A simplified model for coupling Darrieus-Landau and diffusive-thermal instabilities

Abstract

A simplified phenomenological model is proposed to couple the long-wave Darrieus--Landau (DL) instability and the short-wave diffusive-thermal (DT) instability in premixed flames. By identifying a cubic coupling term in the linear dispersion relation, representing the leading-order interaction between hydrodynamic expansion and diffusive transport, this framework moves beyond the traditional treatment of these instabilities in isolation. Two distinct asymptotic regimes are identified: the first recovers the classical Michelson--Sivashinsky equation for order-unity positive Markstein numbers $\mathcal M>0$, the second reveals a distinguished DL-DT crossover regime where both instabilities participate at equal order. In this crossover limit, where the Markstein number is small ($\mathcal M \sim \sqrtε$ with $ε$ measuring thermal expansion), a generalized evolution equation is derived featuring a nonlocal stabilising term controlled by the hydro-diffusive number $\mathcal{N} = \mathcal A/δ_L^2$, where $\mathcal A$ is the hydro-diffusive area -- the characteristic area over which hydrodynamic and diffusive transport processes interact. This term remains active even when Markstein stabilisation vanishes. Numerical solutions in sufficiently large domains based on our model reveal a distinctive chaotic regime in which the characteristic DL cusp structures are in persistent competition with small-scale wrinkles. This minimal unified framework thus captures the essential coupled dynamics governing flame front instability and provides a tractable explanation for the fine-scale cellular structures and accelerated growth rates observed, without recourse to the full complexity of the complete conservation equations.

Figures (6)

• Figure 1: Numerical solution in a relatively large domain ($\nu=0.01$) with $\mu=+1$ (top row) and $\mu=-1$ (bottom row). Plotted are time snapshots of the flame shape and the trend of the propagation speed $U$; the small inset corresponds to the phase portrait, i.e., $U$ versus $dU/dT$.
• Figure 2: Numerical solution in a relatively large domain ($\nu=0.1$) with $\mu=+1$ (top row) and $\mu=-1$ (bottom row). Plotted are time snapshots of the flame shape and the trend of the propagation speed $U$; the small inset corresponds to the phase portrait, i.e., $U$ versus $dU/dT$.
• Figure 3: Numerical solution in a relatively small domains, $\nu=0.2$ (top row), $\nu=0.4$ (middle row) and $\nu=0.6$ (bottom row) with $\mu=-1$. The phase portrait for $\nu=0.4$ shows a homoclinic orbit connecting stable and unstable saddle points at $(U,dU/dT)=(4.54,0)$.
• Figure 4: Numerical solution in a large domain, $\nu=0.0005$, with $\lambda=+1$. Even for positive Markstein numbers, the dynamics becomes chaotic for sufficiently large domain sizes. Remarkably, the chaotic state retains the characteristic single-cusp structure of the DL instability, but the cusp is intermittently destroyed by the emergence of short-wavelength wrinkles and then reforms, giving rise to a complex, recurrent cycle. The supplementary file containing the animation is more insightful.
• Figure 5: Marginal stability curves for different values of $\epsilon=(r-1)/2$ in the limit $\beta\to \infty$ where $\beta$ is the Zeldovich number. The red dashed line corresponds to the formula $l=-2-8k^2$, applicable when $r=1$ (i.e., $\epsilon=0$), obtained by Sivashinsky sivashinsky1977diffusional. The four solid lines correspond to $r=\{1.2,1.5,2,5\}$ (i.e., $\epsilon=\{0.1,0.25,0.5,2.0\}$), obtained by Jackson and Kapila jackson1984effect. The green shaded region indicates the DL--DT crossover regime where the cubic term $-\mathcal{A}|k|^3$ identified in the present work becomes significant; the red shaded region indicates where it is negligible.