Self-similar solutions of unsteady ablation flows in inertial confinement fusion

Abstract

We exhibit and detail the properties of self-similar solutions for inviscid compressible ablative flows in slab symmetry with nonlinear heat conduction which are relevant to inertial confinement fusion (ICF). These solutions have been found after several contributions over the last four decades. We first derive the set of ODEs – a nonlinear eigenvalue problem – which governs the self-similar solutions by using the invariance of the Euler equations with nonlinear heat conduction under the two-parameter Lie group symmetry. A sub-family which leaves the density invariant is detailed since these solutions may be used to model the ‘early-time’ period of an ICF implosion where a shock wave travels from the front to the rear surface of a target. A chart allowing us to determine the starting point of a numerical solution, knowing the physical boundary conditions, has been built. A physical analysis of these unsteady ablation flows is then provided, the associated dimensionless numbers (Mach, Froude and Péclet numbers) being calculated. Finally, we show that self-similar ablation fronts generated within the framework of the above hypotheses (electron heat conduction, growing heat flux at the boundary, etc.) and for large heat fluxes and not too large pressures at the boundary do not satisfy the low-Mach-number criteria. Indeed both the compressibility and the stratification of the hot-flow region are too large. This is, in particular, the case for self-similar solutions obtained for energies in the range of the future Laser MegaJoule laser facility. Two particular solutions of this latter sub-family have been recently used for studying stability properties of ablation fronts.