mark short and philip a. blythe- structure and stability of weak-heat-release detonations for finite...

8/3/2019 Mark Short and Philip A. Blythe- Structure and stability of weak-heat-release detonations for finite Mach numbers

1/15

doi: 10.1098/rspa.2001.0936, 1795-18074582002Proc. R. Soc. Lond. A

Mark Short and A. Blythe Philip

Mach numbersrelease detonations for finheatStructure and stability of weak

Email alerting serviceherecorner of the article or click

Receive free email alerts when new articles cite this article - sign up in the box at the top right-h

http://rspa.royalsocietypublishing.org/subscriptionsgo to:Proc. R. Soc. Lond. ATo subscribe to

This journal is 2002 The Royal Society

on October 22, 2010rspa.royalsocietypublishing.orgDownloaded from
http://rspa.royalsocietypublishing.org/cgi/alerts/ctalert?alertType=citedby&addAlert=cited_by&saveAlert=no&cited_by_criteria_resid=royprsa;458/2024/1795&return_type=article&return_url=http://rspa.royalsocietypublishing.org/content/458/2024/1795.full.pdfhttp://rspa.royalsocietypublishing.org/cgi/alerts/ctalert?alertType=citedby&addAlert=cited_by&saveAlert=no&cited_by_criteria_resid=royprsa;458/2024/1795&return_type=article&return_url=http://rspa.royalsocietypublishing.org/content/458/2024/1795.full.pdfhttp://rspa.royalsocietypublishing.org/cgi/alerts/ctalert?alertType=citedby&addAlert=cited_by&saveAlert=no&cited_by_criteria_resid=royprsa;458/2024/1795&return_type=article&return_url=http://rspa.royalsocietypublishing.org/content/458/2024/1795.full.pdfhttp://rspa.royalsocietypublishing.org/subscriptionshttp://rspa.royalsocietypublishing.org/subscriptionshttp://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/subscriptionshttp://rspa.royalsocietypublishing.org/cgi/alerts/ctalert?alertType=citedby&addAlert=cited_by&saveAlert=no&cited_by_criteria_resid=royprsa;458/2024/1795&return_type=article&return_url=http://rspa.royalsocietypublishing.org/content/458/2024/1795.full.pdf


2/15

10.1098/rspa.2001.0936

Structure and stability of weak-heat-release

detonations for finite Mach numbers

B y M a r k S h o r t1 a n d P h i l i p A . B l y t h e2

1Department of Theoretical and Applied Mechanics,University of Illinois, Urbana, IL 61801, USA

2Department of Mechanical Engineering and Mechanics,Lehigh University, Lehigh, Bethlehem, PA 18015, USA

Received 12 September 2001; accepted 16 October 2001; published online 29 May 2002

The stability of an overdriven detonation wave for a one-step Arrhenius reactionin an ideal gas is examined in the limits of a finite detonation Mach number, smallmodified heat release (1)and large activation energy , limits that are relevant tothe generation of the regular cellular detonation structures observed experimentallyin highly diluted mixtures. Here, is the specific heat ratio and is the formationenergy. The limiting case, = 0, corresponds to a finite Mach number reactionlessshock, which in an ideal gas is always stable. Nevertheless, for small but finite valuesof (1), we will show that detonation instabilities can arise when perturbations inthe reaction rate generated by a small disturbance at the shock front are sufficient inmagnitude to balance the corresponding acoustic fluctuations that are also generated.These acoustic disturbances are also those that govern the stability of a reactionlessshock ( = 0). Such reaction-rate perturbations are only generated for sufficientlylarge activation energies, and it is the precise magnitude of the activation energy thatleads to detonation instability that we identify here for various choices of and . Inparticular, subject to the restriction on the modified heat release (1) 1, threenew weakly exothermic ordered limits are identified that are relevant to detonation

instability: (1) ( 1) = O(1) and ( 1) = O(1); (2) ( 1) = O(1) and( 1) = o(1); and (3) ( 1) 1 and ( 1) = o(1).Keywords: detonations; stability; partial differential equations (PDEs)

1. Introduction

Gaseous detonations are chemically propagating compressible waves consisting of ashock front that is driven by volumetric expansion induced by chemical reaction ofthe shocked material. The steady one-dimensional structure of a detonation wave isdue to Zeldovich, von Neumann and Doring and is known as the ZND wave. Theminimum sustainable steady detonation speed is the ChapmanJouguet (CJ) deto-nation velocity, which, for an irreversible exothermic reaction, is the speed at which

the equilibrium or burnt zone flow is exactly sonic relative to the lead shock wave.Detonation waves travelling above the CJ velocity are called overdriven. Typical det-onation speeds in gases are of the order of 5002000 m s1, with peak pressures of theorder of 10100 atm. For a one-step reaction of Arrhenius form in an ideal gas, the

Proc. R. Soc. Lond. A (2002) 458, 17951807

1795

c 2002 The Royal Society

http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/


3/15

1796 M. Short and P. A. Blythe

detonation structure is characterized by four parameters: the ratio of specific heats;the chemical heat release; the activation energy of the reaction; and the detonationMach number (Fickett & Davis 1979) or, equivalently, the detonation overdrive.

In practice, the idealized stable planar detonation waves described above are sel-dom observed. For instance, detonation waves propagating in a rectangular channelreveal a much-publicized pattern of three-dimensional cellular instability. These arisefrom intense regions of vorticity around shock intersection or triple points associatedwith the unstable detonation structure. The triple points arise at the intersectionof incident and Mach stem shocks, which entail the normal detonation front, witha reflected shock, which propagates transverse to the detonation front. The vortic-ity around the triple points causes their locus to be etched onto soot-covered foillining the channel walls, revealing the cellular or diamond instability pattern. Bysimply varying the initial pressure or mixture fraction of the reactive material, thecells are observed either to be very regular or highly irregular in nature (Fickett &Davis 1979). An ability to understand the reasons why these instabilities occur andto model their development will be important to any practical system that uses therapid energy conversion rates of detonation fronts. A relevant example that is con-currently enjoying much attention is the operation of the pulsed detonation engine

(Kailasanath 2000).Being the simplest unstable structures, many experiments have been conductedon the laboratory scale on the mechanisms which generate the regular cell patterns.These are typically observed when the detonation propagates at some finite near-CJMach number, and the reactant material is highly diluted by as much as 90% withinerts (Fickett & Davis 1979). Irregular structures are observed by decreasing thepercentage of diluent, thereby increasing the overall heat that would be released bycomplete chemical reaction of the mixture. Thus, from an experimental viewpoint,two-dimensionally stable waves might be expected to occur in the limit of very lowoverall heat release. Precisely this trend was confirmed by the early theoretical resultsof Erpenbeck (1964), who conducted a multi-dimensional linear stability analysis ofa planar steady detonation for a one-step Arrhenius reaction. He found that forparameters typical of gaseous mixtures, stable waves can only exist for sufficiently

low values of the heat release. To give a specific example, it is convenient to definequantities Q and E to represent the chemical heat release and activation energy,respectively, scaled with respect to the thermal enthalpy of the unshocked material,or

Q =Q

RT0, E =

E

RT0, (1.1)

where Q represents the dimensional heat release, Ethe dimensional activation energy,R the gas constant and T0 the dimensional temperature of the ambient unshockedmaterial. Erpenbeck (1964) showed that for an activation energy E = 50, typicalof gaseous explosives, a ratio of specific heats = 1.2 and a detonation overdrivef = 1.2, the detonation is stable to two-dimensional disturbances only when the heatrelease is below Q = 0.3.

Although the general detonation stability problem is remarkably complex, from atheoretical viewpoint we have reason to believe that analytical progress on this prob-lem can be made in the asymptotic limit of Q 1, or low heat release, particularlyfor parameters near the neutral stability boundary, where weakly nonlinear stability

Proc. R. Soc. Lond. A (2002)



4/15

Weak-heat-release detonations 1797

analyses combined with the Q 1 limit could be applied. Additionally, because thetransverse wave structures would be weak in such cases, our understanding of thetransverse wave collision process should be greatly improved. On the other hand,from an experimental viewpoint, predicting parameters for generating stable waveswould greatly help our understanding of many detonation phenomena, for example,the detonation-failure problem, without the additional complications of instability.

In the present paper, we will characterize the critical magnitudes of the detonationparameters for low exothermic reactions where one could expect to observe a stableor marginally unstable finite Mach number detonation, i.e. we will characterize thehigh diluent regimes that Erpenbeck (1964) predicted would generate the regular cellstructures observed in the laboratory-scale experiments. In particular, we will showthat this occurs only when a sufficiently large activation energy is assumed, so that anon-trivial coupling between perturbations in the rate and energy equations occurs.

The stability of detonations in the limit of weak heat release has also previouslybeen considered in a recent paper by Short & Stewart (1999). Specifically, theyconsidered the problem in which the heat release scaled with respect to the thermalenthalpy of the shocked material is small, or

=

Q

RTs 1, (1.2)where Ts is the immediate post-shock dimensional temperature. Note that the twodefinitions of the heat releases, and Q, differ by the temperature jump across thedetonation shock 2 = Ts /T0, where

2 =(2D2 + 1)[( 1) + 2/D2]

(+ 1)2, (1.3)

which, for an O(1) Mach number D, is itself O(1). When 1, the steady det-onation structure comprises a uniform state to leading order, i.e. the inert shockstructure, with modifications due to small appearing as a first-order correction tothe uniform state. It should be noted that for finite Mach numbers D, 1 isequivalent to Q 1. They showed that for a detonation propagating with a finiteMach number D, with the overdrive f such that f 1 = O(1), and for

( 1)Q 1, E 1( 1)Q , (1.4)

the leading-order linear stability problem was governed by that of a reactionless shockwith an equivalent D, and thus the detonation is always stable (Majda & Rosales1983). Note that not only does (1.2) include the case where (1) = ord(1), Q 1,but also the case where ( 1) 1, the Newtonian limit, with Q = O(1). Thusdetonations with finite D and the product ( 1)Q small, a quantity we refer toas the modified heat release, only require that the activation energy E be much lessthan 1/(1)Q to be stable to two-dimensional detonations. Short & Stewart (1999)does not say anything about the parameters that characterize marginally unstable

detonations in the laboratory-scale experiments described above when D = O(1),and this is the question we address here.

In this paper we will show that in order to induce instability in a detonationwhen the modified heat release ( 1)Q is small, perturbations in the reaction rate




5/15


that are carried along particle paths must be sufficient in magnitude to balance theacoustic fluctuations to the leading-order uniform post-shock state that are generatedfrom perturbations in the shock state. This only occurs when the activation energyof the reaction is sufficiently large. In particular, we will identify three new weaklyexothermic regimes, defined by the relative magnitudes ofQ, Eand 1, of which twoare able to characterize marginally unstable detonations in the weak exothermicitylimit.

2. Model and steady detonation structure

The rate process is modelled as an irreversible one-step Arrhenius first-order reaction,with

DY

Dt= r = K(1 Y)exp

T

(2.1)

where Y is the reaction progress variable, T is the temperature, K is a suitablepre-factor and = E/RTs is the activation energy. Appropriate conservation lawsare

D

Dt+

u = 0,

Du

Dt+

v

p = 0,

De

Dt+ p

Dv

Dt= 0, (2.2)

with caloric and ideal thermal equations of state

e =T

( 1) Y, T =p

. (2.3)

Here, p is the pressure, is the density, v is the specific volume, e is the internalenergy, u = (u1, u2) is the velocity, is the specific heat ratio and is the for-mation (chemical) energy of the unreacted mixture. Density, pressure, temperatureand velocity are made dimensionless with respect to the steady post-shock density,temperature and sound speed (cs ), respectively. The steady half-reaction length (l

)is used as the length-scale, and l/cs as the time-scale.

Alternative values of and , defined in terms of the pre-shock thermal energy,

are denoted by Q and E, respectively (see (1.1)). The model defined by (2.1)(2.3)supports the steady detonation structure

p = a + (1 a)[1 ( 1)bY]1/2,

u1 =(1p)

Ms+ Ms , u

2 = 0, =

Msu1

,

(2.4)

where Ms is the post-shock Mach number, while a and b are constants defined interms of Ms and (Short & Stewart 1999). Correspondingly, the variation in thereaction progress variable is determined by the first-order equation

YX = r/u1, (2.5)

where X is a steady shock-attached coordinate. Based on the dimensionless length-scale, the pre-factor

K =

1/20

u1(1 Y)exp

p/

dY. (2.6)




6/15


In addition, the shock conditions are

= p = T = 1, u1 = M

s , u

2 = 0, Y = 0. (2.7)

The steady structure is completed by specifying the detonation overdrive f defined asf = (D/DCJ)

2, where D is the detonation Mach number and DCJ is the ChapmanJouguet detonation velocity. Note that

DCJ =

1 +

(2 1)

Q

+

1 +

(2 1)

Q

1

1/2 1/2,

Ms =

( 1)D2 + 2

2D2 ( 1)

1/2,

(2.8)

so that through (1.1)(1.3) and (2.8), it can be seen how the shock Mach numberD enters the picture.

A normal-mode linear stability analysis of the steady-state structure defined byequations (2.4)(2.7) is carried out by defining a shock-attached coordinate systemx = X h(y, t), where h(y, t) represents the shock perturbation. Perturbations tothe steady structure take the form

z = z(x) + z(x)et+iky , h = het+iky , (2.9)

for eigenvalue and wavenumber k. In (2.9), the vector z = [v, u1, u2, p , Y ], and z(x)corresponds to the perturbation eigenfunctions. The differential system governing z

has the form

A ,x + ( + ikB +C) ( + ikB)z,x = 0, (2.10)where = z/h and A, B and C are functions of the steady-state defined inShort & Stewart (1998). The system of five ordinary differential equations definedin (2.10) is supplemented by five shock conditions obtained from the RankineHugoniot relations and a single acoustic radiation condition that eliminates distur-bances propagating upstream from a point where the reaction has effectively termi-nated. This system can be solved numerically for specified choices of , , and fusing a shooting method that integrates either from the shock to the burnt zone (Lee& Stewart 1990; Bourlioux & Majda 1992; Bourlioux et al. 1991; Short & Stewart1998) or from the burnt zone to the shock (Sharpe 1997).

3. Stability of weak-heat-release detonations

(a) 1, ( 1) = O(1), = O(1)As in Short & Stewart (1999), we again consider a weak-heat-release limit for deto-nations starting with the assumption that

1, (3.1)so that Q = O() and, from (2.8), DCJ = O(1). For the present, it is also assumed

that the specific heats ratio ( 1) = O(1) and the overdrive f 1 = O(1) > 0;the latter restriction avoids a complex transonic flow problem that occurs for f = 1.Hence

D2 1 = O(1) > 0. (3.2)




7/15


Consequently, the post-shock flow Mach number Ms = O(1). In fact, it is necessaryto choose D such that Ms < 1/

, in order that a steady solution exists. Under

these assumptions, the steady relations (2.4) may be expanded to O() as

p 1 ( 1)M2s

(1

M2s )Y0 ,

u1 Ms + ( 1)Ms

(1M2s )Y0 ,

v(x) 1 + ( 1)(1M2s )

Y0 ,

T(x) 1 + ( 1)

(1 M2s )(1M2s )

Y0 ,

(3.3)

whereY(x) Y0 (x) + Y1 (x). (3.4)

Technically, equations (3.3) will always arise whenever the modified heat release(

1)

1. At this stage, the expansions are independent of the size of the

activation energy . For f = O(1), Short & Stewart (1999) considered an activationenergy 1. In the present case, we will show that the limit

= O(1) (3.5)

should be assumed in order to generate unstable weakly exothermic detonations. It isonly this scaling for which perturbations in the chemical reaction rate are comparablewith acoustic disturbances that govern reactionless shock stability.

Using (3.3), the exponential factor in (2.1) then becomes, to leading order,

exp

T

exp[]exp[Y0 ], with =

( 1)

(1 M2s )(1M2s )

> 0. (3.6)

For the limit (3.5), = O(1). Consequently, after substituting equation (3.3) intoK K0(1 + K1), we get

K0 = M

s exp( )[Ei(12) + Ei()] = O(e), (3.7)where Ei is the exponential integral. Y0 (x) is determined by the implicit relation

Ei[(Y0 (x) + 1)] = Ei()

1 xEi(

1

2)

Ei()+ 1

. (3.8)

In addition, the rate function r r0(x) + r1(x), wherer0(x) = M

s Y

0,x(x). (3.9)

Note that both Y0 (x) and r

0(x) vary smoothly with x, despite the assumption of a

large activation energy. By expanding the steady relations (2.4) to O(2), an expres-sion for r1(x) can also be evaluated.

By substituting the steady relations (3.3) and (3.4) into the exact stability prob-lem (2.10), the leading-order perturbation eigenfunctions 0(x) are found to




8/15


satisfy the leading-order stability problem

v0,x +

Msv0

1

Ms(u10,x + iku

20) = 0,

u10,x +

Ms

u10 +1

Ms

p0,x = 0,

u20,x +

Msu20 +

ik

Msp0 = 0,

T0,x +

MsT0 + ( 1)

v0,x +

Msv0

=( 1)

Msr0(x)T

0 ( 1)Y0 exp[(Y0 1)][Ei(12) + Ei()],

Y0,x +

MsY0 =

r0(x)

MsT0,

T0 = v

0 + p

0.

(3.10)

Of particular note is that the reaction-rate perturbation Y is times larger thanthe temperature perturbation T, as can be seen from (3.10), i.e.

Y = O(). (3.11)

It is therefore convenient to introduce a new variable Y such that Y = Y. It isalso readily observed that if 1/, the terms on the right-hand side of the energyequation are o(1). However, when = O(1), all non-zero terms on the right-handside of (3.10) are O(1). Thus, under this limit, the leading-order stability problemhas been reduced to one in which the terms on the left-hand side of (3.10) governthe stability of a reactionless shock, while those on the right-hand side of (3.10)relate to forcing terms that are perturbations in the chemical reaction rate broughtabout by the presence of a large activation energy. In particular, the perturbationeigenfunction for the reaction progress variable Y is O() larger than the remain-

ing perturbation eigenfunctions. Without these reaction-rate perturbations, i.e. for 1/, the stability of the detonation is identical to that of the reactionless shock,which is stable.

In order to establish the location of the neutral stability boundary when = O(1),attention is now confined to a weakly unstable problem and the eigenvalue isexpanded as

i0i + (1r + i1i). (3.12)Formally, to solve this problem, the leading-order stability problem defined by (3.10)must now be supplemented by the first-order problem, one defined at O(). Since weare unable to determine an analytical solution to the leading-order problem, we donot pursue this approach further. However, the asymptotic analysis described aboveclearly identifies the parameter regime required to render unstable a weakly exother-

mic detonation. The approach we adopt is to substitute the steady results (3.3)and (3.4) directly into (2.10), which defines a stability problem valid to O(), andthen integrate these relations subject to the standard shock and radiation condi-tions, where, for consistency, the O() reaction rate r = r0 + r

1 is also substituted




9/15


0.4 0.6 0.8 1.0 1.2 1.4 1.6

k

0

0.001

0.002

0.003

0.004

0.005

Re()

0.4 0.6 0.8 1.0 1.2 1.4 1.6

k

0.40

0.60

0.80

1.00

Im(

)

(a) (b)

Figure 1. Detonation stability spectrum for Q = 0.125, = 1.4, f = 1.2 and E = 50. Dashedlines are obtained by the O() approximation to the steady-state from (3.3). Solid lines areexact numerical calculations using (2.10) (see Short & Stewart 1998).

into (2.10). In other words, we do not explicitly expand the eigenvalues and eigen-functions in . The resultant error terms are then o().

Figure 1 shows the variation of Re() and Im(), with wavenumber k obtained by

approximating the steady structure to O() in the manner described above. Solidlines in this figure represent numerical solutions of (2.10) that are determined withoutusing the weak-heat-release approximation for the steady solution (Short & Stewart1998). Parameter values are chosen to correspond to the limit identified in (3.5) withQ = 0.125, = 1.4, f = 1.2 and E = 50; these imply that = 0.08 and = 2.17,numerical values that clearly approximate the asymptotic regime discussed above. Inboth the approximate and exact cases, there is a single oscillatory mode of instabilitydefined over a finite range of wavenumbers. For the approximate solution, the insta-bility band is defined by 0.566 < k < 1.563, and the maximum growth rate occursat k = 0.884. As can be seen, the approximation for Re() is reasonable, while thatfor Im() is very good. This can be expected from (3.12), since Im() is given by atwo-term approximation, but only the first term in the approximation for Re() iscalculated.

Figure 2 shows the eigenfunction structures at k = 0.566, k = 0.884 and k = 1.563obtained by approximating the steady-state structure to O(). Of particular noteis the large amplitude of Im(Y) in comparison with the amplitude of the remain-ing eigenfunctions, a behaviour predicted by our asymptotic analysis through thescaling (3.11). This is associated with a requirement that in order to generate insta-bility, the leading-order perturbations in the reaction rate must influence those inthe pressure and velocity fields when = O(1).

Figure 3 shows the migration with decreasing Q of the two neutral stability pointsidentified in figure 1. Both the approximate and exact solutions have a critical heatrelease Qc below which the detonation is stable. The estimate from the approximatesolution is Qc = 0.1138, compared with the exact value Qc = 0.1157.

(b) ( 1) 1, = O(1), ( 1) = O(1)At finite detonation Mach numbers, the results (3.3) and (3.10) remain valid,

provided that the modified heat release ( 1) 1. Consequently, it is also of




10/15


0 5 10 15 20x

0.4

0.2

0

0.2

0.4

0 5 10 15 20x

4

3

2

1

0

1

Re('

)

Im(

'

)

5

4

3

2

1

0

1

0.4

0

0.4

2

1

0

1

2

6

4

2

0

2

(a) (b)

(c) (d)

(e) (f)

Re('

)

Im(

'

)

Re(')

Im(

'

)

Figure 2. Approximate perturbation eigenfunction structure showing (a) Re(z) versus x and(b) Im(z) versus x at the low-wavenumber neutral stability point k = 0.566, (c) Re(z) versus

x and (d) Im(z

) versus x at the point of maximum growth rate k = 0.884, (e) Re(z

) versus xand (f) Im(z) versus x at the high-wavenumber neutral stability point k = 1.563 for the singleunstable mode when Q = 0.125, E = 50, = 1.4 and f = 1.2. The curves correspond to v

(), u1 ( ), u

2 ( ), p ( ) and Y ( ).

0.114 0.118 0.122Q

0.6

0.8

1.0

1.2

1.4

1.6

k

0.4 0.6 0.8 1.0Im ()

(a) (b)

Figure 3. Migration of the neutral stability points for E= 50, f = 1.2 and = 1.4.




11/15


interest to examine the limit in which

= O(1), but 1 1 and ( 1) = O(1), (3.13)so that a small modified heat release occurs due to the ratio of specific heats beingclose to one, the Newtonian limit, rather than the actual heat release being small.

In this case, it can be shown that, at the shock,

T = ( 1) 4(+ 1)Ms

D4 + 1

D2(2D2 + 1)

. (3.14)

Hence T0 = O(1). It should also be observed from (3.6) that, with (1) = O(1), = O(1) and therefore, from (3.8) and (3.9), r0 and Y

0 remain O(1). It thereforefollows from (3.10) that Y0 = O(1). Consequently, equations (3.10) can be used toestablish an equation for the pressure perturbation of the form

p0,x +

Msp0 +

Ms(u10,x + iku

20)

=( 1)

M

s

r0(x)T

0

(

1)Y 0 exp[(Y

0

1)][

Ei(1

2) + Ei()]. (3.15)

Finally, from this result, using T0 = O( 1), it can be seen that the term on theright-hand side of (3.15) is O( 1) and can be neglected, so that

p0,x +

Msp0 +

Ms(u10,x + iku

20) = 0. (3.16)

This equation, together with the first three results in (3.10), is identical to the sta-bility problem for a reactionless finite Mach number shock. Thus the detonationstructure is stable in the present limit, our second result.

(c) ( 1) 1, = O(1), ( 1) 1

The results discussed in 3 (b), for which the detonation is stable, are valid for( 1) = O(1), but that description does not remain valid for ( 1) 1, i.e. asthe activation energy is increased further. In this case, the right-hand side in thepressure relation (3.15) becomes significant. Consequently, the description no longercorresponds to that of a reactionless shock, and the possibility of instabilities is againpresent. Again, however, provided that the modified heat release ( 1) is small,equations (3.3) and (3.10) still provide valid descriptions for the steady structureand stability problem if(1)2 1. Consequently, these equations appear to holdwhen

1

( 1) ( 1) 1. (3.17)

In this limit, 1, and on the scale of the half reaction length, x = O(1), thestructure is now equivalent to a square detonation wave. A detailed description of

the wave profile will not be attempted in this paper. Nevertheless, it is possible toshow that for the steady profile, the mass fraction change in the induction zone isO(1), where = ( 1), and the temperature change is O(1). Similarly, thesubsequent growth zone, where Y = O(1), is exponentially thin in . The region




12/15


0 1 2 3 4 5k

0.05

0

0.05

0.10

Re()

0 1 2 3 4 5k

0

1

2

3

4

Im(

)

(a) (b)

Figure 4. Detonation stability spectrum (a) Re() and (b) Im() against wavenumber corres-ponding to E= 50, f = 1.2, = 1.2 and Q = 0.4 (), Q = 0.45 ( ) and Q = 0.55 ( ).

1.5 1.6 1.7 1.8 1.9 2.0

k

0.2

0.1

0

0.1

Re()

1.5 1.6 1.7 1.8 1.9 2.0

k

1.0

1.2

1.4

1.6

Im(

)

(a) (b)

Figure 5. Interaction between two modes. The dashed lines correspond to Q = 0.57,dotted lines to Q = 0.58 and solid lines to Q = 0.59.

corresponding to 1, suggested by the above comments, has been investigatedby carrying out a series of numerical calculations based on the full system (2.10).

Figure 4 illustrates the stability spectrum of a detonation having E = 50, = 1.2and f = 1.2 for three values of the heat release Q = 0.4, Q = 0.45 and Q = 0.55,

corresponding to ( 1) = 0.070, 0.078, 0.094 and ( 1) = 8.72, 8.66, 8.55,respectively. These solutions represent the lowest-frequency modes and are obtainedusing the numerical algorithm described in Short & Stewart (1998). For Q = 0.4,there is a single finite range of wavenumbers over which the detonation is unstable.A neutral stability boundary is first crossed at k = 0.387, and crossed again atk = 1.528. However, at Q = 0.45, there are two bands of wavenumbers over whichthe detonation is unstable. Unlike previously calculated detonation spectra (Short& Stewart 1998), these correspond to the same mode (see figure 4). For Q = 0.45,the neutral stability boundary is first crossed at k = 0.308, and crossed again atk = 1.730. However, the mode also becomes unstable at k = 1.833, generates a finiteregion of instability, and crosses the neutral stability boundary again at k = 3.090,after which no more crossings take place. A similar behaviour is observed for Q =0.55. This behaviour, which has to our knowledge hitherto not been observed in the

context of detonation stability, can be explained as a linear interaction between thelow-frequency mode and a higher-frequency mode.

This interaction is illustrated in figure 5, which shows the stability behaviour forQ = 0.57, Q = 0.58 and Q = 0.59 in a wavenumber region in which the mode crossing




13/15


is clearly observed. For these Q, ( 1) = 0.097, 0.099, 0.100 and ( 1) = 8.534,8.524, 8.514, respectively. Dashed lines correspond to Q = 0.57, dotted lines toQ = 0.58 and solid lines to Q = 0.59. At k = 1.5 in figure 5a, the upper branchescorrespond to the set of lower-frequency modes observed in figure 5b at k = 1.5. Otherlines correspond to the next-higher-frequency mode. For Q = 0.57 and Q = 0.58,figure 5b illustrates a crossing of lower- and higher-frequency modes in frequencyspace, which leads to the lower-frequency mode re-emerging in the unstable space,as shown in figure 5a. When Q = 0.59, however, the frequencies of the two modesremain distinct, and the lower-frequency mode crosses the neutral stability boundaryat k = 1.730 and does not re-emerge into the unstable domain. Alternatively, thehigher-frequency mode, identified by the solid line with Re() < 0 at k = 1.5 infigure 5a, crosses the neutral stability boundary at k = 1.980. Crossings of thistype represent the typical behaviour observed in multi-mode detonation instability(Short & Stewart 1998). Any analytical models of detonation stability in the limits( 1) 1, 1 should be able to capture this mode interaction.

It should be noted that the stable modes shown above are valid in the sense thatour model is not an unbounded one. Rather than apply an alternative condition,which on an infinite domain enforces a boundedness condition at infinity, we apply

an acoustic radiation condition at some large but finite distance downstream ofthe shock at a point where perturbations in reactant mass fraction can be ignored.The different models, or closure conditions, lead to unsteady eigenvalues that areessentially indistinguishable, but our model allows the tracking of stable modes. Theactual point where the radiation condition is applied is unimportant, since calculatedeigenvalues begin to converge rapidly, i.e. the perturbations are acoustically radiat-ing, after a distance of only a few half-reaction lengths behind the detonation shock(Short & Stewart 1998). This condition is also one that is more representative of thecondition that is used in many numerical studies on the nonlinear stability of steadydetonation structures (see, for example, Bourlioux et al. 1991). Here, a non-reflectingboundary condition is applied at some large distance behind the steady detonationshock, so that downstream propagating disturbances arising in the burnt zone arenot reflected upstream. Thus, for initially perturbed stable waves, we should see the

development of spatially growing waves in the burnt zone, but since the calculationdomain is spatially confined and the temporal evolution finite, we need never beconcerned with unboundness. Indeed, this behaviour is precisely that observed byBourlioux et al. (1991).

4. Conclusion

The stability of a finite Mach number overdriven detonation wave has been exam-ined in the limit of weak modified heat release ( 1) and large activation energy, parameters that are particularly relevant to the generation of the regular cellulardetonation structures observed experimentally in highly diluted mixtures. Three par-ticular distinguished limits are considered. In case (a), when

1, (

1) = O(1),

= O(1), a finite band of unstable wavenumbers was determined. For case (b),with ( 1) 1, = O(1), ( 1) = O(1), all solutions were found to be stable.This limit, however, suggested that difficulties might arise when ( 1) 1, andthis situation was studied as case (c). All calculations in case (c) were carried out for




14/15


the full linearized stability problem (2.10) and were not based on any formal asymp-totic approach. The results demonstrated some interesting behaviour, including modeinteractions and multiple stability domains. These findings, and the identification oflimits defining the instability of a finite Mach number, weak-heat-release detonation,are relevant to current research on multi-dimensional instability of detonation cellsand, in particular, help to quantify the regions that lead to the generation of thecells in finite Mach number detonations in highly diluted mixtures.

M.S. was supported by the AFOSR.

References

Bourlioux, A. & Majda, A. J. 1992 Theoretical and numerical structure for unstable two-dimensional detonations. Combust. Flame 90, 211229.

Bourlioux, A., Majda, A. J. & Roytburd, V. 1991 Theoretical and numerical structure forunstable one-dimensional detonations. SIAM J. Appl. Math. 51, 303343.

Erpenbeck, J. J. 1964 Stability of idealized one-reaction detonations. Phys. Fluids 7, 684696.

Fickett, W. & Davis, W. C. 1979 Detonation. University of California Press.

Kailasanath, K. 2000 Review of propulsion applications of detonation waves. AIAA J. 38 1698

1708.Lee, H. I. & Stewart, D. S. 1990 Calculation of linear instability: one-dimensional instability ofplane detonation. J. Fluid Mech. 216, 103132.

Majda, A. J. & Rosales, R. 1983 A theory for spontaneous Mach stem formation in reactingshock fronts I. the basic perturbation analysis. SIAM J. Appl. Math. 43 13101334.

Sharpe, G. J. 1997 Linear stability of idealized detonations. Proc. R. Soc. Lond. A453, 26032625.

Short, M. & Stewart, D. S. 1998 Cellular detonation stability I. A normal-mode linear analysis.J. Fluid Mech. 368, 229262.

Short, M. & Stewart, D. S. 1999 Multi-dimensional stability of weak-heat-release detonations.J. Fluid Mech. 382, 109136.




15/15


mark short and philip a. blythe- structure and stability of weak-heat-release detonations for finite...

Documents