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.

Logarithmic nonlinear Schro··dinger equation and irrotational, compressible flows: An exact solution

“…(2) monatomic gas (γ = 5/3) [17,27], or (3) slab geometry [23]. We also verify that our results are different from several existing ones in the dynamics of a spherical gas cloud [28].…”

“…Spherically symmetric flow configurations are then obtained, and precise comparisons with existing works are tabulated in Sec. VI [22][23][24][25][26][27][28]. The advantage of the present approach, in contrast to searching directly for spherically symmetric solutions, is that slightly more general three dimensional solutions are possible by taking λ, μ, and θ as linear functions in time t, i.e., Eq.…”

Principles of the radiative ablation modeling

“…[6][7][8][9][10] These particular solutions remain difficult to compute numerically. For specific boundary conditions, there are simplified solutions expressible in terms of ordinary differential equations.…”

A hydrodynamic analysis of self-similar radiative ablation flows

“…Hydrodynamic instabilities, and in particular ablation flow instabilities, are a key and withstanding issue for laser-driven inertial confinement fusion (ICF). In an ongoing effort to obtain a better description of the early-irradiation flow instabilities of an ICF pellet implosion (IPI), a dedicated approach using, as mean flows, self-similar ablative heat-wave solutions of the gas dynamics equations with nonlinear heat conduction [1] in slab symmetry, has been developed and applied to direct laser irradiation configurations [2,3]. This approach consists in computing, with a spectrally accurate numerical method [4], semi-infinite slab mean flows and their time-dependent linear perturbations.…”

“…[2]). Self-similar radiative ablation flows within this bounding shock-wave approximation have been investigated -for a monatomic gas (γ = 5/3) and the fully ionized gas model of Kramers (µ = 2, ν = 13/2) -by means of an extensive exploration of the space of parameters (B p , B ϕ ) for solutions computable with the numerical method detailed in[4].…”