S. D. Watt and G. J. Sharpe- One-dimensional linear stability of curved detonations

8/3/2019 S. D. Watt and G. J. Sharpe- One-dimensional linear stability of curved detonations

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doi: 10.1098/rspa.2004.1290, 2551-25684602004Proc. R. Soc. Lond. A

S. D. Watt and G. J. SharpeOne-dimensional linear stability of curved detonations

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10.1098/rspa.2004.1290

One-dimensional linear stability

of curved detonations

B y S . D. W a t t a n d G. J . S h a r p e

School of Mathematics and Statistics, University of Birmingham,Edgbaston, Birmingham B15 2TT, UK (simon [email protected])

Received 25 April 2003; accepted 30 January 2004; published online 2 June 2004

In this paper, a one-dimensional stability analysis of weakly curved, quasi-steadydetonation waves is performed using a numerical shooting method, for an idealizeddetonation with a single irreversible reaction. Neutral stability boundaries are deter-mined and shown in an activation temperaturecurvature diagram, and the depen-dence of the complex growth rates on curvature is investigated for several cases. It is

shown that increasing curvature destabilizes detonation waves, and hence curved det-onations can be unstable even when the planar front is stable. Even a small increasein curvature can significantly destabilize the wave. It is also shown that curved deto-nations are always unstable sufficiently near the critical curvature above which thereare no underlying quasi-steady solutions.

Keywords: pulsating instability; shock waves;

reactive flow; neutral stability boundaries

1. Introduction

Detonation waves are rapid (supersonic) combustion waves, in which a strong shock

ignites the fuel and the heat released by the burning then drives the shock into freshfuel. Experiments show that detonations usually propagate in an unsteady, unstablemanner (Fickett & Davis 1979). In some cases the front pulsates as it propagates, suchas when blunt bodies are fired into reactive gases (e.g. Lehr 1972), or in tubes withsquare cross-sections (Haloua et al. 2000). In these cases the front speed oscillateswith a regular or irregular period. Much progress in understanding such pulsatingdetonation fronts has been achieved by considering the one-dimensional stabilityof the underlying steady detonation wave, either by linear stability analyses (Lee &Stewart 1990; Sharpe 1997, 1999; Short & Dold 1996), weakly nonlinear theories (Yao& Stewart 1996; Short 2001) or by fully nonlinear numerical simulations (Bourliouxet al. 1991; Short & Quirk 1997; Short et al. 1999; Sharpe & Falle 2000a, b; Short& Sharpe 2002). These previous works consider the stability of a planar detonation

wave. However, in many cases the underlying steady or quasi-steady front is curved,such as at the tip of the bow shock ahead of supersonic blunt bodies. Hence one mayask what effect curvature of the front has on the stability of the detonation wave.

Curvature of detonation fronts is important in many applications, such as in spher-ically or cylindrically expanding detonation fronts (e.g. Sharpe 2000a), detonationsin condensed-phase explosive sticks (Bdzil 1981; Stewart 1998), where expansion of

Proc. R. Soc. Lond. A (2004) 460, 25512568

2551

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2552 S. D. Watt and G. J. Sharpe

the detonation products behind the front causes the front to be curved, or detona-tion diffraction problems where the detonation exits from the end of an open tube(Schultz & Shepherd 2000). The effects of weak curvature on the structure and prop-agation speed of detonation waves has been extensively investigated using steady orquasi-steady quasi-one-dimensional analyses (Bdzil 1981; Klein 1991; Klein & Stew-

art 1993; Stewart & Bdzil 1988; Yao 1996; Yao & Stewart 1996; Stewart 1998; Sharpe2000a, b; Short & Sharpe 2004). In these approximations the front is assumed to beweakly curved and slowly varying, such that the length-scale of the reaction zone ismuch shorter than the radius of curvature and that the front is evolving on a muchlonger time-scale than the time it takes a particle to traverse the reaction zone. Theseanalyses show that curved detonations travel slower than the planar detonation, areof the eigenvalue type with a frozen sonic point inside the reaction zone (Fickett& Davis 1979) and have backward S-shaped detonation speedcurvature (Dn)diagrams, with an upper branch corresponding to propagating detonations and anassociated critical curvature above which quasi-steady detonations cannot propa-gate. Comparisons of the results of the quasi-steady, quasi-one-dimensional analysiswith full, time-dependent numerical simulations of stable curved detonations, e.g. forspherically or cylindrically expanding detonation fronts (Sharpe 2000a) or detonationdiffraction (Aslam & Stewart 1999), are in very good agreement.

In this paper we perform a one-dimensional linear stability analysis of steadilyor quasi-steadily propagating weakly curved detonations with a single, irreversiblereaction. A referee pointed out that a similar analysis to that in this paper was donepreviously in the thesis by Yao (1996). However, Yao (1996) gives results for onlyone parameter set (corresponding to a case where the planar detonation is highlyunstable). The main purpose of this paper is to determine one-dimensional neutralstability boundaries and whether in general increasing curvature has a stabilizingor destabilizing effect on the detonation, including how rapidly the wave is stabi-lized/destabilized as the curvature increases, points for which Yao (1996) did notgive results or discuss. Determining stability boundaries and the effect of differentparameters on the stability of the wave is the major role of a linear stability analysis.

While unstable detonations frequently have a multi-dimensional structure, so-called cellular detonations, if a detonation becomes more unstable in one dimension,it also becomes more multi-dimensionally unstable (Sharpe 1997; Short & Stewart1998); hence it is sufficient to consider the pulsating instability for this purpose.Indeed, this is a philosophy being adopted for determining the effects of parameterson detonation stability for models with realistic multi-step chemistry representingspecific fuels, such as the effect of argon dilution on acetyleneoxygen detonation sta-bility (Radulescu et al. 2002). It is computationally prohibitive to perform accuratemulti-dimensional numerical simulations for realistic chemistry due to the orders ofmagnitude difference between the wavelength of the instability (detonation cell size)and the reaction lengths, the need for very high resolution in order to obtain accurateresults and the very long nonlinear evolution time of the cellular instability (Sharpe

& Falle 2000c; Sharpe 2001). However, if one is merely interested in how a parameteror initial state affects the stability of the detonation, then it is sufficient to performone-dimensional calculations, for which it is currently feasible to obtain accurateresults, of the pulsating instability for different values of the parameter. More irreg-ular oscillations then correlate with a more irregular cellular structure (Radulescu etal. 2002).

Proc. R. Soc. Lond. A (2004)



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Stability of curved detonations 2553

The plan of the paper is as follows: the governing equations are given in 2 and thequasi-steady solutions are described in 3; the linearized equations are derived in 4and solved in 5; the results and conclusions are given in 6 and 7, respectively.

2. Governing equations

Consider a detonation propagating through a reactive ideal fluid with a single, irre-versible reaction. The governing equations to model this case for a curved detona-tion are derived in Yao & Stewart (1996), using intrinsic, shock-attached curvilinearcoordinates. In this paper we consider quasi-one-dimensional, weakly curved detona-tions ( 1), i.e. we assume that the reaction zone is much shorter than the radiusof curvature and that the curvature changes very slowly along the shock front onthe reaction zone length-scale, so that the reaction zone structure depends spatiallyonly on the distance normal to the front, n. Note that for spherically symmetricor axisymmetric cylindrically expanding detonations the problem is completely onedimensional (rather than just quasi-one dimensional), i.e. it depends only on theradial direction (n is then the distance from the shock front in the radial direction).Under this quasi-one-dimensional approximation, the governing equations in Yao &

Stewart (1996) reduce, to O(), to

t+

n(un) + (un + Dn) = 0, (2.1)

unt

+ Dn + ununn

+1

p

n= 0, (2.2)

e

t+ un

e

n

p

2

t+ un

n

= 0, (2.3)

t+ un

n+ exp( /T) = 0, (2.4)

where is the density, p is the pressure, un is the fluid velocity in the direction

normal to the front, Dn is the shock velocity, is the reaction progress variable( = 1 for unburnt and = 0 for burnt), e is the energy per unit mass, given by

e =p

( 1) q(1 ),

is the curvature, q is the heat of reaction, is the activation temperature, and is the rate constant. These equations have been non-dimensionalized such that thedensity in the initial (upstream, unburnt) state is unity, the shock velocity for theplanar ( = 0) front is unity, the unit length is given by the half-reaction length ofthe planar wave (Erpenbeck 1964) and the temperature is given by

T =p

=

c2

,

where c is the dimensionless sound speed. Detonation stability results are usuallygiven in terms of Erpenbecks (1964) scales for the activation temperature and heatrelease, E and Q, i.e. those scaled with the upstream temperature. See Sharpe (1997)for a conversion between these scalings and ours.

In this paper we set Q = 50 and = 1.2, and vary E and .




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3. The quasi-steady solutions

In this section we consider the underlying steady, or more strictly quasi-steady, solu-tions of the governing equations (2.1)(2.4), i.e. we assume that the detonation frontis evolving on a time-scale which is much longer than the time it takes a particle totraverse the reaction zone, so that we are considering the case where both the speedand structure of the wave are slowly varying. Note that, for curved detonations prop-agating in condensed-phase explosives, the underlying wave can be strictly steady inthe front rest frame.

The governing equations then become

d(0un0)

dn+ 0(un0 + Dn) = 0, (3.1)

un0dun0

dn+

1

0

dp0dn

= 0, (3.2)

de0dn

p020

d0dn

= 0, (3.3)

un0 d0

dn+ 0 exp( /T0) = 0, (3.4)

where a 0 subscript denotes quantities in the quasi-steady detonation. Note thatthese are just the leading-order equations when the variables are asymptoticallyexpanded in the slow time-scale (see the appendix).

The jump conditions across the shock are

+un+ = Dn,

p+ p = D2n

1

1

+

,

e+

+p+

++ 1

2u2n+

= e +p +1

2D2

n,

+ = 1,

where the minus () subscript denotes quantities in the upstream, unshocked stateand the plus (+) subscript denotes quantities immediately behind the shock, with

p the smaller root of 2q(2 1) = (1 p)

2.The four equations (3.1)(3.4) can be reduced to one equation,

dun0d0

=

c20 u2n0

un0W0

,

where W0 = 0 exp(/c20),

= [( 1)qW0 + c20(un0 + Dn)]

is the (modified) thermicity and

c20 = p + ( 1)[(D2n u

2n0)/2 + q(1 0)].




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0 0.005 0.01

0.75

0.80

0.85

0.90

0.95

1.00

Dn

Figure 1. TheDn

curves forE

= 20 (dotted line),E

= 25.

26 (dot-dashed line),E = 31.05 (dashed lines) and E = 35.84 (solid line).

The generalized ChapmanJouguet condition is that u2n0 = c20 at = 0. This con-

dition yields the normal velocity, uCJn0 , and the mass fraction, CJ0 , at the CJ point.

Expanding about the CJ point, the derivative (dun0/d0)CJ can be found. Using

this to determine the solution near the CJ point, the complete solution can be foundnumerically, by integrating away from the CJ point to the shock. For a given curva-ture, , there is a value of the shock velocity, Dn, such that the shock conditions aresatisfied. This value is not necessarily unique, but only one is on the upper branchof the Dn curve, which is relevant to propagating detonations. This numericalprocedure is described in Sharpe (2000a). For fixed E, Q and , there is a criticalcurvature above which there is no quasi-steady solution. This is the turning point onthe Dn curve. Figure 1 shows Dn curves for the four activation temperatures(E) used in 6. We can see that as the activation temperature increases, the rangeof curvature for which a quasi-steady solution exists decreases. Note, however, thatfor the case of cylindrically or spherically expanding detonations the quasi-steadyassumption must formally break down at the critical curvature point, since Dn willchange rapidly near this point.

4. The linearized equations

In this section we determine the linear stability equations governing the leading-orderquasi-steady solutions (cf. Yao 1996). Note that linear stability analysis of quasi-

steady waves (not just strictly steady waves) is standard, a related example in thearea of reactive flow is the linear stability of spherically expanding flames (Addabboet al. 2003). A formal analysis involving the slow time-scale of the underlying quasi-steady wave is given in the appendix, where it is shown that the small time-dependentterms ignored in determining the leading-order quasi-steady solutions in the previoussection do not affect the linear stability of these solutions to leading order.




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To analyse the linear stability of the quasi-steady solution found previously, weperturb the solution so that the position of the (perturbed) shock has the form

n(t) = et, 1,

i.e. a perturbation which is normal to the front. Note that for spherically or cylin-drically expanding fronts this corresponds to a symmetric perturbation in the radialdirection. We transform to a frame moving with the perturbed shock: n = n et

and t = t, where the velocity of the perturbed shock in the n direction is

Dn = Dn + et with Dn =

2et.

The governing equations (2.1)(2.4) are then written in this perturbed shock frame,and the primes are subsequently dropped.

We assume perturbations of the form

q(n, t) = q0(n) + q1(n)et,

where q is one of , p, un or and we expand W as

W = W0(n) + W01(n)et + W0pp1(n)e

t + W01(n)et + ,

where W0 = W0/0, etc.Using these and substituting the expressions for the perturbed quantities into the

governing equations, after linearizing in , the equations can be written in the form

0du

dn= Au + s,

where

u = (1, un1, p1, 1)T, 0 = u

2n0 c

20,

A =

0

un00

1

un00

0 un01

00

0 0c20 un0 0

0 0 0 0

un0

+

un0(un0 + Dn) 0un0 +00

u

2

n0

(un0 + Dn)

un0

(un0 + Dn) 0

c200

(un0 + Dn) c20

0(un0 + Dn) 0

0c20un0 0c

20un0 un0(un0 + Dn) 0

0 0 0 0




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+

q( 1)

un0(2W0 + 0W0) +

dun0d0

W00W0

u3n0(2u2n0 + 0)

dun0d0

q( 1)

0(2W0 + 0W0)

dun0d0

W00un0

(u2n0 + 0) 2W0dun0d0

un0q( 1)(2W0 + 0W0) +dun0d0 u

2n0W00W0

un0

dun0d0 (c

20 + u2n0)

0W0un0

W00

u2n0

W0

u2n0

dun0d0

0q( 1)W0p

un0

0qW0( 1)

un0

dun0d0

W0

0un0+ qW0p( 1) qW0( 1)

dun0d0

W0 un00qW0p( 1) un00qW0( 1)

W0p0un0

W00un0

,

s = 2(0,un0, 0c20, 0)

T + (0un0, c20,0c

20un0, 0)

T.

The 0 subscript represents the quasi-steady solution. Note that at the CJ point thereis a singularity, as CJ0 = 0.

When the quasi-steady solutions were found, they were determined as a functionof the reaction progress variable 0 and it is convenient to also choose this as theindependent variable for the perturbed equations. The linear system then becomes

0W0un0

du

d0= Au + s. (4.1)

Finally, the boundary conditions of the perturbed equations at the shock are foundto be

1(0) =4Dn(+ 1)p

(2p+ ( 1)D2n)2

, un1(0) =

+ 1

2p

D2n + 1

,

p1(0) =4Dn

+ 1, 1(0) = 0.

The boundary conditions at the generalized CJ point are that the solutions arebounded there.

5. Determining the eigenvalues

From the previous section, the linearized equations form a fourth-order ODE system.This system will have four independent solutions with four boundary conditions atthe shock. However, due to the physical constraint that the solution must be boundedat the CJ point, the eigenvalues will be such that the solution is independent ofan unbounded solution.




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(a) Analysis about the CJ point

At the CJ point, (4.1) is singular, as CJ0 = 0. Let us consider the behaviour nearthe CJ point.

Rewrite (4.1) by introducing a new independent variable

w = 0 CJ0

as

wdu

dw= Au + s. (5.1)

Note that w = 0 (at 0 = CJ0 ) is a regular singular point of equation (5.1). This

equation has four independent homogeneous solutions which to leading order in w areof the form wivi, where i and vi are the eigenvalues and eigenvectors, respectively,of

(A)CJ =

aCJ1 aCJ2 a

CJ3 a

CJ4

un00

aCJ1 un00

aCJ2 un00

aCJ3 un00

aCJ4

u2n0aCJ1 u2n0aCJ2 u2n0aCJ3 u2n0aCJ4

0 0 0 0

,

where

a1 = un0(un0 + Dn) +dun0d0

W0 2qW0( 1)

un0

0qW0( 1)

un0,

a2 = 0un0 + 0 +20W0

un0

dun0d0

,

a3 =

un0

W0

u2n0

dun0d0

un0(un0 + Dn)

0qW0p( 1)

un0,

a4 = q0W0

un0.

The general solution is then

u =4

i=1

aiwivi + up as w 0,

where the ai are (complex) constants and up is the particular integral. For thissystem, the eigenvalues are 0, 0, 0, h, where

h = aCJ1 un0

0a2

CJ

+ (u2n0a3)CJ.

Note that for Re() > 0, h < 0, hence the solution corresponding to this eigenvaluewill be unbounded at w = 0. Thus we need to find the eigenvalues such that thissolution is not present, or a4 = 0, while simultaneously satisfying the perturbedshock conditions.




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(b) Solution method

There are two possible methods of solving this problem. The first is to use thethree bounded asymptotic solutions about the CJ point as initial conditions forintegrating towards the shock and then to attempt to satisfy the four shock conditionswith a linear combination of these bounded solutions. This method is developed and

described in Sharpe (1997, 1999).The second method, due to Lee & Stewart (1990), is to integrate from the shock

towards the CJ point and apply a boundedness, or equivalently a radiation, condition.The required eigenvalue will give a solution which is bounded at the singularity. Inthe original method of Lee & Stewart (1990), a finite domain was used, with 0 as theindependent variable, as is done here. The integration was stopped one grid pointfrom the CJ point and the boundedness condition applied there. Note that laterversions of this method involved using infinite domains with the distance behindthe shock, x, as the independent variable (Short & Stewart 1998; Sharpe 1999). Inthis paper, however, we follow closely the original method described in detail inLee & Stewart (1990), but note that the boundedness condition to be employed isdifferent for the curved detonation case. We introduce a measure of how unbounded

the solution is, or how large a4 is,

M = l4 (u() up()),

where

l4 = (aCJ1 , a

CJ2 , a

CJ3 , a

CJ4 )

is the left eigenvector of (A)CJ corresponding to eigenvalue h. As is a complexnumber, we need to take the modulus of M to give an absolute measure. If M = 0near the CJ point, then u is independent of the unbounded solution and is aneigenvalue. Note that Yao (1996) derived a similar condition.

As a further verification and check of the method, as well as using the verificationsdescribed in Lee & Stewart (1990), the results are compared with those of the planar

case (Lee & Stewart 1990; Sharpe 1997), including the neutrally stable values of E.In the limit 0 the results agree. Note, however, that for = 0 the CJ point isthen an irregular singular point of (5.1) since then W0 = 0 at w = 0 = 0 (see Sharpe1997).

6. Results

In this section we show neutral stability boundaries and dispersion relations forvarious activation temperatures and curvatures. In practice, only the lowest frequencymodes appear in the full nonlinear problem for detonations (Short & Wang 2001),and hence we will only consider the first three modes.

(a) Neutral stability

In the full time-dependent problem, we are interested in whether the detonationis unstable to perturbations or not. Thus we are interested in when the eigenvalueschange from having a negative real part to a positive real part, the neutral stabilityboundary.




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0 0.005 0.010 0.01515

20

25

30

35

E

Figure 2. The neutral stability boundaries for the fundamental mode (solid line), first overtone(dashed line) and second overtone (dot-dashed line). Also shown is the critical curvature for thequasi-steady solutions (dotted line).

To find this boundary, first we set E, Q and . The one parameter left to vary isthe curvature, or equivalently the detonation speed. By scanning the complex plane,we can find the approximate location of the eigenvalue where M = 0. By improvingthe resolution, we can converge onto the eigenvalue. In general, increasing the curva-ture, or decreasing the detonation speed, moves the eigenvalue to the right and viceversa. Repeating this process, a more accurate guess can be made for the curvaturewhen the eigenvalue is strictly imaginary. This method is repeated for various val-ues of the activation temperature, E, and for the first three eigenvalues. In figure 2we have plotted the neutral stability boundaries for each of the three eigenvalues,as well as the critical curvature for the quasi-steady waves, in an (E)-plane. Thedetonation is one-dimensionally unstable to each eigenmode above and to the rightof the corresponding boundary curve and stable below and to the left of them. Notethat as is increased with E fixed, each mode becomes unstable in order of ascend-ing frequency, so that the detonation is only stable to one-dimensional perturbationsbelow the neutral stability curve of the fundamental mode. Note also that the neu-tral stability boundaries asymptote to the critical curvature locus. For fixed E, thefundamental mode is always unstable sufficiently near the critical curvature. Hencewhen the curvature is near critical, the detonation will not propagate quasi-steadilybut in an unstable pulsating manner.

(b) Dispersion relations

We track the unstable eigenvalues as a function of curvature for four different acti-vation temperatures. The four we choose are E = 20, for which the planar detonationis one-dimensionally stable, and E = 25.26, E = 31.05 and E = 35.84, which are theactivation temperature when the first, second and third eigenmodes in the planarcase ( = 0) change stability, respectively (Sharpe 1997). The dispersion relations(Re() and Im() versus diagrams) for these cases are shown in figures 3, 4, 5 and6, respectively.




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0

0.005

0.010

0.015

0.020

0.010 0.011 0.012 0.0130

0.05

0.10

0.15

0.20

0.25

(a)

(b)

Re(

)

Im()

Figure 3. (a) The real and (b) the imaginary parts of the fundamental mode (solid line) andfirst overtone (dashed line) as a function of for E = 20.

(i) E = 20Figure 2 shows that for this activation temperature, the detonation is one-

dimensionally unstable only when the curvature is above = 0.00976. The fun-damental mode becomes unstable at = 0.009 76, and figure 3 shows that as isincreased further, the growth rate (Re()) of this mode increases rapidly, while thefrequency (Im()) decreases. Hence increasing the curvature quickly destabilizes thedetonation. The frequency of the fundamental mode becomes zero at = 0.012 58,and the eigenvalue subsequently bifurcates into two real parts. One branch increasesextremely rapidly with curvature, while the other decreases, until the critical cur-vature for the quasi-steady wave is reached at = 0.0126. The second mode onlybecomes unstable sufficiently near the critical curvature, at = 0.012 33. However,the growth rate of this mode increases very rapidly with curvature once it becomes

unstable.

(ii) E = 25.26

From figure 2, we see that for this case the fundamental mode is neutrally stable forthe planar case ( = 0) and unstable on the whole upper branch of the Dn curve for




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0

0.01

0.02

0.03

0.04

0 0.002 0.004 0.006 0.008 0.010

0

0.1

0.2

0.3

0.4

(a)

(b)

Re(

)

Im(

)

Figure 4. (a) The real and (b) the imaginary parts of the fundamental mode (solid line), firstovertone (dashed line) and second overtone (dot-dashed line) as a function of for E = 25.26.

> 0 (the critical curvature for the quasi-steady waves is = 0.008 95). The secondmode becomes unstable only for sufficiently large curvature > 0.007 08, while thethird mode is only unstable near the critical curvature, > 0.008 87. Figure 4 showsthat as increases from zero the growth rate of the fundamental mode increases.The frequency becomes zero at = 0.0084, and again this eigenvalue subsequentlybifurcates into two real values. Once the second mode becomes unstable, its growthrate increases and its frequency decreases very rapidly with further increases of .The third mode is even more sensitive to curvature. Figure 4 also shows that nearthe critical curvature the growth rates of the two lowest frequency modes becomecomparable.

(iii) E = 31.05

From figure 2, we see that in this case the first and second eigenmodes are unstableon the whole of the upper branch of the Dn curve, with the second mode neutrallystable at = 0. The critical curvature for E = 31.05 is = 0.00672. Figure 5shows that as the curvature increases, the growth rate of the fundamental modeis almost constant while that of the second mode increases. Eventually the growth




14/19


0

0.01

0.02

0.03

0.04

0.05

0.06

0 2 4 6

0

0.2

0.4

0.6

0.8

(a)

(103)

(b)

Re

()

Im(

)


rate of the second mode becomes greater than that of the fundamental mode andhence this is the linearly dominant (most unstable) mode for a range of curvature(0.003 42 < < 0.004 71). Again the first eigenvalue bifurcates into two real values(at = 0.004 56). Note that the growth rates of the first and second modes are verysimilar near the critical curvature. Note also that the growth rates of the first andsecond modes decrease slightly as the critical curvature is approached. The thirdmode is only unstable for a small part of the upper Dn branch ( > 0.005 06), butit rapidly becomes more unstable as increases further.

(iv) E = 35.84

From figure 2, we see that the first three eigenmodes are unstable for all of the

allowable curvature, with the third mode neutrally stable in the planar case. Thecritical curvature is = 0.005 53. As in the previous examples, the first eigenvaluebifurcates, but this time when = 0. The second mode is the dominant one for smallenough curvature, but the growth rates of the first and second modes become verysimilar for the rest of the Dn upper branch with nearly constant growth rates untilthe curvature is near critical, where the growth rates decrease sharply. The growth




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0

0.02

0.04

0.06

0.08

0.10(a)

0 2 4 6

(103)

0

0.2

0.4

0.6

0.8

1.0

1.2

(b)

Re()

Im()


rate of the third mode increases rapidly with curvature. One thing to note is thatnow the real parts of all three eigenvalues are very close at the critical curvature.

7. Conclusions

In this paper we have investigated the one-dimensional linear stability of weaklycurved detonations. The results show that even weak curvature has a significantdestabilizing effect on detonation waves. Secondly, we find that the fundamentalmode is always unstable sufficiently near the critical curvature for quasi-steady waves,even if the planar wave is quite stable. Moreover, we have found that the fundamentallinear mode usually bifurcates into two real values before the critical curvature is

reached. Whenever the fundamental mode bifurcates, it appears that in the fullynonlinear case the detonation is always far into a regime of very large amplitude,highly irregular oscillations (or even beyond a one-dimensional detonability limit formore complex kinetic models) (Short & Quirk 1997; Sharpe & Falle 2000a, b). Hencethe analysis suggests that the detonation actually becomes highly unstable as thecritical curvature is approached.




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These results agree with one-dimensional numerical simulations of spherically andcylindrically expanding detonation fronts, directly initiated by a point-source blastwave (He & Clavin 1994; Sharpe 2000a; Ng & Lee 2002). In these simulations, oncethe detonation has formed, it initially pulsates even when the planar wave is stable,but the amplitude of the pulsations decreases as the shock radius increases, indicating

that the detonation becomes more stable with decreasing curvature. Indeed, recenthigh-resolution numerical simulations of expanding detonations (Watt & Sharpe2004) show that the leading-order neutral stability results determined from the linearanalysis predict well those found in the fully nonlinear simulations.

For condensed-phase rate-sticks, the critical curvature is often associated withthe critical diameter, below which the rate-stick cannot be detonated (Bdzil 1981;Stewart 1998). Since we have found that the detonation is unstable near the criticalcurvature this suggests that as the critical diameter of the rate-stick is approachedthe transition between steady detonation propagation and no detonation will not besharp but there will be a regime of unstable, unsteady propagation, where the frontpulsates by repeated failure and re-ignition of the detonation, a possibility suggestedby Stewart (2002). However, for rate-sticks, multi-dimensional effects may be impor-tant, even in the steady wave (Short & Bdzil 2003), and material inhomogeneitieswill also be important.

Since the purpose of this paper is to determine the stabilizing/destabilizing effectof curvature of detonation waves, it was sufficient to consider a one-dimensionalperturbation analysis. Indeed, a general multi-dimensional linear stability analysisis not straightforward for a curved detonation due to the complexity of the multi-dimensional shock-attached equations (Yao & Stewart 1996). However, for specificcases, such as spherical or cylindrical waves, a multi-dimensional analysis may bemore amenable. We intend to investigate this in the future.

This work was funded by the EPSRC and the DSTL (formerly DERA) under the Joint GrantScheme. The authors are also grateful to a referee for pointing out to us the work done in thethesis by Yao (1996).

Appendix A.

Consider a quasi-steady detonation, such that the speed and structure of the waveevolve on the slow time-scale t = t with 1. Denote this quasi-steady solutionby an s subscript, i.e. uns uns(n, t), Dns Dns(t), etc. Here we will just considerthe momentum equation (2.1) for the sake of briefness. The quasi-steady solution ishence governed by

uns

t+

dDnsdt

+ unsuns

n+

1

psn

= 0. (A1)

Suppose that the exact quasi-steady solution is perturbed such that the position of

the perturbed front is n = e

t

, 1.It is necessary to transform to a frame moving with the perturbed front (Erpenbeck1964): n = n et, t = t. The speed of the perturbed front is Dns(t) + e

t andhence its acceleration is dDns/dt +

2et. Note that the one-dimensional momen-tum equation written in the shock attached frame for a general accelerating shockfront is of the form (2.1) (Yao & Stewart 1996), and hence in the frame moving with




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the perturbed shock the momentum equation is

unt

+ dDns

dt+ 2et + un

unn

+1

p

n= 0, (A2)

where un is the fluid velocity with respect to the perturbed shock. The primes are

subsequently dropped for convenience. The perturbed dependent variables then havethe form

q(n, t) = qs(n, t) + qp(n, t)et.

Substituting into (A 2) gives

unst

+ dDns

dt+ uns

unsn

+1

psn

+ et

unp

t+ unp +

2 + unpuns

n+ uns

unpn

+1

s

ppn

p2s

pnn

+ o() = 0.

Note that the term in square brackets is identically zero by (A 1) and hence theleading-order terms in (A 2) are actually O(). Note also that (A 1) and (A 2) are

exact; we have made no asymptotic truncation as yet. Linearizing (A 2) in , thelinear stability equation is hence

unp

t+ unp +

2 + unpuns

n+ uns

unpn

+1

s

ppn

p2s

psn

= 0. (A3)

However, since 1, let us solve (A 1) and (A 3) in using an asymptotic expansionin , i.e. expand as

qs(n, t) = q0(n) + O(), qp(n, t) = q1(n) + O(), Dns = Dn0 + O(),

where q0 and q1 depend only parametrically on t. To leading order the quasi-steadysolution is hence governed by

un0dun0dn + 10dp0dn = 0,

while the leading-order linear stability equation is

un1 + 2 + un1

dun0dn

+ un0dun1

dn+

1

0

dp1dn

120

dp0dn

= 0.

The main points from the above are that the quasi-steady and linear stability solu-tions in this paper are the leading-order solutions in an expansion in the slow time-scale, and that the small O() time-dependent terms ignored in determining thequasi-steady solutions and the associated Dn relations will give only a correspond-ingly small correction to the linear stability results presented here.

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S. D. Watt and G. J. Sharpe- One-dimensional linear stability of curved detonations