## Two-Dimensional Instabilities in Thermonuclear FlamesWithin the star there are numerous mechanisms that have the potential to accelerate a deflagration wave. Landau-Darrieus(LD) instabilities can lead to wrinkling of the flame. Because the lighter ash lies below the heavier carbon-oxygen fuel, the flame interface is also subject to Rayleigh-Taylor(RT) and Kelvin-Helmholtz instabilities. Finally, the flame can be accelerated by interaction with turbulence arising from convective instabilities within the flame as well as turbulence generated by the deflagration itself. ## Hydrodynamic Instabilities
Landau and Lifschitz (Fluid Mechanics, 2nd Edition, § 128),
describe conditions for stability of a planar flame front under taking
into account the influences and gravity and capillary forces. The
conditions for instability under small deviations of the planar
interface are given by an dispersion relation in horizontal wave
number
where
## Landau-Darrieus InstabilityIn the limit where the acceleration due to gravity can be ignored, the roots of Landau's dispersion relation are both real, one positive and one negative; therefore the flow at the flame sheet is unstable against small perturbations. Effects of finite flame thickness, and the variability of rate of combustion as a function flame front curvature modify the growth of the flame sheet perturbations.
The following figure shows the late form appearance of a LD cusp
for the case of Temperature Movie: [ MPEG, MOV ] The following time snapshots show that LD cusps coalesce with time. Vertical Velocity Movie [ MPEG, MOV ] ## Rayleigh-Taylor InstabilitiesLandau's dispersion relation, in the limit of large gravity, yields the classical RT dispersion relation
Which is unstable when the density of the fuel is greater that the density of the ash.
Rayleigh-Taylor (RT) instabilities arise when a less dense material
supports a denser material against the action of some body force, such
as gravity. The unique feature of these 2D simulations is the
inclusion of the additional effect of the nuclear combustion of dense
carbon into the less dense magnesium ash. In this numerical
experiment,
## Transitional LD/RT Regime
In the region in which the gravitational contribution to the Landau
dispersion relation are of the same order as the other terms when the
wavelength of the perturbations,
In the following simulation the
## References
J. B. Bell, M. S. Day, C. A. Rendleman, S.
E. Woosley, and M. A. Zingale, "Adaptive low Mach number
simulations of nuclear flame microphysics",
LBNL Report 52395,
J. Comp. Phys,
J. B. Bell, M. S. Day, C. A. Rendleman, S. E. Woosley,
and M. A. Zingale, "Direct Numerical Simulations of Type Ia
Supernovae Flames I: The Landau-Darrieus Instability",
LBNL Report LBNL-54088,
Astrophysical Journal, J. B. Bell, M. S. Day, C. A. Rendleman, S. E. Woosley, and
M. A. Zingale, ``Direct Numerical Simulations of Type Ia
Supernovae Flames II: The Rayleigh-Taylor Instability,'',
LBNL Report 54300,
Astrophysical Journal,
M. Zingale, S. E. Woosley, C.A. Rendleman, M. S. Day, and J. B. Bell,
``Three-dimensional Numerical Simulations of Rayleigh-Taylor
Unstable Flames in Type Ia Supernovae'',
LBNL Report LBNL-56966,
submitted, |