Diffusive–thermal instability

Diffusive–thermal instability is an instrinsic flame instability that occurs both in premixed flames and in diffusion flames and arises because of the difference in fuel diffusion coefficient and thermal diffusivity, characterized by non-unity values of Lewis numbers. The instability mechanism that arises here is the same as in Turing instability, although the mechanism was first discovered by Yakov Zeldovich in 1944 to explain the cellular structures appearing in hydrogen flames. Quantitative stability theory for premixed flames were developed by Gregory Sivashinsky (1977), Guy Joulin and Paul Clavin (1979) and for diffusion flames by Jong S. Kim (1997).

Dispersion relation for premixed flames

Diffusive-thermal instability diagram

To neglect the influences by hydrodynamic instabilities such as Darrieus–Landau instability, Rayleigh–Taylor instability etc., the analysis usually neglects effects due to the thermal expansion of the gas mixture by assuming a constant density model. Such a approximation is referred to as diffusive-thermal approximation or thermo-diffusive approximation. With a one-step chemistry model and assuming the perturbations to a steady planar flame in the form $e^{i\mathbf {k} \cdot \mathbf {x} _{\bot }+\omega t}$, where $\mathbf {x} _{\bot }$ is the transverse coordinate system perpendicular to flame, $t$ is the time, $\mathbf {k}$ is the perturbation wavevector and $\omega$ is the temporal growth rate of the disturbance, the dispersion relation $\omega (k)$ for one-reactant flames is given implicitly by

$2\Gamma ^{2}(\Gamma -1)+l(\Gamma -1-2\omega )=0$

where $\Gamma ={\sqrt {1+4\omega +4k^{2}}}$, $l\equiv (Le-1)/\beta$, $Le$ is the Lewis number of the fuel and $\beta$ is the Zeldovich number. This relation provides in general three roots for $\omega$ in which the one with maximum $\Re \{\omega \}$ would determine the stability character. The conditions for stability are given by the following equations

$8k^{2}+l+2>0,\quad 256k^{4}+(-6l^{2}+32l+256)k^{2}-l^{2}+8l+32>0$

describing two curves in the $l$vs.$k$ plane. The first condition is associated with condition $\Im \{\omega \}=0$, whereas in the second condition $\Im \{\omega \}\neq 0.$ The first curve separates the region of stable mode from the region corresponding to cellular instability, whereas the second condition indicates the presence of traveling and/or pulsating instability.

See also

• Turing pattern
• Darrieus–Landau instability