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^
'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 owhence 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 , bCl = %i =^ ' •
4.1c
<K "• = T 4.2 u«. 4 . 3whence (3.13) gives
4.4This 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-^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
-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 / E5.10
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
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 ^ - ^ < ^ '-^
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 where7.14
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)
= 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.
i
£
Piston ^^^^f^,,
x^(t)
Shock
Distance , x '
x;(t')
Figure 1M
SIM,
0.4
0.8
1.2
0.2 0.4 0.6 ^ 0.8
U
Figure 2 Pl "in JU20
10° «^
10-; v ^
10-2^
10"^ 1
hf
1 1A = 69x^
^ ^ ^ 1 • /u^
, 1d^
1 _""^oTl
»x ^
- ^ ^ 10.4
0.3
0.2
0.1
.046<
S
m/ï -- ^ •i'
A = 6 0 /
4 0 ^
1 —y
X
20^
/ /-0 = A .
1.0 1.2 1.4
Shock , K^t
0.4
r
0.7
r
-1.0
.(1)
-1.5
2.0
-- 2 . 5 --J-3.5
J I ' •0 0.1
Figure 80.2 . 0.3
0.4 0.5
c -s
ID