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.

Exact similarity solutions for inviscid compressible ablative flows in slab symmetry with nonlinear heat conduction are proposed for studying unsteadiness and compressibility effects on the hydrodynamic stability of ablation fronts relevant to inertial confinement fusion. Both the similarity solutions and their linear perturbations are numerically computed with a dynamical multidomain Chebyshev pseudospectral method. Herewith the first analysis of laser-imprinting based on a dynamic solution is presented, showing that maximum perturbation amplification occurs for a laser-intensity modulation of zero transverse wave number, with growth dominated by the mean flow stretching.

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The classical Cauchy–Poisson problem, of water waves generated by an impulsive disturbance on the free surface, is treated in three dimensions for finite constant depth. The conventional solution is in the form of a Fourier-Bessel transform. We wish to find its asymptotic behaviour at large distances r and large times t . Difficulties arise at the wavefront, where r/t is equal to the maximum group velocity. In the analogous two-dimensional problem the waves near the front are associated with two nearly coincident points of stationary phase and described asymptotically by Airy functions. In the three-dimensional problem the solution is expressed as a double integral, where there are four nearly coincident points of stationary phase. A systematic procedure is given for successive terms in an asymptotic expansion, involving the square of an Airy function and its derivatives. In both two and three dimensions it is important to use an appropriate transform of the variable(s) of integration, to achieve uniform validity of the asymptotic expansion. Calculations are performed to illustrate the utility of the asymptotic results.