Stretch and instabilities

3.3.1 Flame stretch effects

Practical flames can be wrinkled and unsteady and they can exist in flow fields that are non-uniform and unsteady. Therefore a propagating flame front is subjected to a strain and curvature effect, which together constitute flame stretch and change the frontal area. The rate of stretch is defined as the normalized rate of change of an infinitesimal area element of the flame.

α= 1 A

dA

dt (3.6)

Flame stretch can increase or decrease the burning velocity significantly. For small to moderate rates of stretch (< 1000s⁻¹) its effect can be expressed to first order by [138]:

u_l−un= Lα (3.7)

Where L is a Markstein length. This length is often normalized to obtain a dimensionless Markstein number Ma:

Ma= L δ_l =Lu_l

ν (3.8)

Equation 3.7 can be non-dimensionalized by dividing the stretch α by the chemical time δl/ulto obtain the Karlovitz stretch factor Ka:

u_l−un

u_l = KaMa (3.9)

The Markstein length is a physicochemical parameter embodying the effect of flame stretch. It is important to measure its value and u_lsimultaneously to enable the derivation of the stretch free burning velocity and because it is a relevant measure for the stretch sensitivity of flames. A prerequisite for this derivation is a well defined stretch rate.

3.3.2 Flame front instabilities

A laminar flame can grow unstable through several mechanisms [138–140]. Some of these are illustrated in Figure 3.2. The different mechanisms have been expounded by Verhelst [27]. For completeness, part of the discussion is literally repeated here.

Figure 3.2: Structure of a wrinkled flame front showing the hydrodynamic streamlines and the diffusive fluxes of heat and mass [108]

When a flame front is regarded as a passive, infinitely thin interface between unburned and burned gases, a wave-like perturbation will increase the volumetric burning rate through increased flame area and will have the following additional effects:

• The discontinuity of density across the flame front (ρu→ ρb) causes a hydrodynamic instability, known as the Darrieus-Landau instability. As illustrated in Figure 3.2 a wrinkle of the flame front will cause a widening of the streamtube to the protrusion of the flame front into the unburned gases, resulting in a locally decreased gas velocity. This will cause a further protrusion of this flame segment, since the flame speed remains unaffected if the effect of stretch on the structure of the flame is neglected. So when only hydrodynamic stretch is considered, the flame is unconditionally unstable.

• The lower density of burned gases compared to unburned gases produces a second instability arising from gravitational effects. This buoyant instability, known as the Rayleigh-Taylor instability, arises when a less-dense fluid is present beneath a more-dense fluid, which is the case in e.g. an upwardly propagating flame.

• Finally, flame instability can be triggered through unequal diffusivities. As the flame propagation rate is largely influenced by the flame temperature, and this is in turn influenced by the conduction of heat from the flame front to the unburned gases (full arrows in Figure 3.2) and the diffusion of reactants from the unburned gases to the flame front (dashed arrows), a perturbation of the balance between diffusivities will have important effects.

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Three diffusivities are of importance: the thermal diffusivity of the unburned mixture (D_T), the mass diffusivity of the deficient reactant(D_M,lim) and the mass diffusivity of the excess reactant (D_M,exc). In a lean flame the deficient reactant is the fuel, in a rich flame it is oxygen. The ratio of two diffusivities can be used to judge the stability of a flame when subject to a perturbation or flame stretch.

The ratio of the thermal diffusivity of the unburned mixture to the mass diffusivity of the deficient reactant is called the Lewis number Le:

Le= D_T DM,lim

(3.10) If the Lewis number is greater than unity, the thermal diffusivity exceeds the mass diffusivity of the limiting reactant. When this is the case, a wrinkled flame front will see its parts that are ’bulging’ towards the unburned gases lose heat more rapidly than diffusing reactants can compensate for. The parts that recede in the burned gases, on the contrary, will increase in temperature more rapidly than being depleted of reactants. As a result, the flame speed of the ’crests’ will decrease and the flame speed of the ’troughs’ will increase, which counteracts the wrinkling and promotes a smooth flame front. The mixture is then thermo-diffusively stable. Similar reasoning shows that a Lewis number smaller than unity indicates unstable behavior.

• Another mechanism involving unequal diffusivities is the following: when the limiting reactant diffuses more rapidly than the excess reactant (D_M,lim>

D_M,exc), it will reach a bulge of the flame front into the unburned gases more quickly and cause a local shift in mixture ratio towards stoichiometry, which will increase the local flame speed. Thus, a perturbation is amplified and the resulting instability is termed preferential diffusion instability. This mechanism is easily illustrated by the propensity of rich heavier-than-air fuels (e.g. propane/air, iso-octane/air) and lean lighter-than-air fuels (e.g. methane/air, hydrogen/air) to develop cellular flame fronts [140].

The selective diffusion of reactants can be viewed as a stratification of the mixture. Both mechanisms involving unequal diffusivities are sometimes called differential diffusion instabilities, or instabilities due to non-equidiffusion.

The combined effect of instability mechanisms shows when the flame is stretched and it depends on the magnitude of the stretch rate. For example, a spherically expanding flame is subject to positive stretch (see 3.10), so a thermo-diffusively stable mixture (indicated by a positive Markstein length) will start out smooth as wrinkles tend to be smoothed out. When the stretch rate becomes too small to stabilize the hydrodynamic instability, cellular structures will develop.

3.3.3 Relevance to alcohol engines

Although methanol has the same molecular mass as O₂, it is a slightly bulkier molecule, which makes for a lower mass diffusivity. As a result, rich methanol-air mixtures will be destabilized through preferential diffusion (D_M,O2> DM,CH3OH).

Figure 3.3 compares the Lewis number of methanol-air mixture at 5 bar and 350 K to that of other fuels. The mass diffusivities of CH₃OH and O₂ in N₂ were considered to simplify the calculation. For rich mixtures the Lewis number of methanol-air is less than unity, indicating diffusional instability. For ethanol-air mixtures the situation is comparable. The diffusional instability of rich mixtures can be expected to be even more pronounced since ethanol is a larger molecule than methanol (D_M,C2H5OH< DM,CH3OH < DM,O2). This instability was recently confirmed experimentally for both methanol [141, 142] and ethanol [108, 143]. At 5 bar and 360 K both fuels exhibited negative Markstein lengths for φ> 1.2.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

Le

Φ

methanol ethanol methane iso-octane

Figure 3.3: Calculated Lewis numbers for various fuels at 0.5 MPa, 360 K. Rich methanol-and ethanol-air mixtures are thermo-diffusively unstable [142].

The instability region, characterized by negative Markstein lenghts, has been shown to grow with pressure [27, 108]. A first reason is that the hydrodynamic instability intensifies. This instability is mainly determined by the difference between unburned and burned densities, represented by the density ratio ρ, and the density gradient across the flame, influenced by the flame thickness δl. Although the density ratio only slightly increases with pressure, the flame thickness significantly decreases, strongly enhancing the gradients and the associated hydrodynamic instability. Secondly, the range of unstable wavelengths is bounded by a multiple of the flame thickness, thus a decrease in δlextends this range [27].

The lower wavelength limit results from quenching by convection and maximum laminar burning velocity [140].

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A decrease in thermo-diffusive stabilizing effects has also been suggested as a cause for the growing instability ratio with pressure [144]. Although the Lewis number remains fairly constant with pressure, recombination reactions become increasingly important with pressure, influencing preferential diffusion [145], which is not reflected in the simple concept of a Lewis number. The Markstein number, however, decreases with pressure and is thus a better parameter to characterize stability.

No conclusive trend has been reported for the evolution of instabilities at higher unburned mixture temperature. The density ratio ρ and flame thickness both decrease, rendering the effect on hydrodynamic instability unclear. Also, the Markstein length varies only slightly with temperature [143, 146].

The above means that at engine-like conditions, alcohol-air flames will be unstable, except maybe for very lean mixtures. This will have important effects on the turbulent burning velocity (see chapter 4) but also means that it is practically impossible to directly measure u_lat these conditions.