arX

iv:p

hysi

cs/0

5051

02 v

1 1

4 M

ay 2

005

The Rate of Expansion of Spherical Flames

V. Karlin∗ and G. Sivashinsky†

Abstract

In this paper we investigate the acceleration of the expansion of pre-mixed spherical flames and evolution of the cellular patterns on their sur-faces. An asymptotic model is used for the simulations and a spectral numer-ical algorithm is employed to study flames over large time intervals. Numer-ous numerical experiments indicate that for large enough time intervals theacceleration of two-dimensional expanding flame ceases andthe expansionrate stabilizes to a value significantly exceeding the burning rate. The impor-tance of the effect of forcing was also confirmed and the validity of sectorialsimulations of closed flames was studied in order to justify prospective useof the Fourier spectral model for the three-dimensional spherical flames.

1 Introduction

In [1] extensive experimental data on premixed expanding spherical flames havebeen analysed and it was concluded that starting from a certain moment the av-eraged flame radius<r > grows as<r > ∝ t3/2 rather than<r > ∝ t. Thephenomenon was linked to the cellularization of flames, which was well knownfrom experiments too. Indeed, the appearance of cellular patterns increases theflame surface area, hence the fuel consumption, and hence theaveraged flameexpansion rate.

The cellularization of flame fronts was, in its turn, associated with intrinsiccombustion instabilities. The effect of the hydrodynamic combustion instabilityon expanding spherical flames was studied in [2] using the linear perturbationtheory combined with phenomenological assumptions. Later, the approach wasfurther improved and freed from the phenomenological assumptions [3]. Theselinearized solutions confirmed the onset of the instabilityof the flame front butcould not quantify its cellularization and acceleration because the latter phenom-ena are essentially nonlinear.

∗University of Central Lancashire, Preston, UK†Tel Aviv University, Tel Aviv, Israel

A simple, yet physically reasonable, nonlinear model of hydrodynamicallyunstable planar flames was suggested in [4]. In [5] it was extended to expandingspherical flames and studied numerically confirming that there is a time instancet∗, such that the flame expansion rate behaves like<r> ∝ t3/2 for t > t∗. Thesame model was also suggested in [6], where similar to the planar flames [7],analytical pole solutions were obtained and studied as well.

Because of their physical origins simple nonlinear models of expanding flames[5, 6] are expected to be valid only locally. Apparently, theresults obtained whenapplying them to the whole flame are instructive indeed, but still inconclusive andcannot be accepted as the adequate theoretical model of cellularization and accel-eration. A physically consistent global model of flames of any smooth enoughgeometry was developed in [8]. Mathematically, the approach projects the gov-erning equations to the flame surface reducing the mathematical dimension of theproblem by one. However, the resulting equation is still extremely costly fromthe computational point of view and only two-dimensional simulations have beencarried out so far.

A compromise between universality and computability was suggested in [9],where consideration was limited to a narrow but still very practical case of flameswhich do not deviate from the spherical ones significantly. On the technical sidethe model combines the operator of the linearized problem obtained in [2] forthe expanding spherical flame in terms of spherical harmonics expansions and aHuygens type nonlinearity specific to the local nonlinear model [5,6]. Physically,model [9] is consistent with [8] and is robust and plausible enough to simulatethe cellularization of expanding spherical flames in three spatial dimensions. Atthe time of writing of this paper, the flame sizes we were able to reach in ourcomputations do not significantly exceed those reported in [9] and are not largeenough to match our two-dimensional calculations. However, our investigationsshow that numerical studies of the expanding three-dimensional flames on thetime scales required for comparison with the two-dimensional calculations arepossible.

In the following sections we specify the mathematical models and numericalalgorithms to solve them. In Section 4 we report our results on flame front be-haviour on long time intervals and on the effect of external forcing. Also, wepresent our attempts to simulate the three-dimensional flames and assess the pos-sibility of simulation of closed flames via their finite segments.

2 Mathematical Models

Let us consider an expanding flame front and assume that its surface is closeenough to a sphere and that every point on the flame surface is uniquely defined

2

by its distancer = r(θ, φ, t) from the origin for0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π, andt > 0. It is convenient to represent such a flame as a perturbationΦ(θ, φ, t) of aspherical surface of a reference radiusr0(t), i.e. r(θ, φ, t) = r0(t) + Φ(θ, φ, t).Then, the Fourier image of the governing equation of the flamefront evolution inthe nondimensional notations suggested in [5,6] can be written as

dΦk

dt=

{−

θ2π

[r0(t)]2|k|2 +

γθπ

2r0(t)|k|

}Φk

−θ2

π

2[r0(t)]2

∞∑

l=−∞

l(k − l)ΦlΦk−l + fk(t). (1)

Here |k| < ∞, t > 0, fk(t) are the Fourier components of the properly scaledupstream perturbations of the unburnt gas velocity fieldf(φ, t), and initial valuesof Φk(0) = Φ

(0)k are given. By construction, equation (1) holds in the sector

0 ≤ φ ≤ 2π/θπ with a large enough integerθπ.Models of [5] and [6] differ by an additive term proportionalto δ0,k/r0(t).

This term adds justO(ln t) to Φk for k = 0 only. This is not essential and the termis not included in our conception.

Equation (1) was obtained as a local model of a curved expanding flame. How-ever, we will use it globally, on the whole flame surface, withθπ = 1. In orderto justify such an action, let us first note that if transformed back to the physicalspace, the equation takes the form

∂Φ(φ, t)

∂t=

1

[r0(t)]2∂2Φ(φ, t)

∂φ2+

γθπ

2πr0(t)

∂

∂φ

2π/θπ∫

0

Φ(θ, t) cotφ − θ

2dθ

+1

2[r0(t)]2

[∂Φ(φ, t)

∂φ

]2

+ f(φ, t), (2)

which is rotation-invariant forf(φ, t) ≡ 0 andθπ = 1.On the other hand, we may rewrite the geometrically invariant equation ob-

tained in [8] in the coordinate form as follows:

∂r

∂t=

√1 +

(rφ

r

)2{

1 − ǫr2 + 2(rφ)

2 − rrφφ

[r2 + (rφ)2]3/2

−γ

2

1 +1

π

2π∫

0

p(φ, θ)dθ

, (3)

3

where

p(φ, θ) =

[r(φ) − r(θ)] · n(φ)

|r(φ) − r(θ)|2√

[r(θ)]2 + (rθ)2 |θ − φ| > 0,

−[r(φ)]2 + 2(rφ)

2 − r(φ)rφφ

2{[r(φ)]2 + (rφ)2}|θ − φ| = 0,

(4)

n(φ) =[−(rφ sin φ + r cos φ), (rφ cos φ − r sin φ)]√

r2 + (rφ)2, (5)

andr(φ) = [r(φ) cosφ, r(φ) sinφ]. (6)

Assuming again thatr(φ, t) = r0(t) + Φ(φ, t) and linearizing all but the Huygensterms, one arrives to the equation

∂Φ(φ, t)

∂t=

ǫ

[r0(t)]2∂2Φ(φ, t)

∂φ2+

γ

2πr0(t)

∂

∂φ

2π∫

0

Φ(θ, t) cotφ − θ

2dθ

+1

2[r0(t)]2

[∂Φ(φ, t)

∂φ

]2

+ 1 −dr0(t)

dt−

ǫ

r0(t)

+

{ǫ

[r0(t)]2−

γ

r0(t)

}Φ(φ, t) +

γ

2πr0(t)

2π∫

0

Φ(θ)dθ. (7)

which differs from (2), forθπ = 1, by a few non-essential terms only.Comparison of (2) and (7) reveals the detailed relationshipbetween the sim-

plified model [5, 6] and the comprehensive one [8]. In particular, it validates useof (2) as a global model of the whole spherical flame as long as the perturbationsremain small enough.

Equations of the three-dimensional model [9] can be writtenin terms of thespherical harmonics expansion coefficients

Φn,m(t) =

2π∫

0

π∫

0

Φ(θ, φ, t)Yn,m(θ, φ) sin θdθdφ, |n| < ∞, |m| ≤ |n|

of Φ(θ, φ, t) as

dΦn,m(t)

dt= ω(n, t)Φn,m(t) +

1

2[r0(t)]2Nn,m(t) + fn,m(t). (8)

4

Here |n|, |m| < ∞, t > 0, fn,m(t) are the spherical harmonics coefficients ofthe properly scaled upstream perturbations of the unburnt gas velocity field, andinitial values ofΦn,m(0) = Φ

(0)n,m are given. The expression for the linear response

ω(n, t) =2n(n − 1)

(2n + 1)r0(t)−

(n − 1)(n + 2)

r20(t)

, (9)

emerges from the analysis of [2] andNn,m(t) are the spherical harmonics coeffi-cients of the nonlinear Huygens term

N (Φ) = ‖∇SΦ‖2 =

(∂Φ

∂θ

)2

+1

(sin θ)2

(∂Φ

∂φ

)2

. (10)

Choice ofr0(t) in the models introduced in this Section may be based on avariety of principles. In equation (7), which governs smallperturbations of (3)-(6), choice ofr0(t) is more or less arbitrary. However

r0(t) =1

2π

2π∫

0

r(φ, t)dφ (11)

is probably the most appropriate one because it minimizes the perturbationsr(t)−r0(t) = Φ(φ, t). On the other hand, equation (1), or (2), was obtained as thatone,which governs perturbations of a uniform and steadily expanding spherical flamer(t) ≡ t. Therefore, choice of

r0(t) ≡ t (12)

is more reasonable in this case, as this is the solution around which the lineariza-tion is performed. Equation (1) was already considered with(11) in [5]. Here wecomplement the analysis of (1) by considering it with (12).

3 Computational Algorithms

System (1) is solved numerically by neglecting the harmonics of orders higherthan a finite integer numberK > 0. Then, the nonlinearity can be representedas a circular convolution and evaluated effectively with the FFT. Also, we foundthat the stability of the numerical integration scheme suggested in [5] can be im-proved significantly by calculating the contribution from the linear terms in (1)analytically. Thus, the linear terms, i.e. the source of physical instability, aretackled exactly and only the nonlinear ones, with the dumping effect, are approx-imated. This improvement allowed us to continue the calculations for up to tentimes further in time than in [5].

5

Using the notation

gk[Φ] = −θ2

π

2

K∑

l=−K

l(k − l)ΦlΦk−l − V δk,0 + fk(t), (13)

equation (1) can be written as

dΦk

dt=

(−

θ2π

r20(t)

|k|2 +γθπ

2r0(t)|k|

)Φk +

1

r20(t)

gk, |k| ≤ K. (14)

and we will search for its solutions in the form

Φk(t) = Ck(t)Yk(t), (15)

whereY (t) is the solution of the uniform equation.Straightforward evaluations yield

Yk(t) = e∆ωk(t), ∆ωk(t) = −θ2π|k|

2

t∫

t0

dt

r20(t)

+1

2γθπ|k|

t∫

t0

dt

r0(t). (16)

Then, the equation forCk(t) is

dCk

dt= r−2

0 (t)eθ2π|k|

2t∫

t0

dt

r20(t)

− 12γθπ |k|

t∫t0

dtr0(t)

gk, (17)

wheret0 is an arbitrary real of which the final result (15) does not depend.In order to integrate the ODE’s (17) over the interval[tn, tn + ∆t] or [tn −

∆t, tn + ∆t], we interpolate values ofgk(t) ≡ gk[Φ] polynomially:

gk(t) =

mp∑

µ=0

Gk,n,µtµ, t ∈ [tn − ∆t, tn + ∆t]. (18)

For example, the first order extrapolation gives

gk(t) ≡ Gk,n,0 = gk(tn), (19)

and the second one results in

gk(t) = gk(tn)t − tn + ∆t

∆t− gk(tn−1)

t − tn∆t

= Gk,n,0 + Gk,n,1t, (20)

where

Gk,n,0 =−gk(tn)(tn − ∆t) + gk(tn−1)tn

∆t, Gk,n,1 =

gk(tn) − gk(tn−1)

∆t. (21)

6

In what follows we provide formulas for the first order extrapolation only. For-mulas of higher orders of accuracy are a bit bulky, but can be obtained straightfor-wardly.

The interpolation results, in general, in

Φk(tn + ∆t) = e∆ωk(tn+∆t)

[Φk(tn) +

mp∑

µ=0

κk,n,µGk,n,µ

](22)

with

∆ωk(t) = −θ2π|k|

2

t∫

tn

dt

r20(t)

+1

2γθπ|k|

t∫

tn

dt

r0(t), (23)

and

κk,n,µ =

tn+∆t∫

tn

ξµr−20 (ξ)e−∆ωk(ξ)dξ. (24)

In order to move any further, we will now assume that

r0(t) ∝ tα. (25)

Then (23) can be written as∆ωk(t) = ωk(t) − ωk(tn), where

ωk(t) =θ2

π|k|2t2n

r20(tn)t

+γθπ|k|tn ln t

2r0(tn), α = 1. (26)

and

ωk(t) =θ2

π|k|2t2α

n

(2α − 1)r20(tn)t2α−1

−γθπ|k|t

αn

2(α − 1)r0(tn)tα−1, α 6= 1. (27)

Further, (24) for the first order of approximation will become

κk,n,µ =tµn

r20(tn)

∆t. (28)

Then, the integration formula (22) is transformed into

Φk(tn + ∆t) = eωk(tn+∆t)−ωk(tn)

[Φk(t0) +

∆t

r20(t0)

gk(tn)

], (29)

with ωk(t) given by (26) and (27).A computational algorithm based on numerical spherical harmonics transfor-

mation (SHT) [10], similar to [9], was used in this work in order to solve (8). Inaddition, the stability of the numerical integration scheme was improved by eval-uating the contribution from the linear terms analytically, as shown above, and thecode was parallelized in order to speed up the computations and to use larger datasets.

7

4 Computational results

4.1 Saturation on the long time intervals

Typical shapes of the flame fronts governed by (1) over large time intervals areillustrated in Fig. 1. The graph of[r(φ, t)−<r >]/<r > for t = 7.65 × 104

shows that the wrinkle amplitudes are up to 10% of the averaged flame radius.The explicit forcing was not applied in this example.

1.0 0.5 0 0.5 1.0

1.0

0.5

0

0.5

1.0

r(φ,t)

r(φ,

t)

× 105

× 1050.1 0.05 0 0.05 0.1

0.1

0.05

0

0.05

0.1

(r−<r>)/<r>(r

−<

r>)/

<r>

Figure 1: Evolution of a spherical flame governed by (1). Herevalues ofr(φ, t)are on the left and[r(φ, t)− <r>]/<r> for t = 7.65 × 104 are on the right.Positive values of the latter are in blue and negative ones are in red;γ = 0.8,f(φ, t) ≡ 0.

Permanent growth of the size of a spherical flame as it expandsprompts studiesof the effect of the size of a planar flame on its propagation speed as the firststep towards the understanding of the acceleration mechanism of the expandingflames. The investigation of the dynamics of planar flames shows that the spatiallyaveraged flame speed

<Φt>=1

L

L∫

0

∂Φt

∂tdx (30)

of a flame sizeL stops growing and begins to oscillate irregularly around its timeaverage<<Φt>> for large enought. A definite correlation between the size of theflame and its stabilized spatially averaged propagation speed<<Φt>> was estab-lished, see e.g. [11]. The effect was explained by proving the high sensitivity ofplanar cellular flames to particular types of linear perturbations, see [11–14]. Bycontinuing calculations reported in [11] for even larger planar flames, we estab-lished that their propagation speed no longer grows after a certain critical flame

8

size is reached, see graph on the right of Fig. 2. In this paperwe are interested inextending these findings for planar propagating flames to theexpanding ones. Inparticular, we are studying the possibility of a stabilization of the expansion ratefor large enough time intervals, when the flame size grows sufficiently large.

Stabilized velocities<< Φt >> of the planar flames and averaged velocities<Φt>= 1

2π

∫ 2π

0∂Φt

∂tdφ of the spherical flames are compared in Fig. 2. Power law

approximations(t − t∗)α for the expansion rate of the spherical flame are also

depicted there. Sudden increase of<Φt> begins fromtc ≈ 2.2 × 103. For thewhole considered time interval[2.2 × 103, 7.65 × 104] the optimalα ≈ 0.34. Forearlier timest ∈ [2.2×103, 2.0×104], the best approximation is withα ≈ 0.47, i.e.almost1/2 as obtained in experiments. However, as time goes by, the expansionrate slows down and fort ∈ [3.0 × 104, 7.65 × 104] we gotα ≈ 0.23.

0 1.5 3 4.5 6 7.5

x 104

00.10.20.30.40.5

t

<Φ

t> α≈ 0.34α≈ 0.47α≈ 0.23

0 0.75 1.5 2.25 3

x 104

00.10.20.30.40.5

L

<<

Φt>

>

Figure 2: Averaged flame front velocity for spherical (left)and planar (right)flames versus timet and flame sizeL respectively. Hereγ = 0.8 andf(φ, t) ≡ 0.Markers on the right graph show the calculated cases.

A tendency towards stabilization of<Φt> to a constant fort → ∞ is evidentin the graph too. A change in morphology of the flame front atts ≈ 2.5 ×104 is even more obvious in the graph of the variation of the perturbation of theaveraged circular flame relatively to the radius of this averaged flame, i.e. ofmax

0≤φ≤2π[r(φ, t)−<r>]/<r>, which is shown on the left of Fig. 3.

The number of cells per unit length of the averaged flame contour is depictedon the right of Fig. 3. It stabilizes to a cell size of about200, which is exactlythe same as for large enough planar flames. Eventually, Fig. 4illustrates thespectral distribution of energy of the perturbationΦ(φ, t), which stabilizes byts ≈2.5 × 104 too.

Stabilization of the averaged perturbation gradient∇Φ = ∂Φ/∂φ can also beseen on the left of Fig. 5. Besides the saturation of the gradient the graph illus-trates the importance of the number of Fourier modesK involved in the numericalsimulation. The jumps in the graph correspond to the instances when we doubledK in order to match the continuously increasing size of the flame. The graph is

9

0 1.5 3 4.5 6 7.5

x 104

00.050.1

0.150.2

t

Va

r φ(Φ)/

<r>

0 1.5 3 4.5 6 7.5

x 104

00.0050.01

0.0150.02

t

N/[

2π<

r>]

Figure 3: Ratio of the maximal amplitude of perturbation andof the averagedflame radius (left) and number of cells per unit length of the averaged flame con-tour (right). Hereγ = 0.8 andf(φ, t) ≡ 0.

0 2 4 6

x 104

−8−6−4−2

0

k

log 1

0Ek

0 4 8 12

x 104

−8−6−4−2

0

k

log 1

0Ek

Figure 4: Spectral distributions of energy of the perturbation Φ(φ, t) for t =2.4928 × 104 (left) andt = 7.65 × 104 (right).

getting smoother ifK is adjusted more gradually and is kept much larger than theratio of the critical wavelengthλcr = 4π/γ to the circumferential length of theflame. The graph on the right of Fig. 5 depicts the energy of thehighest orderFourier harmonic involved in the simulation.

0 1.5 3 4.5 6 7.5

x 104

00.5

11.5

2

t

<|∇Φ

|>

0 1.5 3 4.5 6 7.5

x 104

0

2

4x 10

−3

t

log 1

0EK

Figure 5: Averaged gradient of the perturbation (left) and energy of the highestorder harmonic involved in the simulation (right).

10

According to the data obtained in numerical simulations we may summarizethe mechanism of flame front expansion in the Fourier space asfollows. Theenergy of relatively long wave harmonicsλ > λcr permanently grows according tothe Darrieus-Landau instability. This gained energy is transferred towards shorterwavelengths via the nonlinear effects and dissipates through the modes of shortenough wavelengthsλ < λcr. As flame expands the number of short enoughangular modes reduces and the overall dissipation rate may become insufficientto counterbalance the generation of energy due to the Darrieus-Landau instability.This results in an accumulation of energy in the short wavelength spectrum, seeFig. 5 (right), and leads to the blow-up of the numerical solution if the numberKof used harmonics is not increased in time. This effect explains a slight elevationof the short wave tail of the spectral energy distribution onthe right of Fig. 4in comparison to the graph on the left. The latter one corresponds to the timemoment soon afterK was doubled, though the former one is just approaching themoment whenK needs to be increased.

In general, the process of stabilization of the expanding spherical front to asaturated state is very similar to the planar flame. The only distinctive differenceis that the transitional period for the expanding flame is much longer. Using therealistic set of dimensional parameters from [5], e.g. planar flame speed relative tothe burnt gasesub = 0.5 m/sec and thermal diffusivityDth = 2.5×10−5 m2/sec,one may interpret our findings in dimensional terms as follows. First cusps beginto appear on the flame surface for<r> ≈ 0.1 m; flame acceleration with the rate<r> ∝ t3/2 starts for<r> ≈ 0.4 m; acceleration rate begins to slow down for<r> ≈ 5 m, and the acceleration ceases for<r> ≈ 20 m. Most of experimentssummarized in [1] and reported in more recent works [15, 16] were carried outin enclosures and were affected by essential pressure rise and acoustics. Thus,there might be a considerable discrepancy between the characteristic flame radiijust reported and those measured in realistic combustion experiments. Eventually,of course there is a chance that the stabilization of the expansion rate is just aconsequence of the perturbative nature of the governing model used in this work.

4.2 Effect of forcing

A random point-wise set of perturbations uniformly distributed in time and in theFourier space is a suitable model for both the computationalround-off errors anda variety of perturbations of physical origins. In general,such a model would looklike

f(x, t) =

M(t)∑

m=1

am cos(ξmφ + ϕm)δ(t − tm), (31)

11

wheream, tm, ξm, andϕm are non-correlated random sequences. It is assumedthat t1 ≤ t2 ≤ · · · ≤ tm ≤ · · · ≤ tM(t) ≤ t, 0 ≤ ϕm ≤ 2π, andξm ≥ 0,m = 1, 2, . . . , M(t). However, in practice we use only two harmonics withξ =ξ∗ = floor [γ<r>/2] andξ = ξ∗ + 1 weighted according to their closeness to thecritical wavenumberξcr = γ<r>/2:

f(x, t) ≈ f0[(1− ξcr + ξ∗) cos ξ∗φ+(ξcr − ξ∗) cos(ξ∗ +1)φ]∞∑

m=0

δ(t− tm). (32)

These two harmonics approximate the critical planar flame harmonic of the wave-lengthλcr = 4π/γ, which was shown to contribute the most in the fastest growingperturbations in the Sivashinsky-type models of flame dynamics, see [17].

The sign of the noise amplitudef0 in (32) was either plus or minus for everym with the probability1/2. The delta functionδ(t − tm) was approximated by(πτ)−1/2e−(t−tm)2/τ with a small enough value ofτ . The impulse-like noise (32)is used here for the sake of simplicity. Some arguments towards its validity weresuggested in [18]. Validation of models like (31), (32) and of the present numer-ical approach was carried out in [17]. More sophisticated and physically realisticmodels of temporal noise characteristics can be used with (1) as well.

The importance of the effect of the amplitude of noise and frequency of ap-pearance of the impulse-like perturbations on the flame speed is illustrated in Fig.6.

0 1.5 3 4.5 6 7.5

x 104

00.20.40.60.8

t

<Φ

t>

0 1.5 3 4.5 6 7.5

x 104

0

0.2

0.4

0.6

0.8

t

<Φt>

f0=10−12

f0=10−6

f0=10−3

f0=10−1

Figure 6: Effect of forcing amplitude and frequency on the averaged flame expan-sion speed for<tm+1 − tm>= 50 (left) and<tm+1 − tm>= 10 (right).

More details of the effect of noise are presented in Fig. 7. Onthe left weplotted graphs of the time instance at which the flame begins to accelerate ver-sus frequency of appearance of the impulse-like perturbations for a variety of theperturbation amplitudef0. Similar graphs of the nearly saturated flame expansionspeed are given on the right.

12

−3 −2.5 −2 −1.5 −1500

1000150020002500

−log10

<tm+1

−tm>

t *

−3 −2.5 −2 −1.5 −10.4

0.5

0.6

−log10

<tm+1

−tm>

<Φt>

f0=10−12

f0=10−6

f0=10−3

f0=10−1

Figure 7: Effect of forcing amplitude and frequency on the beginning of acceler-ation (left) and stabilized flame expansion speed (right).

4.3 Three-dimensional numerical experiments

In order to integrate (8)-(10) we used an algorithm which is very similar to thatone developed in [9]. In addition, we parallelized the algorithm and used a fewtechniques to improve its stability and accuracy, see [19].

The basis functions in the spherical harmonics transformation (SHT) are theorthonormal eigenfunctions of the Laplace operator in spherical coordinates. How-ever, unlike the Discrete Fourier Transformation (DFT), the discrete SHT of a dataset is its approximation rather than an equivalent representation. Namely, it linksa full matrix withN2 elements representing a function on a regular spherical meshof N2 nodes in physical space and a triangular matrix ofN2/2 spherical harmon-ics coefficients. Thus, every back/forward cycle results inloss of information.This loss is similar to the effect of a short wave filter. In ourapproach it is associ-ated with the nonlinear term only.

Coordinate singularities at the sphere poles result in accumulation of approx-imation errors near the poles and weaken numerical stability. In order to preventaccumulation of approximation errors near the poles, we rotate the coordinate sys-tem around an axis in the equatorial plane by an angleω from time to time. Thecoordinate transformation formulas for such an axis passing through the pointsφ = ±π/2 are

cos θ′ = cos θ cos ω + sin θ sin ω cos φ

cotφ′ = cot φ cos ω − csc φ sinω cot θ

Their structure precludes use of the addition theorem for spherical polynomials totransform the coefficientsΦn,m to the new coordinate system without the globalSHT back to the physical space. On the other hand, the back/forward SHT entailsthe application of a short wave filter to the whole solutionΦn,m, rather than just tothe nonlinear termNn,m. Hence, these rotations should not be done too frequently.Also, it is useful to combine them with rotations around the axis passing throughthe poles. Implementation of the latter ones is trivial.

13

Message Passing Interface (MPI) parallelization paradigmwas implementedto allow the computational work to be distributed to a numberof processors. How-ever, the distributed data needs to be exchanged between these processors, whichcreates the biggest problem in modelling dynamics of large radius flames. Un-like the multidimensional DFT, discrete SHT does not possess a structure of atensor product of one-dimensional transformations. Therefore, the data transmis-sion required by the truly distributed discrete SHT is much more sophisticated andintensive than just back/forward row/column transposition of the global solutionarray required by the DFT, see e.g. [20]. This results in a very fast growth of thecommunication overheads when the number of processors increases.

Spherical computational algorithms based on parallel Legendre-Fourier trans-formations are much less efficient than the standard planar Fourier methods. How-ever, expanding spherical flames can be successfully simulated by the parallel SFTmethod. An example of the evolution of a random three-dimensional perturbationof a spherical flame is illustrated in Fig. 8.

Figure 8: Evolution of a spherical flame;t ≈ 40 (left) andt ≈ 72 (right). Hereγ = 0.764.

4.4 Dynamics of curved flame segments

Our numerical experiments showed that long time simulations of the three-dimensionalexpanding flames using discrete SHT might be possible on a parallel computerwith large enough physically shared memory. However, thereis an alternativeapproach based on simulations of the three-dimensional flame segments and fol-lowing extension to the whole surface by periodicity. In order to validate this ideawe applied it to the two-dimensional flames first.

14

Results of two-dimensional numerical simulations of the dynamics of sectorsθ ∈ [0, 2π/θπ] of a spherical flame are illustrated in Fig. 9. One may see thatthe narrowing of the sector does not affect the flame expansion rate in absence ofexplicit forcing and that a correlation between this rate and the size of the segmentbecomes apparent as forcing strengthens. Similar observations were obtained forother parameters discussed earlier in this Section.

0 0.5 1 1.5 2 2.5

x 104

0

0.2

0.4

t

<Φ

t>

0 0.5 1 1.5 2 2.5

x 104

0

0.2

0.4

t

<Φt>

θπ=1

θπ=2

θπ=4

θπ=8

Figure 9: Averaged flame segment expansion speed forf0 = 0 (left) andf0 = 0.1,<tm+1 − tm>= 10 (right).

Our calculations also show that there is a critical value ofθπ above which sec-torial simulations no longer represent the whole sphericalflame. For example, wewould not present the results of our simulations withθπ = 16 as an approximationfor the whole spherical flame. However, based on our two-dimensional results, itlooks like simulations of the three-dimensional sphericalflames using the Fourier,rather than the Fourier-Legendre, spectral model of the Sivashinsky type in thesector0 ≤ θ, φ ≤ π, or even in0 ≤ θ, φ ≤ π/2, are safe.

5 Conclusions

Long time interval simulations of a simplified model of the expanding sphericalflames indicated that their expansion rate slows down as the flame size grows. Thesaturation of the planar flame propagation speed as their size grows was estab-lished too. Hence, a hypothesis of stabilization of the spherical flame expansionrate over a finite time interval is proposed.

Further similarities with the propagating planar flames achieved by the ex-panding spherical flames on large time intervals were established in the studiesof the effect of forcing, revealing a clear correlation between the strength of theforcing and the flame expansion rate. This supports the idea that the accelerationof both planar and expanding flames results from explicit and/or implicit forcing,which is always present both in experiments and calculations at least as noise ofvarious physical origins.

15

In spite of many benefits of the simplified flame dynamics models, they havenot been constructed to cope with significant perturbationsof spherical flames.This casts a reasonable doubt in the possibility of extending the effects observedfor the simplified model to realistic flames. In order to verify the hypothesis,numerical simulations of a more sophisticated model are required. In particular,model [8] is valid for flames of any geometry if thermal gas expansion due tocombustion does not generate significant vorticity, i.e. for γ ≪ 1. However, evenin the coordinate form (3)-(6) the governing equation of model [8] is extremelydifficult to solve numerically, because, in contrast to (2),its nonlocal term has noconvolution structure.

Using parallelized spherical harmonics transformation, the evolution of a three-dimensional expanding spherical flame has been successfully simulated to a stagewhen wrinkles appear and form a well developed cellular structure. However,computational problems associated with the spherical harmonics transformationmake it difficult to extend these calculations on time intervals which would matchthose attained in our two-dimensional simulations. On the other hand, it was alsonoticed that the simulations of the three-dimensional spherical flames using theFourier, rather than the Fourier-Legendre, spectral modelof the Sivashinsky typein the sector0 ≤ θ, φ ≤ π, or even in0 ≤ θ, φ ≤ π/2, are reasonable.

Acknowledgements

This work was supported by the EPSRC (Grant GR/R66692, UK), the US-IsraelBinational Science Foundation (Grant 200 2008), the German-Israel Foundation(Grant G-695-15.10/2001), and the Israel Science Foundation (Grants 67-01 and278-03). One of the authors is grateful to the Royal Society,The Academy StudyGroup on Israel and the Middle East, and to the Tel Aviv University for theirsupport when working on this paper.

References

[1] Y.A. Gostintsev, A.G. Istratov, and Y.V. Shulenin. A self-similar regimeof free turbulent flame propagation in mixed gaseous mixtures. Physics ofCombustion and Explosion, 24(5):63–70, 1988.

[2] A.G. Istratov and V.B. Librovich. On the stability of gasdynamic discontinu-ities associated with chemical reactions. The case of a spherical flame.ActaAstronautica, 14:453–467, 1969.

16

[3] J.K. Bechtold and M. Matalon. Hydrodynamic and diffusion effects on thestability of spherically expanding flames.Combustion and Flame, 67:77–90,1987.

[4] G.I. Sivashinsky. Nonlinear analysis of hydrodynamic instability in laminarflames - I. Derivation of basic equations.Acta Astronautica, 4:1177–1206,1977.

[5] L. Filyand, G.I. Sivashinsky, and M.L. Frankel. On self-acceleration of out-ward propagating wrinkled flames.Physica D, 72:110–118, 1994.

[6] G. Joulin. Nonlinear hydrodynamic instability of expanding flames: Intrinsicdynamics.Physical Review E, 50(3):2030–2047, 1994.

[7] O. Thual, U. Frisch, and M. Henon. Application of pole decomposition toan equation governing the dynamics of wrinkled flame fronts.Le Journal dePhysique, 46(9):1485–1494, Septembre 1985.

[8] M.L. Frankel. An equation of surface dynamics modeling flame fronts asdensity discontinuities in potential flows.Phys. Fluids A, 2(10):1879–1883,1990.

[9] Y. D’Angelo, G. Joulin, and G. Boury. On model evolution equations for thewhole surface of three-dimensional expanding wrinkled premixed flames.Combustion Theory and Modelling, 4:1–22, 2000.

[10] http://www.scd.ucar.edu/css/software/spherepack. Spherepack 3.0. Univer-sity Corporation for Atmospheric Research.

[11] V. Karlin. Cellular flames may exhibit a nonmodal transient instability.Pro-ceedings of the Combustion Institute, 29(2):1537–1542, 2002.

[12] G. Joulin. On the hydrodynamic stability of curved premixed flames. J.Phys. France, 50:1069–1082, Mai 1989.

[13] V. Karlin. Nonmodal instability as the mechanism of acceleration of cellularflames. In A.B. Movchan, editor,IUTAM Symposium on Asymptotics, Sin-gularities and Homogenization in Problems of Mechanics, pages 549–556.Kluwer Academic Publishers, 2003.

[14] V. Karlin. Estimation of the linear transient growth ofperturbations of cel-lular flames. 14(8):1191–1210, 2004.

17

[15] D. Bradley, R.A. Hicks, M. Lawes, C.G.W. Sheppard, and R. Woolley. Themeasurement of laminar burning velocities and Markstein numbers for iso-octane-air and iso-octane-n-heptane-air mixtures at elevated temperaturesand pressures in an explosion bomb.Combustion and Flame, 115:126–144,1998.

[16] G. Rozenchan, D. L. Zhu, C. K. Law, and S. D. Tse. Outward propagation,burning velocities, and chemical effects of methane flames up to 60 atm.Proceedings of the Combustion Institute, 29(2):14611469, 2002.

[17] V. Karlin. Detailed analysis of a pseudoresonant interaction between cellularflames and velocity turbulence. PreprintarXiv:physics/0502036, athttp://arxiv.org, February 2005. 20 pp.

[18] G. Joulin. On a model for the response of unstable premixed flames to tur-bulence.Combustion Science and Technology, 60:1–5, 1988.

[19] J. Mai and V. Karlin. Numerical study of premixed laminar spherical flames.In Proceedings of the European Combustion Meeting (ECM 2005), Louvain-la-Neuve, Belgium, April 3–6 2005. paper 090.

[20] V. Karlin, V. Maz’ya, and G. Schmidt. High accuracy periodic solutions tothe Sivashinsky equation.Journal of Computational Physics, 188(1):209–231, 2003.

18