Swaminathan N., Bray K.N.C. (eds.) Turbulent Premixed Flames

Подождите немного. Документ загружается.

3.1 Instabilities in Flames 163

time (s)

Figure 3.9. Computed F and p 1 km from the centre of the ﬁreball for large spherical

atmospheric explosions of hydrogen (broken lines), φ = 0.5 and propane–air (solid lines),

φ = 1.06, up to maximum radii of 100 m.

for the propane–air ﬂame, φ = 1.06, that was studied in the large-scale experimental,

atmospheric explosions of [22]. Relevant properties are Ma

= 5.5, u



= 0.41 m/s,

B = 0.242, σ = 8.303, and

c = 333.7m/s. The computed values of F and p are

plotted against the time from the onset of the instability by the solid lines in Fig. 3.9.

The time lag in the arrival of the pressure pulse is apparent. Values of p are for

a distance of 1 km from the centre of the ﬁreball. The computed ﬂame radii extend

to a maximum of 100 m, when the ﬂame terminates. With σ = 8.303, for this radius

of a completely burned mixture, the original radius over which the initial unburned

mixture extended, prior to ignition, would be 49.4 m. The maximum computed

pressure of about 0.3 kPa is not severely damaging, although it would be higher

closer to the ﬂame and is rising rapidly as the ﬂame propagates. It is close to a value

at which glass windows would fail. In the experiments of [22], the maximum ﬂame

radius was 3.75 m after 0.51 s.

In contrast, the broken lines give computed values of F and p in an atmospheric

explosion of a more unstable mixture of hydrogen–air, φ = 0.5, Ma

=−8, u



0.65 m/s[6], B = 0.59, σ = 5.01, and

c = 379.8m/s. Again, the maximum ﬂame

radius was 100 m and d was 1 km. This explosion is more rapid, with F reaching a

maximum value of 32 and p one of about 2 kPa. Clearly, larger explosions would

be even more damaging and there is no apparent limit to the acceleration of large

unstable ﬁreballs, other than that provided by an increasing radiative energy loss

from the burned gases [22]. This is an area worthy of further study.

RAYLEIGH–TAYLOR INSTABILITIES. The pressure waves generated by DLTD insta-

bilities can create additional instabilities and overpressures. Taylor instabilities

164 Combustion Instabilities

arise when pressure gradients are aligned orthogonally to the high-density gra-

dients at the ﬂame surface, with vorticity generation by means of the baroclinic

term, ∇(1/ρ) ×∇p [25]. Markstein [26] demonstrated this instability by wrinkling

a smooth spherical laminar ﬂame surface with an impacting shock wave. When the

planar pressure pulse hit the sphere, it accelerated the lower-density burned gas

preferentially. This phenomenon was also observed in vented explosions [14, 27]

and ﬂame propagation along tubes [28], where it can produce large increases in the

burning rate. The Rayleigh instability occurs when the combustion energy release

feeds directly into the positive phase of a pressure wave, amplifying it according

to Rayleigh’s [29, 30] criterion: The amplitude of an acoustic wave will increase if

heat is added in phase with the pressure. The Rayleigh source term





dV [see

Eqs. (3.39) and (3.40)] in the acoustic energy equation can be highly destabilising,

and the unsteady rate of heat release arising from ﬂuctuations in φ during the com-

bustion of lean premixtures also can affect the oscillation frequency [31]. In practice,

Rayleigh–Taylor thermoacoustic instabilities are usually coupled and are hereafter

designated by RT. The thermoacoustic instabilities and their control strategies and

simulation methods are discussed in Sections 3.2 and 3.3, respectively.

Conﬁned and semiconﬁned ﬂames create overall increases in pressure and gen-

erate pressure waves that reﬂect at containing walls. Transitions between different

instabilities in conﬁned explosion ﬂames were studied in an explosion bomb of 385-

mm diameter with three pairs of orthogonal windows of 150-mm diameter [32].

Simultaneous ignition at two diametrically opposite sparks at the wall of the bomb

created two near-identical imploding ﬂames that could be viewed at the central win-

dow. The advantage of this conﬁguration was that the ﬂames propagated at higher

pressures and Peclet numbers than would have been possible with central ignition.

The technique was used to measure laminar burning velocities at high pressures in

[33]. Some experimental results from [32] for explosions of three different mixtures,

initially at 0.5 MPa and 358 K but with different degrees of instability, are presented

in Table 3.1. These values were measured at the maximum values of u

.Thethree

mixtures ranged from a relatively stable stoichiometric i-octane–air mixture to a very

unstable rich mixture, φ = 1.6, of the same gases.

It was possible to explain the maximum enhanced burning velocity

(

)

max

of the

ﬁrst and most stable stoichiometric mixture entirely in terms of DLTD instabilities

after Pe

was attained. The value of p was too small to be measured. Values of

F were found from Eqs. (3.14) and (3.16) and of u

from Eq. (3.19). The second

mixture was of H

–air with φ = 0.4, and this created a small primary thermoacoustic

instability, with p = 0.8 kPa at

(

)

max

.AfterPe

was attained, burning velocities

initially could be attributed entirely to DLTD instabilities until just after the onset of

ﬂame oscillations, when the associated cellularity began to decline. The oscillations

developed in tandem at both of the imploding ﬂames and seemed to cause the

reduction in both ﬂame cellularity and u

These changes are explained by reference to Fig. 3.2. The Taylor effect suggests

that, with this conﬁguration, a pressure higher on the unburned than on the burned

side, the leading edge of burned gas would be preferentially accelerated towards

the burned gas, reducing the cellularity. Conversely, with a higher pressure on the

burned side, the same leading edge of burned gas would be preferentially accelerated

towards the unburned gas, now increasing the cellularity. Thus the Taylor generation

of vorticity can either stabilise or destabilise the ﬂame. Laser-sheet observations and

3.1 Instabilities in Flames 165

Table 3.1. Summary of experimental ﬁndings [32] at maximum u

, with all values for this

condition. PA is primary acoustic and SA is secondary acoustic oscillations. estimated value,

is Rayleigh-Taylor stabilisation

pT u



at (u

)

max

F at p at Type of

Mixture (MPa) (K) (u

)

max

(m/s) Ma

(m/s) (u

)

max

)

max

(kPa) instability

i-C

–air 1.8 502 0.295 2 0.66 2.2 0 DLTD

φ = 1.0

–air 0.97 440 0.34 −11 2.19 6.4 0.8 DLTD,

φ = 0.4 PA, RT

i-C

–air 2.06 508 0.10 −20

∗

3.75 37.5 300 DLTD,

φ = 1.6 RT , PA

the changing values of u

suggest the stabilising, or laminarising, tendency to be

the stronger, with initially a pronounced reduction in cellularity. Subsequently this

was followed by the development of a ‘ﬁnger-like’ surface structure. Particle image

velocimetry also showed that, with unburned-gas ﬂow into the ﬂame surface, the

surface was smoothed and with the ﬂow away from the surface it tended to form the

ﬁnger-like structure [32]. Kaskan [34] observed similar stabilising and destabilising

area changes in near-ﬂat ﬂames propagating along a tube and subjected to pressure

oscillations of small amplitude. Later experiments showed cellular ﬂames to be

laminarised by low-intensity primary acoustic oscillations with a maximum p of

about 0.5 kPa during ﬂame propagation along an open tube [35].

There is no such effect with the more severe, secondary thermoacoustic oscilla-

tions generated in the third, highly unstable, rich iso-octane–air mixture in Table 3.1.

This is clearly manifest as a severe secondary thermoacoustic instability, encouraged

by a very negative Ma

and triggered by the original DLTD instabilities. It yields

high values of p = 300 kPa and of F = 37.5at

(

)

max

. The values of u

are more

than ﬁve times greater than those predicted for DLTD instabilities alone. This and

the high value of p conﬁrm the generation of strong RT instabilities, which are

often referred to as thermoacoustic instabilities. These are addressed in detail in the

following two sections of this chapter.

Figure 3.10 shows the developing amplitude of the pressure oscillations as the

pressure rises, as well as increases in u

and the changing rate of change of the heat

release rate. In the initial stages of the DLTD instabilities, the theory of the earlier

subsection on the peninsula of instability was able to predict the values of u



, indi-

cated by the diamond symbols. This instability seeded the RT instabilities through

the rate of increase of u



and associated increase in p, which further increased

values of u

. The further growth of p provided a powerful feedback mechanism

for further increases. Diagrams similar to Fig. 3.10 are provided in [32] for the other

two mixtures of Table 3.1.

In ﬂame propagation from the end of a closed tube, pressure waves are generated

by the impulsive start to ﬂame propagation at the closed end. These were analysed

numerically in [36]. A leading shock was found to be reﬂected at the open end and

the reﬂected wave then reﬂected at both ends, generating an increasingly complex

sequence of reﬂections. With high burning velocities, a rapid acceleration to a high

ﬂame speed generates higher values of p in secondary thermoacoustic oscillations.

166 Combustion Instabilities

Figure 3.10. Changes in u

in ﬁgure) and pressure oscillations during inward propagation

of two near-simultaneous i-octane–air, φ = 1.6, ﬂames. The dotted curve shows the rate of

change of heat release rate [32].

Experiments showed that high ﬂame speeds eliminate reversals in the direction of

ﬂame propagation that arise from the interaction of the ﬂame with the oscillating col-

umn of gas at lower ﬂame speeds [37]. These reversals led to the laminarisation of the

ﬂame. At the higher ﬂame speeds, secondary thermoacoustic instablities with higher-

pressure amplitudes p of up to 20 kPa ultimately destroyed the cellular structure.

This was succeeded by one more akin to that of a highly turbulent ﬂame, with a

burning velocity many times greater than the laminar burning velocity [35]. Theo-

retically, Bychkov [38] and Aldredge [39] identiﬁed a range of acoustic amplitudes

in which ﬂames are stabilised by primary acoustic velocity oscillations. This regime

is intermediate between those of primary and secondary thermoacoustic instabilites.

Unconﬁned ﬂames also can be subjected to RT intabilities, arising from the

generation of oscillations in the burner manifold ahead of the ﬂame. Strong noise

can radiate because of the acoustic–combustion coupling between the ﬂame and

the manifold’s resonant modes. For the conﬁgurations considered in [40], pressure

perturbations in the manifold were almost in phase with perturbations of the heat

release rate. The power generated by the Rayleigh source term minus the small

damping losses in the system determined the acoustic radiation to the far ﬁeld,

which is explored more in Section 3.2.

3.1.2 Turbulent Burning, Extinctions, Relights, and Acoustic Waves

When mild velocity ﬂuctuations are imposed on an unstable laminar ﬂame, the burn-

ing velocity enhanced by instabilities persists [41, 42]. Paul and Bray [43] developed

a ﬂamelet model for moderately turbulent combustion, based on the numerical sim-

ulations in [41]. It accounts for the persistence of DLTD instability, together with

stretch rate and Markstein number effects. The predicted burning velocities are in

good agreement with experimental measurements. As the spectrum of turbulence

3.1 Instabilities in Flames 167

develops with an increasing rms turbulent velocity u

rms

, the length scales increas-

ingly subsume those of the original unstable wavelengths. In the experiments in [32]

this process seemed to be completed at about u

rms



= 3 and the original pressure

ﬂuctuations that are due to instabilities largely disappeared. At higher turbulence

intensities, values of Markstein numbers continue to inﬂuence the burning rate and

ﬂame extinction [44].

A practical deﬁnition of the turbulent burning velocity u

is that when multiplied

by the unburned-gas density and the appropriate ﬂame area it gives the mass rate at

which burned gas is formed. Deﬁning this area is problematic because of the very

different ﬂow geometries in the various burners and explosion bombs that are used

to measure u

. In addition, as correlations involve u

rms

, this should be the same

over the entire ﬂame surface. Vital information on the turbulent ﬂame bush can be

revealed by laser-sheet measurements of the distribution of reactants, products, their

interfaces, and the mean reaction progress variable

c. Values of c have been used to

deﬁne appropriate ﬂame surface areas for u

[45–47]. Further progress on this topic

is discussed in Section 2.3.

An earlier correlation of experimental turbulent burning velocities expressed

rms

as a function of the turbulent Karlovitz stretch factor K and Le [48]. For

isotropic turbulence, K is based on the rms strain rate on a randomly orientated

surface, u

rms

/λ, where λ is the Taylor scale of turbulence. When normalised by the

chemical time, δ



, this gives

K =

rms



λ u



. (3.23)

With δ



approximated by ν/u



and λ related empirically to the integral length scale

 by λ

/ = A



ν/u

rms

, where A



is an empirical constant, here assigned a value of

16, Eq. (3.23) gives

K = 0.25 Re

−1/2



rms





, (3.24)

where Re

is the turbulent Reynolds number based on .

When the ﬂame surface area of spherical explosion ﬂames is deﬁned by a surface

at which the mean reaction progress variable is

c, the associated burning velocity

measured u

in the isotropic turbulence of a fan-stirred bomb was correlated by [47]:

rms

= A

α K

, (3.25)

in which A

is a constant that is dependent solely on the surface selected to deﬁne

the turbulent burning velocity based on the mass rate of burning. For the surface at

which the mass unburned behind it is equal to the mass burned ahead of it, A

unity and

c is usually close to 0.59. For a leading edge with c = 0.05, the value of A

is 0.75. Values of α and β are expressed by ﬁrst-order equations in Ma

α =

0.022

(

30 − Ma

)

for + ve Ma

0.0311

(

30 − Ma

)

for − ve Ma

β =

0.0105

(

− 30

)

for + ve Ma

−0.0075

(

+ 30

)

for − ve Ma

. (3.26)

168 Combustion Instabilities

A limit to the ever-increasing turbulent burning velocity with u

rms

is provided by

localised ﬂame extinctions at sufﬁciently high values of K [49].

Flame-extinction stretch rates in laminar ﬂow at different φ were obtained with

experimental opposed jet ﬂames. Values are plotted in Fig. 3.1 by the solid lines for

and CH

[50] and H

[51], with air, under atmospheric T and p. The high values

for lean H

mixtures are particularly noteworthy. When turbulent ﬂames make very

brief, localised excursions into stretch rates that are higher than those that would

extinguish a laminar ﬂame, they do not necessarily extinguish [52]. Because of this

apparent time lag and other theoretical complexities [44], data on turbulent ﬂame

quenching are best found experimentally. The probability of an initial ﬂame kernel

continuing to propagate, p

, was measured in fan-stirred bombs at different rms

turbulent velocities, u

rms

. These experiments were with mixtures of CH

, and

i-octane with air and very lean H

–air mixtures, all in the pressure range 0.1–1.5 MPa

and at room temperature. Values of K for probabilities of 0.8 and 0.2 were expressed

in terms of Ma

by [49]

0.8

(

+ 4

)

1.8

= 34.4forp

0.8

, (3.27a)

0.2

(

+ 4

)

1.4

= 37.1forp

0.2

, (3.27b)

for −3.0 ≤ Ma

≤ 11.0. Values of K

0.8

are plotted against Ma

in Fig. 3.6.

Localised turbulence-induced extinctions followed by relights emerge as a prob-

lem when lean hydrocarbon–air mixtures are leaned off in low-NO

burners. The

associated rapid changes in dV/dt can induce pressure oscillations with frequencies

as high as 100 Hz [53]. A combined computational and experimental study of pre-

mixed turbulent combustion of CH

–air in a swirl burner showed a region of high

stretch rate between inner and outer recirculation zones [54]. As the mixture was

leaned off and K increased, reaction in this region was locally extinguished at about

φ = 0.59. However, reaction continued elsewhere, stabilised by the hot gas in an

inner recirculation zone in a different ﬂame conﬁguration. Video ﬁlms revealed os-

cillations at a frequency of about 20 Hz between the two quasi-steady solutions with

the generation of acoustic waves. With a further reduction in φ, the redistribution of

strain rates generated a ﬁnal ﬂame conﬁguration, after which further reduction in φ

below 0.56 extinguished the ﬂame completely.

In another important area, localised ﬂame extinctions by high stretch rates limit

the continuing acceleration of runaway turbulent ﬂames in ducts with turbulence-

inducing obstacles. Under these conditions, the turbulent burning velocity attains a

maximum value [55]. If this and the generated shock waves [36] are high and strong

enough to auto-ignite the unburned mixture between the shock wave and the ﬂame,

a transition from deﬂagration to detonation can ensue [55].

3.1.3 Auto-Ignitive Burning

AUTO-IGNITIVE PROPAGATION VELOCITY. Where the intended combustion mode is

either a premixed laminar or turbulent ﬂame, a stochastic transition to auto-ignitive

burning can be perceived as an instability. A familiar example is knocking combustion

in a gasoline engine. Efﬁcient engine operation, with a high burn rate on the fringe

of severe knock, can be nulliﬁed by occasional cyclic excursions into such a knock

3.1 Instabilities in Flames 169

as a result of the sensitivity of auto-ignition to comparatively small changes. In

laminar and turbulent ﬂames, molecular-transport processes are integral parts of the

mechanism of ﬂame propagation. This is not the case with auto-ignition.

In a homogeneous mixture, auto-ignition manifests itself as a spontaneous re-

action throughout the mixture, without any propagating front and occurs after the

auto-ignition delay time τ

has elapsed. Where there are spatial gradients of τ

a radial direction r, because of those of either T or φ, or both, an auto-ignition

propagation velocity can arise, given by

dτ

. (3.28)

When u

is close to the acoustic velocity

c, the two can become coupled in a

strong shock wave, with the creation of a new entity, a detonation wave, with a

distinctive high-pressure peak and a very high detonation velocity. An important

parameter is the ratio ξ =

c/u

. Because of inevitable inhomogeneities in mixtures,

auto-ignitive propagation usually originates at a hot spot at which the initial radius

of the hot spot must exceed the critical radius given by thermal-explosion theory

[56]. This condition is usually satisﬁed in engines. The rate of change of the rate of

volume generation again generates pressure waves, as given by Eq. (3.17).

From Eqs. (3.17) and (3.20), with r equal to the radius of an auto-ignitively

propagating spherical hot spot, replacing u

with u

, and with ξ =

c/u

p(t) =

4π d









t−t

(σ −1)

ρrc

(σ −1)



dξ

−1



t−t

(3.29)

With

c =

√

γp/ρ,Eq.(3.29) becomes

p(t)

r γ

(σ −1)



dξ

−1



t−t

. (3.30)

The last term within the square brackets, unlike that with unstable ﬂames, is likely

to be relatively small, and the fractional pressure increase that is due to the auto-

ignition, characterised as knock, is inversely proportional to ξ

. This can be as high as

0.4MPaat1.4 MPa for a single hot spot [57]. To derive this magnitude requires values

of u

. These can be found most conveniently if the mixture composition is assumed

uniform and the gradient of reactivity arises entirely as a temperature gradient. In

that case, Eq. (3.28) can be re-expressed as



∂T

∂r



−1



∂τ

∂T



−1

. (3.31)

Values of τ

(

T, p

)

differ between different fuels and change in different ways

with T and p . This is illustrated by the plots of τ

(

T, p

)

, at the indicated pressures,

against 1000/T in Fig. 3.11, taken from [58]. Four of the curves are for primary ref-

erence fuels (PRFs), which are used in attempts to simulate the knocking behaviour

of gasolines. The octane number, ON, is the percentage by volume of i-octane mixed

with n-heptane in a PRF. Only at the high temperatures in Fig. 3.11 do they exhibit

a similar straight-line Arrhenius relationship to the other three fuels. The variation

with pressure is an inverse one, expressed by p

−n

, where n is a pressure exponent.

170 Combustion Instabilities

Figure 3.11. Ignition delay times τ

(T, p) of various fuels with air at φ = 1.0, unless otherwise

stated. See [58] for source references.

It has a value of about 1.7 for PRFs and one closer to unity for many non-PRFs,

for T < 1000 K [59]. At higher temperatures, values of τ

(

T, p

)

become closer for

different PRFs.

Values of ∂τ

/∂T were derived for the PRF with ON = 60 for a stoichiometric

mixture at 4 MPa from the τ

versus 1/T data in [60]. To obtain u

, it is necessary

to assume a value of ∂T/∂r. Temperature gradients arise at cooled surface boundary

layers and, in the body of the charge, due to turbulence. An assumed value of

∂T/∂r =−1K/mm gave the values of u

shown by the cross symbols in Fig. 3.12

and, from the corresponding values of

c, it also gave those of ξ. The 60 ON curve

in Fig. 3.11 shows a region where ∂τ

/∂T is close to zero between about 800 and

860 K, which implies an inﬁnite value of u

and ξ = 0. It is apparent from Fig. 3.11

that, for the PRFs, ∂τ

/∂T increases numerically with ON. Hence u

will decrease and

ξ will increase. Clearly, from Eq. (3.30), low values of ξ imply large values of p(t)/p,

and severe knock would be anticipated for the conditions of Fig. 3.12 in a gasoline

engine. It would become less severe at about 950 K, but severe high-temperature

knock would begin to develop above 1100 K.

The computations of u

are for the given initial boundary conditions of T and

p. In practice, these correspond best to auto-ignitions in shock tubes and rapid-

compression machines. For slower compressions in engines when auto-ignition is

3.1 Instabilities in Flames 171

Figure 3.12. Variations of ξ and u

with temperature at 4 MPa, stoichiometric 0.60 i-octane,

0.40 n-heptane, and air, derived from τ

values in [55], dT/dx =−1K/mm. From [58].

not completed at ﬁxed values of T and p , the mixture will have partially reacted

during compression. Consequently the mixture will auto-ignite rather more rapidly

than is suggested by the tabulated values of τ

for T and p. With a reduced effective

in Eq. (3.31), u

would be increased.

MODES OF AUTO-IGNITION. A value of ξ close to unity is particularly signiﬁcant

because it suggests a resonance between the acoustic speed

c of the pressure wave

and the propagation velocity u

of the auto-ignitive front. The auto-ignition delay

time is followed by a much shorter excitation time τ

, during which most of the heat

release rate occurs [61]. A high rate of chemical heat release at small τ

, continually

feeding into the pressure front, will reinforce the front and increase the probability

of a detonation. The residence time of a pressure pulse in the hot spot is r

c = t

and a dimensionless measure of this energy input into the pulse is

ε =

. (3.32)

Here ε is the number of excitation times feeding into the pressure pulse during its

residence in the hot spot.

In a detonation wave the coupling of the chemical reaction with the shock wave

results in a very damaging high-pressure spike, and this single entity propagates at

the detonation velocity. In addition to this mode of auto-ignitive propagation, there

are a variety of other modes, and these depend on the values of ξ and ε. They were

identiﬁed in DNSs of auto-ignitions at spherical hot spots. The simulations involved

detailed chemical kinetics over wide ranges of initial values of ∂T/∂r and r

[62, 63].

Equimole CO–H

mixtures with air at different φ were chosen for this study because

of their relatively simple detailed chemical kinetics.

172 Combustion Instabilities

εε

DEVELOPING

DETONATION

Figure 3.13. Developing detonation peninsula on plot of ξ against . Other modes are thermal

explosion below ξ

, supersonic and subsonic auto-ignitive fronts propagate in regions P and

B, respectively. From [62].

It was ﬁrst necessary to compute values of τ

and τ

for different φ, p, and T .The

ensuing simulations ultimately identiﬁed the upper and lower limits of detonation,

and ξ

. These are indicated in Fig. 3.13,from[62], by the open triangle and

cross symbols. These deﬁne the boundaries of a developing detonation peninsula

within which detonations develop within a hot spot. Peak detonation pressures,

when normalised by the ambient pressure, increase with ε. Under these conditions

Eq. (3.30) cannot be used to predict the pressure pulse. At very low ε, as its value

increases, so does the range of ξ over which a detonation can develop at a hot spot.

At the highest values of ε this range does not appear to change. The smallest values

of ε are associated with the least-reactive lean mixtures and the largest values with

the most reactive, with φ = 1.0 and T = 1200 K.

With even more reactive mixtures, with values of ε of about 70, the detonation

can continue to propagate outside the hot spot in the absence of any gradient of

reactivity [64]. In addition, as the reactivity increases, so does the pressure ratio

at the detonation front and the hot-spot number density. Initially separate waves

from hot spots interact and are reinforced, inducing further auto-ignitions that can

generate severe engine knock [65]. Intersecting shock waves from closely packed

neighbouring hot spots can create enlarged (though not necessarily spherical) hot

spots that, because of their increased size and higher temperature and pressure,

have higher values of ε. Further study is needed to identify the initiating conditions,