Self-similar solutions of unsteady ablation flows in inertial confinement fusion (original) (raw)
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Self-similar solutions of unsteady ablation flow
We exhibit and detail the properties of exact self-similar solutions for inviscid compressible ablative flows in slab symmetry with nonlinear heat conduction 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 that 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 this is the most relevant case for ICF. A physical analysis of these unsteady ablation flows is then provided where the associated dimensionless numbers (Mach, Froude and Péclet numbers) are calculated. Finally we show that these solutions do not satisfy the constraints of the low Mach number approximation, implying that ablation fronts generated within the framework of the present hypotheses (electronic conduction, growing heat flux at the boundary, etc.) cannot be approximated by a steady quasi-incompressible flow as it is often assumed in ICF. Two particular solutions of this family have been recently used for studying stability properties of ablation fronts [Abéguilé et al., PRL 97, 035002 (2006)], since they are representative of the flows that should be reached on the future French Laser MégaJoule facility.
Linear perturbation response of self-similar ablative flows relevant to inertial confinement fusion
Journal of Fluid Mechanics, 2008
A family of exact similarity solutions for inviscid compressible ablative flows in slab symmetry with nonlinear heat conduction is proposed for studying unsteadiness and compressibility effects on the hydrodynamic stability of ablation fronts relevant to inertial confinement fusion. Dynamical multi-domain Chebyshev spectral methods are employed for computing both the similarity solution and its time-dependent linear perturbations. This approach has been exploited to analyse the linear stability properties of two self-similar ablative configurations submitted to direct laser illumination asymmetries (Abéguilé et al., Phys. Rev. Lett., vol. 97, 2006, 035002). Linear perturbation temporal and reduced responses are analysed, evidencing a maximum instability for illumination asymmetries of zero transverse wavenumber as well as three distinct regimes of ablation-front distortion evolution, and emphasising the importance of the mean flow unsteadiness, compressibility and stratification.
Stability of ablation flows in inertial confinement fusion: optimal linear perturbations
HAL (Le Centre pour la Communication Scientifique Directe), 2021
Most amplified initial perturbations are found for an unsteady compressible ablation flow with nonlinear heat conduction that is relevant to inertial confinement fusion (ICF). The analysis is carried out for a self-similar base flow in slab symmetry which is descriptive of the subsonic heat wave bounded by a leading shock front, that is observed within the outer shell, or ablator, of an ICF target during the early stage, shock transit phase, of its implosion. Three dimensional linear perturbations as well as distortions of the flow external surface and shock front are accounted for. The optimisation is performed by means of direct-adjoint iterations. The derivation of a continuous adjoint problem follows from the Lagrange multipliers method. The physical analysis of these optimal perturbations reveals that nonmodal effects do exist in such an ablation flow and that transient growth may dominate the flow stability until the end of the shock transit phase, even at short wavelengths which are held as innocuous from the classical standpoint of the ablative RichtmyerMeshkov instability. Perturbations are not only amplified in the ablation front, but in the whole flow which is found to be especially sensitive to perturbations in the compression region. This fact points out ablator bulk inhomogeneities as the most detrimental defects regarding the shock transit phase of an ICF implosion.
Stability of ablation flows in inertial confinement fusion: Nonmodal effects
Physical Review E
Fast transient growth of hydrodynamic perturbations due to non-modal effects is shown to be possible in an ablation flow relevant to inertial confinement fusion (ICF). Likely to arise in capsule ablators with material inhomogeneities, such growths appear to be too fast to be detected by existing measurement techniques, cannot be predicted by any of the methods previously used for studying hydrodynamic instabilities in ICF, yet could cause early transitions to nonlinear regimes. These findings call for reconsidering the stability of ICF flows within the framework of non-modal stability theory.
Kovásznay modes in stability of self-similar ablation flows of ICF
EPL (Europhysics Letters), 2008
The history of the linear perturbations in a "laser imprinting" configuration in inertial confinement fusion is described. The time-dependent mean flow is provided by a self-similar solution of gas dynamics equations with nonlinear heat conduction for semi-infinite slabs of perfect gases. The analysis is conducted with the Kovásznay modes, namely the vorticity, acoustic and entropy modes. Exact propagation equations for these three basic modes are derived. Both the similarity solutions and their linear perturbations are numerically computed with an adaptive multidomain Chebyshev method. In particular, the dynamics of the shock wave is detailed. Compressibility effects and possible implications to inertial confinement fusion experiments are emphasized.
Linear perturbation amplification in ICF self-similar ablation flows
Journal de Physique IV (Proceedings), 2006
The stability of an ablative flow is of importance in inertial confinement fusion (ICF). Here, we exhibit a family of exact self-similar solutions of the gas dynamics equations with nonlinear heat conduction for semi-infinite slabs of perfect gases. Such self-similar solutions which arise for particular but realistic initial and boundary conditionsboundary pressure and incoming heat flux follow time power laws-are representative of the early stage of the ablation of a pellet by a laser, where a shock wave propagates upstream of a thermal front. Following the work of Boudesocque-Dubois et al. , these solutions are computed using a dynamical multidomain Chebyshev pseudospectral method . A wide variety of ablation configurations may thus be obtained. Linear stability analyses of such time-dependent solutions are performed by solving an initial and boundary value problem for linear perturbations. The numerical methods are based on a dynamical multidomain Chebyshev pseudo-spectral method and an operator splitting between a hyperbolic system and a parabolic equation. Here, focusing on the laser imprint problem, we have considered boundary heat flux perturbations and obtained space-time evolutions of flow perturbations for a wide range of wavenumbers. By contrast with steady low Mach number models of ablation front instabilities, we obtain that: (a) maximum perturbation amplitudes in the thin ablation layer are reached for transverse wavenumber k ⊥ = 0, (b) the damping of ablation front perturbations is closely related to thermal diffusion, (c) ablation front perturbations seem to persist although the transverse wave number increases, (d) a complex wave-like structure arises between the ablation front and the shock-wave front. *
Growth rates of the ablative Rayleigh–Taylor instability in inertial confinement fusion
Physics of Plasmas, 1998
A simple procedure is developed to determine the Froude number Fr, the effective power index for thermal conduction , the ablation-front thickness L 0 , the ablation velocity V a , and the acceleration g of laser-accelerated ablation fronts. These parameters are determined by fitting the density and pressure profiles obtained from one-dimensional numerical simulations with the analytic isobaric profiles of Kull and Anisimov ͓Phys. Fluids 29, 2067 ͑1986͔͒. These quantities are then used to calculate the growth rate of the ablative Rayleigh-Taylor instability using the theory developed by Goncharov et al. ͓Phys. Plasmas 3, 4665 ͑1996͔͒. The complicated expression of the growth rate ͑valid for arbitrary Froude numbers͒ derived by Goncharov et al. is simplified by using reasonably accurate fitting formulas.
Indiana University Mathematics Journal, 2008
In this paper, we study a linear system related to the 2d system of Euler equations with thermal conduction in the quasi-isobaric approximation of . This model is used for the study of the ablation front instability, which appears in the problem of inertial confinement fusion. The heat flux Q is given by the Fourier law T −ν Q proportional to ∇T , where ν > 1 is the thermal conduction index, and the external force is a gravity field g = −g ex. This physical system contains a mixing region, in which the density of the gaz varies quickly, and one denotes by L0 an associated characteristic length. The fluid velocity in the denser region is denoted by Va. * CEA/DM2S,
A hydrodynamic analysis of self-similar radiative ablation flows
Journal of Fluid Mechanics, 2018
Self-similar solutions to the compressible Euler equations with nonlinear conduction are considered as particular instances of unsteady radiative deflagration – or ‘ablation’ – waves with the goal of characterizing the actual hydrodynamic properties that such flows may present. The chosen family of solutions, corresponding to the ablation of an initially quiescent perfectly cold and homogeneous semi-infinite slab of inviscid compressible gas under the action of increasing external pressures and radiation fluxes, is well suited to the description of the early ablation of a target by gas-filled cavity X-rays in experiments of high energy density physics. These solutions are presently computed by means of a highly accurate numerical method for the radiative conduction model of a fully ionized plasma under the approximation of a non-isothermal leading shock wave. The resulting set of solutions is unique for its high fidelity description of the flows down to their finest scales and its e...