h e views expressed herein are those of the authors alone and do not **
**necessarily represent those of the Institute." **

### College of Aeronautics Report No. 8616

### March 1986

### Dynamic Behaviour of Combustible Gases

### Between a Shock Wave and a Following Flame

### by J F Clarke and Z W Wang"

### College of Aeronautics

### Cranfield Institute of Technology

### Cranfield, Bedford MK43 OAL, England

### ISBN 0 947767 43 6

### £7.50

* Present address: Department of Engineering Mechanics, Xian Jiaotang University, Xian, China.

**"The views expressed herein are those of the authors alone and do not **
**necessarily represent those of the Institute." **

### re-ignition of a combustible gas in the induction domain between a strong

### precursor shock wave and a following premixed flame. The re-ignition process

### takes place immediately ahead of the flame, and involves both local

### temperature and local pressure increases which are strongly linked to increases

### in precursor shock strength; the results are compatible with long-standing

### observations of the events known as "explosion within the explosion".

1. Introduction

We consider the one-dimensional flow field created by a piston or

contact surface that moves into a region of otherwise unbounded cold

combustible gas and, in doing so, creates a precursor shock of sufficient

strength to switch on a significant rate of chemical activity. The situation

has been described and discussed before by Clarke (1978), Cant (1984), Clarke

& Cant (1984), and Jackson & Kapila (1985). In the present case we introduce

the additional element of a premixed flame, that propagates in the space between

piston and shock wave, and implies by its presence that the initiating event

creates both the flame and the shock simultaneously, such as happens in the

case (for example) of a local electrical discharge or spark. It will be

assumed that in the very early time intervals following initiation of these

processes the field is "conical" in the sense that all of the variables depend

not on distance and time separately but only on the quotient

distance-divided-by-the-time. The implications of these assumptions will be worked out in §4 below,

after the equations and boundary conditions have been established in §§2 and 3.

Following establishment of the initial field, the rapid chemical activity

generated by the "switch-on" shock wave begins to take over, and the field

departs from its initially conical character. The flame is modelled as a

discontinuity, whose (possibly turbulent) propagation speed is

temperature-sensitive, and this also contributes to departure from conicality. It is

necessary to analyse both the hot inert all-burnt gas-flow between piston and

flame and the particularly interesting and complex induction domain that lies

between flame and shock. Perturbation methods are adequate for these purposes,

as has been demonstrated in the references quoted earlier, although numerical

integration of the induction-domain equations is necessary in view of the

intrinsically non-linear character of the domain. The technqiue used here to

iterative method based on an integral equation for the temperature

perturbation.

Fig, 1 is a sketch of the configuration in at'^t' (or distance, time)

space. In view of the number of particular domains and locations that must

be dealt with in an analysis of the present problem Fig. 1 is helpful in

defining the meaning of the various subscripts that are used to identify the

several quantities needed for a description of the whole field. Section 5

introduces the perturbation ideas and establishes perturbation conditions

at the accelerating flame sheet. The burnt-gas domain is described in

§6, while §7 deals with the induction domain and with some matters that

arise from the need to incorporate conditions at the accelerating flame

surface into numerical solution of the temperature integral equation.

Finally, Section 8 presents results, discusses their significant, and

### 2. Equations

### Temperature T , density ƒ , pressure-p , sound speeds, and mass fraction c

### are all measured in units of their dimensional values T . ,

* f'.*

### , >' A ' i C '

### immediately downstream of the initial shock wave. Local gas velocity M, is also

### measured in units of A'. . The induction time t of the combustion reaction

s<. I.

### immediately behind the initial shock is given by

### where

* , *

### E ^ E ' / R ' T '

### and

* ^*

### is the activation energy per mole of the simple irreversible reaction

### F — * - ? * that converts fuel T into product P . Q is the heat of reaction

### per unit mass of reactant, W is reactant molecular weight, C is the

### specific heat at constant pressure, assumed constant, and B' is the

### pre-exponential frequency factor of the reaction; R.' is the Universal Gas

### Constant.

### Time t is measured in units of t and spatial coordinate x' is then

### naturally made into a dimensionless coordinate

* -x.*

### by dividing it by

* o\X-r' *

### It is convenient to formulate the present problem in terms of a

### Lagrangian

* y^l*

### coordinate system, where

### . ^'*- ' 2 3

### The piston location in ^ , t space is given by '^.^^t) , so that

* y-O *

### locates the piston in Lagrangian space.

### Frozen sound speed immediately downstream of the initial shock wave is

### given by

### '/,. 2.4

### where V is the ratio of frozen specific heats; V will be assumed

### constant. Both the flame and the shock wave will be treated as

### discontinuities in the present exercise, since their thicknesses are both

### very small indeed on the scale of A C . We therefore use the conservation

### equations in Lagrangian form without viscous, heat-conduction and mass-diffusion

### effects, as follows:

**f****•¥****f****KK****-Si O ^****^-^**

### Vu^ 4- > ^ O , 2.6

### Y^T . V^Q.^

* '(S'-^\*

### -o. 2.7

**E Q ^ ^ H_ ^)c^JE:(;\-v/r)]^ ^ o > 2.8 **

*^*

### = ^ T . 2.9

### In order, the equations describe conservation of mass, momentum, energy,

### reactant species and, finally, the thermal equation of state. The quantity

### Q , defined by

### is the ratio of the available energy of combustion to a typical thermal

### enthalpy of the system.

3. Boundary Conditions

There are three boundaries in the present problem, at the piston or

contact surface, at the flame, and at the shock. At the piston* yy =. o *

and

* > 3.1

where* \^lk^* must be prescribed.

For any given locus* -K. ~ "x.Ak)* it can be shown with the aid of

(2.3) and (2.5) that

**JI^**

**JI^**

** 'i^ ^^**

**'i^ ^^**

** ^ Ï 3.2 . **

where '</',,/' ^ w are y ,* fi* and \A evaluated at* 'x.J\r) .* Thus

*df }(kt* is proportional to the velocity of the* "x..* locus relative to the

local gas velocity. If the locus •ac» represents a particle path, as it

does in the case of a contact surface,* d'x.. f^* and u.. are simply two

ways of writing the same thing and* yj.* is then a constant. When the solid

piston is replaced by a contact surface the latter can therefore still be

situated at 'y/ = O j the conditions that must replace (3.1) at such a surface

express continuity of gas velocity u. and pressure p , namely

lA^o-H^f^ = v A ( o - , e )* ^* w^('<r> , 3.3a

f./G-«-,t) = > / o - > ^ ) ^ 7^^(:t> , 3.3b

in an obvious notation.

The relation (3.2) must apply on either side of the discontinuity that

represents the shock wave, where* y.* is equal to* yy (f)* and oc. is acCt-*^;

therefore

where* P* and u are dimensionless density and gas velocity immediately

behind the shock, and* P* is dimensionless density in the stagnant gas

### ^

* (^' A\:'*

### A ' ^ t 3.5

o o### whence it follows that (3.4) with (2.4) gives

### The relationships between pressure and density ratios and M for normal

### Rankine-Hugoniot shock waves with fixed or frozen chemical composition (e.g.

### Liepmann & Roshko, 1957, §2.13) can be used to show that

### where M.^- is the initial value of M . It will also be necessary to know how

**St s •' **

### pressure and density behind the shock, namely

* -^ ^ f*

### , change with

### changes in NA ; using Liepmann & Roshko (1957) again it can be shown that

**^**

**^**

** - ^ ^ ^ N ^ ~ ' ^ / { ^ ^ ' < ^ ^ - ' ^ . 3.9 **

### We now apply (3.2) to either side of the flame locus oc c

* "^Jt)*

### j

V T

### in the notation defined in Fig. 1 this means that

### Now

* Av. l^t*

### is the speed of the flame over the fixed 'ground' in the present

### case and we can evidently define the propagation Mach number of the flame as M ,

### where

### Air ** t ^ 3.11

### and

* ^*

### is the dimensionless sound speed immediately ahead of the flame. Thus

### (3.10) makes

**i(cK)-'^u^< **

### or, since pressure will be assumed to be the same on either side of the

### flame in the present model,

### % -

### ^

### l

### » -^

* \('^i/^^)-^]^^^(. '*

### 3.13

### This equation, together with

### \ " ?v.

* '*

### 3.14

### will constitute the appropriate statement of conditions at the flame.

### In the present case the flame is assumed to convert chemical energy

### into thermal energy in a constant-pressure constant-enthalpy process, so

### that

**^i = \ - -."^ • **

**4. Initial Conditions; The Initial Problem **

**It is one thing to remark that, as tr-^o-*- the initial state **
**approaches that of an ideal "conical" field in which quantities depend **
**only upon the values of -x/t , and quite another to establish the **

**character of such a field. To start with we imply that the**** (• \ k C**** )-(-icWs» **
**defined in Fig. 1, are spatially uniform in the t-^O-f limit. Thus **

**and **

**Also, since**

**it follows that**

**VA, . = OL . • w. . ^ lA . 4 . 1 a , b**

### Cl = %i =^ ' •

### 4.1c

*4.2*

**<K****"• = T****u«. 4 . 3**

### whence (3.13) gives

4.4**This relation determines the value of the dimensionless velocity immediately **
**behind the shock wave as t -> 0 + , provided that u. . , M , ^ "T .**** ^^^ **

**^ i f-i b t **

**all either given or are deducible. **

**In view of the simple model of the flame outlined in the previous sections, **
**and exemplified in (3.15), it is clear that **

**\ i**

**\ i**

** '**

** i-^Q **

**4.5 **

**since **

**'^'w.l - > > 4.6 **

**by definition. Thus T . is known since we presume that CJ) is specified. **
**For the present we simply state that **

M ^ ^* K^^}S(^B/I.T^)* 4.7

**where n is a known pre-exponential number and E is defined in (2.2). **
**For simplicity in what follows A will be assumed to be a constant, so **

that

and M ^ is then known since all of the quantities on the righthand side

of (4.8) are known. With tt. specified in the case of the piston-driven

situation described in §3 it is clear that i^^j^ in (4.4) can always be found.

Then (3.7) relates this value of «A . to the initial shock Mach number M _ . ^

namely

**V . ^ 2K-'^f(^>^M^^^^ii.,>y(MMi.o] . **

### 4.9

In setting out to choose appropriate initial conditions, that are bothconsistent with the ideas laid out above and physically sensible, it is

important to remember that all quantities are rendered dimensionless in terms

of the set of quantities that apply immediately downstream of the initial

shock. It is therefore E . and not E that is the same for a given

dimensional piston speed and various flame speeds or values of A in a given

combustible medium, for example, since the shock strength and hence i. will

change from one case to another. Likewise u.. will be different for each

different A-value or flame speed, even though the dimensional piston speed w.'.

is chosen to be constant, since* ^\* will differ from case to case in a

similar way.

For the results to be presented below we have chosen situations that

consist of several values of the constant A allied to a number of values of

the real dimensional piston speed. Since A can be taken in a general way to

indicate different speeds of flame propagation at a given T . value, one can see

that A is an empirical number that could be used to distinguish between

laminar and turbulent modes of propagation, for example.

It will be assumed that Q defined in (2.10), has the value 2. when T .

is equal to fcooitjthis is the temperature found behind a shock of Mach number

M . equal to S'S", propagating into an atmosphere whose temperature T ^ is

in (2.2), is So, which corresponds to a fairly typical value of 35.77

kcal/mole for E . Since (2.10) and (4.5), together with the numbers

just described, make

we can use the results of elementary normal-shock theory to show that

*\ l ^* ^ + i4-4SM'^[2.-^M^-0'4]"'[^2. + o-4M^*. ] " ' 4.10

for the chosen situation. In a similar way

The dimensional piston speed is given by w. A ' or, equivalently, by

>«• Si

4.12

where the last equality defines u . ; it is then IA that must be kept

constant if piston speed is to remain similarly fixed.

We can now adopt the following simple procedure to ascertain the

conditions that must exist in the initial "conical" flow fields described in

the Introduction. First choose values for «A. and for M . . and then

calculate u-^. from (4.12) and w.,. from (4.9), as well as T. . ^ E

from (4.10) and (4.11). Combination of (4.4) with (4.8) makes

### ( \ r ' ) A - T ( - E / i \ - > =. V. " V 4.13

. A

and A can then be calculated for the selected VA^. and M . values.

P* St

Only positive values of A are acceptable, of course, so that it is necessary

to have

as can be seen from (4.9) and (4.12). Thus NA..> | is necessary in any

event and the least allowable M .-value rises as w. increases, as one

£•1 p-<.

expects from elementary physical ideas. Fig. 2 displays the way in which E ,

values of the constant A , including the "no flame" value zero. When Q

in (2.10) is equal to 2 ^T^^is 600K the group* <^'.Q/C'* is equal to Hot) (C ;

in other words

***' ^**** cU,iM^**** 4.14 **

and this is assumed to be constant throughout the examples illustrated in

Fig. 2. The values M " ^ S«2i> for E ^ T that are marked on Fig. 2 are

those appropriate to a flame travelling into the undisturbed atmosphere.

In reality any flame must be preceded by a shock-wave so that the flame-speed

worked out from ambient undisturbed atmospheric conditions will be somewhat

lower than its true, post-shock, value. This is illustrated on Fig. 2 for

the single case of A =• ^ &< ^A . = O .

Finally in this Section we remind the reader that selection of an

A 1 i n i t i a l piston-speed value w. and flame-speed number /\ w i l l be

*s u f f i c i e n t to specify the values of a l l of the q u a n t i t i e s M^> > T i . T - k ^ *

»t Vi. 4i

that are required for calculation of induction-domain behaviour in the sections

### 5. Perturbations and Conditions at the Flame

### In the induction domain, where conditions are denoted by a subscript-<5\.

### (see Fig. 1 ) , the various quantities are written as perturbations of order C

### from their initial values as follows:

**/.>**

** ^ ^ , . .**

**^ ^ , . .**

** -<•'> 5 . 1 **

### Similarly, in the domain C ) , (Fig. 1) between flame and piston we write

### The thermal equation of state (2.9) defines

* f^.*

### via the relation

**f**

**f**

** -r **

**-r**

### Since

**Vfc 4UA. • ^ **

### 5.2

**r, -r = . . **

### (3.13) can be written in the form

### where NA. has been written as

### Mf^ = M ^ 4- A M f . . 5.6

### Making use of (4.4), (5.5) gives

### and it is evident that there must be some close connection between <r', A M ^ ,

### and also M , . if the present formulation is to be both consistent and interesting,

### 5.8

The burnt-gas temperature is given (3.15); combining this equation with(4.5) and using the series (5.1) we find that

Then (5.8) and (4.7) make

where (5.6) shows that* tiA* is M^. times the terms in the brace brackets f ]

in (5.8). Therefore when o-E is about one A M - is of the same order

of size as N^,. , which we shall assume is of the same general magnitude as* C, *

These various facts can be summarised by writing

### <S- ^

I / E### 5.10

5.9

& , f r o m ( 5 . 7 ) ,

**(A ** **=s K ('N **

**Uè» **

**Au. **

### 5.11

*where h\^ , defined as follows *

**H = (\;-'^£^nffK4"^ **

**"S."'*(T^-'V"'-**

**OJi^****^Tj **

— » _{5.12 }

has a magnitude roughly of order unity. Thus (5.12) is the appropriate

translation of condition (3.13) into the present perturbation terminology; the

accompanying condition (3.14) goes over directly to the statement

### (0

**AVk **

**- r **

**- r**

**It)**### 6. Behaviour in the Burnt-Gas Domain

### When equation (5.2) are substituted into the set (2.5), (2.6) and (2.7)

### we find that

### 'At ^ii

* diyy ^ *

### In similar fashion (2.9) gives

('> _r- <o ^ 0 >

### and it then easily transpires that

*T . ip... = 0?. . 6.5 *

### where

*t l '/>ltt IcKy/y ' *

### ^ \ < Y ' Q* Mt 6.6

### The solution of (6.5) must obey conditions at both flame and piston.

### At the latter location y/ = 0 and (3.1) and (5.2) together can be

### interpreted as

### ^ ^ ' T» 6.7

### where u"^ is some given funciton of time. As a consequence of (6.7),

**where f is an as-yet unknown function. **
**Since **

**; [ t ^ - ^ < ^ '-^**

**; [ t ^ - ^ < ^ '-^**

** fu u <: > **

**and since both**** P ^ A.**** are of orderunity while M» is of order 5", or i/E , **
**it follows that y ^ is of order c" . The dimensionless time t will **
**be found to be less than one during the interesting intervals of time in **
**in the present study so that, evaluating (6.8) and (6.9) at y . - in order **
**to apply conditions at the flame, it will be sufficient to approximate to **
**these relations in the forms **

**vc^' - iA^'Ytr^) - lA''Vt) , 6.10 **

' ON

**6.11 **

**In other words the space between flame and piston is predominantly one **
**within which gas velocity is uniformly equal to the piston velocity, and pressure **

**is also spatially uniform and changing only with time. The actual way in which **
**•p varies with time can only be found after behaviour in the induction domain **
**between flame and shock has beeen analysed. **

7. Behaviour in the Induction Domain

The set of equations (5.1) in (2.5), (2.6) and (2.7) can be shown to

lead to the results

**7.1 **

where
**<f^ = ^^1) - 4 ^ [ »H(6rvï'.i>i^^. **

7.2
and
**? = t ~ **

**r **

**s| = t + **

y-7.3
7.4
7.5a,b
**<•>**

**c«>**

We have already seen in (5.11) that IA ' is given by the sum of u

and A u , defined in (5.12); with (6.10), these results combine to give

To be consistent with the derivation of (6.10) and (6.15) it is clear that

one must find u (»>

Ats.

It then follows that

from (7.1), (7.2) and (7.3) by making* y =:. O* , so that

### T = t -

### v^

### .

### 7.7

where IA is the equivalent piston-speed increment, defined in (7.6), thatamalgamates the influence of a real change in piston speed IA*'^ with the

equivalent "displacement" influence of the changes in flame speed that occur

as a result of developments in the induction domain.

In order to find the function f one must use the condition at the

shock wave, where

with sufficient accuracy in the present case. The constant K is given by
**A **

**I/a **

The requisite condition at the shock is easily shown to be

### 7.9

where

7.10

7.11

and this condition can be translated into the requirement

that must hold when

As a consequence f obeys the following functional equation,

### 7.12

### 7.13

**ItVI Itvi**where

### 7.14

**1^ = ll^r)l(\-^>r) « I . **

**1^ = ll^r)l(\-^>r) « I .**

_{7.15 }

The extreme inequality in (7.15) can be verified by direct calculation using

(7.11).

The functional relation (7.14) can be solved by assuming that

### f = f

* + ±fC(-^*

### 7.16a

V(.B(

whence it readily follows that

*fill) = -^<ri> -4-'^r^r{Üi^>i^^ ' *

**r ' •* k.v^**

7.16b

7.16c

**7.18 **
**Each of the functions**** ^ (\)**** represents a disturbance reflected between **

**the shock-wave and the flame surface in the time interval O i t"^ ^ • **
**The use of (5.1) in (2.5) - (2.9) leads to the equation **

**or, using (7.4) and integrating once with respect to the time t , to the **
**relation **

**The function of integration F can be found by evaluating (7.18) when **

**In view of (7.1), (7.2) and (7.7) this means that **

**Combination of (7.18) with (7.1), (7.2), (7.7) and (7.19) together with **
**(7.16) leads to an integral equation for the function 9 . Formally it is **
**the same as the one derived by Clarke (1978), the iterative numerical solution **
**of which has been fully described by Cant (1984). Some of these solutions have **
**been discussed by Clarke and Cant(1984) for piston-alone cases and by Cant(1984) **
**for additional piston-alone situations as well as for contact-surface-driven **
**flows. The general iterative method used here to obtain solutions for 9 **

**is the same, and therefore needs no amplification. However, the presence of the **
**flame surface does introduce one new element into the problem which must be **
**described before some results of the numerical evaluation of 9-" and their **
**consequences are discussed. **

### The flame-provoked velocity increment A I A that appears in u""'

### (see (7.6)) depends upon both © J^ c"'' and these quantities must be known

**AIA. a.\A. **

### before A > A can be calculated. Since subscript-A implies evaluation at

*y = y^^ s.C>(^)*

### it is evidently accurate enough to evaluate 9- ^ c-^'"*

### at y^s-O for any time t in order to find 8^

* i*

### c-^' • In particular, when

### 1/- fc O and therefore ^ a:t =- vj , (7.19) gives

### From (7.16)

* ^ *

### But u ^ % ^ is zero by hypothesis. Since 8J^'^ s ^ " V ^ , o ) , <t-J^ = c^'Vt.o)

### are also zero when t is zero it follows from (5.12) that A u (^o) is

### zero; thus u^' (o> vanishes and so, finally,

**rh)**

**rh)**

** = o **

### too. Therefore (7.18) with (7.1) and (7.2) makes

### having used (7.7) to eliminate d(0 • But equations (7.16) show that

### where Ic is defined in (7.13), and so

### If one now uses (7.6) and (5.12) this last relation can be written in the

### form

and this relation is almost in a form that provides an integral equation to

solve for ö J ^ V O , given* ^A.^\t) , \ s ^ y ^ , ^ k* W ; it only

remains to eliminate «^'^ .

The species equation (2.8) can be integrated once in the present

small-disturbance situation (cf (5.1)) to give

### t

A

since <:. vanishes at the shock. In view of (4.5) it now readily follows

that

Combining (7.20) and (7.21) gives

and this is now in a form that can be solved iteratively numerically to give 8^'\

The C* 2* symbols identify integral paths as indicated in Fig. 3, the lettering

of which corresponds to an earlier sketch of the y,t plane given by

Clarke and Cant (1984).

If integration of the system is complete up to and including time t )

and it is required to advance to a new time t-•-At , note first that integral

C B ^ . O F J is known only as far as H and EAP^cF] is known only as far as J >

where J is the first mesh point to the right of H at point -y-= 0 , t .

It is therefore only necessary to estimate the value of G at (T,!*)

in order to find the resultant increment in the integral C B T * ^ O F 3 • Since

& will be known at J it follows that integral 11^^)^^3 can be

incremented in the same way, where we note that if the increment in t between H

8«* (T\^)* is At the increment in* %* is* T^^t* . The estimated increments are

then substituted into the right-hand side of (7.22) which is used to calculate

. <•>

At very early times 9" will itself be small and so (7.22) reduces to

Thus & ft-") - 1 is easily found for* t <:< \* if one can also deal with the "51.

or reflection term. In view of the fact that* k* is small it is sufficiently

accurate to start the solution for very small times by ignoring the X - t e r m

al together.

With the procedures just outlined Aw., can be calculated at each new

time step and the whole 6-field is then evaluated in exactly the same way as

8. Results and Discussion

We choose one case from the variety of situations covered by the parameter

values in Fig. 2 in order to illustrate detailed behaviour. Since it is broad trends

rather than fine detail that we wish to bring out with this analysis a single

case will be adequate for this purpose.

A ,

With «A chosen to have the value l-O and A the value 4 0 the

initial shock Mach number M . is "l.^f. flame Mach number M . . is 0.^9-,

St. t-i '

whilst activation energy number E and flame temperature T,. are SS-OS" i\ i-94

respectively. The flame speed relative to the gas ahead of it is l4'-Cvwv./s

under these conditions, so that it is essentially a turbulent flame propagation

mode that is being modelled here. One must remember that the flame is

propagating into gas whose temperature has been raised by shock heating to roughly

*(oiO\C \* temperature behind the initial flame is therefore I^^LO VC •

Figs. 4 to 9 inclusive show, respectively, perturbations of temperature,

pressure,density, gas velocity, fuel mass fraction and shock Mach number.

Note that density, unlike the other flow variables, is presented as curves of ^ '

versus t for given* y^* values, in order to make behaviour near 'final' times

easier to appreciate. Numerical calculations proceeded until the value of the

temperature-perturbation number first exceeded the value of the activation

energy number E ^ at which time induction was deemed to be concluded and ignition

imminent. The rationale for this decision is precisely the same as that which

underlies the similar view of events in a simple Semenov explosion (e.g. Strehlow,

1969). Indeed, earlier studies (Clarke & Cant, 1984; Cant, 1984) bring out

the close resemblance between behaviour in the element of gas whose 6-^

value first exceeds E in the present spatially nonuniform environment and

behaviour in the spatially uniform situation that exists in Semenov's model of

a thermal explosion. Fig. 10 is a plot of t, .measured as a fraction of the

M and flame-speed parameter A (cf Fig. 2 ) . Although fcs itself does

vary with these conditions the variations are mild and generally speaking

ignition times are about 80% of u .

The most significant feature of the present predictions is the

association of increases in local pressure with increases in local

temperature (cf Figs. 4 and 5 ) . In the domains within which ignition is about

to take place,namely near* y^^Oy* Fig. 6 shows that density actually decreases

somewhat from its initial value. Hence it is evident that local chemical heating

of the gas is raising local pressures as a consequence of inertial confinement.

That these elevated local pressures propagate signals throughout the domain of

interest is clear from the character of the differential equation satisfied by

w- . This equation is not given here explicitly, but it is fully discussed

in a recent essay on wave propagation in chemically active mixtures (Clarke,

1985). The most important consequence of this propagation of pressure signals

is the steady growth in the strength of the leading shock wave, as can be

seen from Fig. 9. Growth of shock strength in its turn leads to a general

elevation of pressures, densities and temperatures within the induction domain

The present perturbation theory therefore reveals key elements in the

physics of shock-generated ignition and in particular, of course, it makes

plain the links that exist between the various elementary processes.

It is now crucial to appreciate that this shock-induced ignition event

takes place just ahead of the flame. In the absence of additional disturbances,

generated from inputs other than the ones that have been described or implied

here, the fact that ignition takes place in the very-close neighbourhood of the

flame sheet must be taken most seriously in the light of the observations made

by Urtiew and Oppenheim (1966) of what they call the "explosion with an

explosion". Of course there can be no ignition behind the flame, since all of

the fuel species has been consumed in those regions in present circumstances.

However, it is not therefore trivial to remark that ignition occurs ahead of the

fact that the flame exists at all implies an ignition event of some kind at

some stage; in the present model the implication is that this first ignition

event takes place at the initial instant of time, together with the birth of the

initial shock wave. The re-ignition event, whose existence constitutes one of

the most important conclusions to be drawn from the present analysis, is

evidently of a very different character from the implied initial occurrence.

Its direct association with a growing pressure field and the consequent

amplification of the precursor shock implies that locally high rates of chemical

power deposition into the system are about to occur. The relationships between

power deposition and compression wave strength have been considered in some

detail by Clarke, Kassoy and Riley (1984a,b) for inert gases and, more recently,

by these same authors (1985) for a combustible medium. It is instructive to

References

Cant, R.S. (1984) "Gasdynamics and ignition behind a strong shock wave".

PhD Thesis, April, 1984, Cranfield Institute of Technology.

Clarke, J.F. (1978) "On the theoretical modelling of the interaction between

a shock wave and an explosive gas mixture". CoA Memo No 7801,

Cranfield Institute of Technology.

Clarke, J.F. (1985) "Finite amplitude waves in combustible gases". Chapter V

in "The Mathematics of Combustion" Ed. John D. Buckmaster, SIAM, Pa.

Clarke, J.F. and Cant, R.S. (1984) "Nonsteady gasdynamic effects in the

induction domain behind a strong shock wave". Prog, in Astro. &

Aeronautics, 95, 142-163.

Clarke, J.F., Kassoy, D.R. and Riley, N. (1984a,b) "Shocks generated in a

confined gas due to rapid heat addition at the boundary. I. Weak

shock waves. II. Strong shock waves" Proc. Roy. Soc. (Lond.), A,

393, 309-329, 331-351.

Clarke, J.F., Kassoy, D.R. & Riley, N. (1985) "On the direct initiation of a

plane detonation wave." (Submitted).

Jackson, T.L. and Kapila, A.K. (1985) "Shock-induced thermal runaway". SIAM J,

Appl. Math., 45, 130-137.

Liepmann, H.W. and Roshko, A. (1957) "Elements of Gasdynamics." John Wiley,

New York.

Strehlow, R.A. (1969) "Fundamentals of Combustion." International Text

Book Co., Scranton, Pa.

Urtiew, P.A. and Oppenheim, A.K. (1966) "Experimental observations of the

transition to detonation in an explosive gas." Proc. Roy. Soc.

Figure Captions

Fig. 1 Configuration and definition of the meaning of subscripts. In

the Lagrangian space «.'('t') coincides with Y ' = ^ ^nd «,',«'

with Y' ,* y* respectively.

Fig. 2 Dependence of activation energy number E , initial burnt-gas

temperature* T* .initial shock Mach number M . and initial flame

**ki '**** «t **

Mach number W,. on piston-speed number tc . for various

pre-exponential numbers A (cf (4.7)). Ambient temperature

T" t 2Jto« 6 4 t,chemical temperature rise (<:(-C4-K^^ = (IffO VC ,

activation energy E / = IS"* T"? kcal/mole and the speed of sound

ahead of the shock is 340 m/s.

Fig. 3 The V ) t plane and various features, described in the text,

that help to calculate conditions immediately ahead of the

accelerating flame sheet.

Fig. 4 Temperature perturbations versus y for various times up to the

end of the induction-time interval for the case u- .= ' " 0 , A = 4 ^

(see text).

Fig. 5 Pressure perturbations for the conditions of Fig. 4.

Fig. 6 Density perturbation versus t for various* y^ {y-s^O* is the

approximate location of the flame sheet); conditions as for Fig. 4.

Fig. 7 Gas velocity perturbations for the conditions of Fig. 4.

Fig. 8 Fuel mass-fraction perturbations for the conditions of Fig. 4.

Fig. 9 Increase in precursor shock-wave Mach number versus time; conditions

### 10 Time to conclusion of the induction events t. as a fraction of

;

### the time t defined in (2.1) versus initial shock Mach number for

### various values of pre-exponential number

* k .*

### Calculations cease

### at the last time step before 9 locally exceeds E ; as a consequence

### ignition will be heralded at some time within the compass of the various

### symbols each one of which denotes a particular set of conditions.

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