A hydrodynamic analysis of self-similar radiative ablation flows

Abstract

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 extensive exploration of external pressure and radiative flux ranges. Two different dimensionless formulations of the equations of motion are put forth, yielding two classifications of these solutions which are used for carrying out a quantitative hydrodynamic analysis of the corresponding flows. Based on the main flow characteristic lengths and on standard characteristic numbers (Mach, Péclet, stratification and Froude numbers), this analysis points out the compressibility and inhomogeneity of the present ablative waves. This compressibility is further analysed to be too high, whether in terms of flow speed or stratification, for the low Mach number approximation, often used in hydrodynamic stability analyses of ablation fronts in inertial confinement fusion (ICF), to be relevant for describing these waves, and more specifically those with fast expansions which are of interest in ICF. Temperature stratification is also shown to induce, through the nonlinear conductivity, supersonic upstream propagation of heat-flux waves, besides a modified propagation of quasi-isothermal acoustic waves, in the flow conduction regions. This description significantly departs from the commonly admitted depiction of a quasi-isothermal conduction region where wave propagation is exclusively ascribed to isothermal acoustics and temperature fluctuations are only diffused.

Investigation of supersonic heat-conductivity hyperbolic waves in radiative ablation flows

“…Here we extend the analysis of Ref. [25] by considering and numerically solving the exact system of evolution equations for three-dimensional linear perturbations about self-similar ablation waves [19,20]. The entire deflagration structure of a non-uniform and unsteady ablation wave is described, smoothly, from the fluid external surface where an incoming radiation flux and an external pressure are applied, up to the fore-running shock front.…”

“…and for an initial state given by (ρ, v, T ) = (1, 0, 0) for m ≥ 0 [19,22]. For certain values of the boundary parameters (B p , B ϕ ), such solutions present the features of an ablation wave extending from the flow external surface (m = 0) through an ablation front up to the fore-running isothermal shock front [24,25]. This shock front is preceded by an infinitesimal radiation wavelet penetrating the cold fluid [22].…”

“…Such solutions, known since Marshak [22], present the advantage over isobaric, isothermal or stationary solutions of rendering, without further approximations, compressibility, nonuniformity and unsteadiness of ablation waves. They have been exploited to model ICF-type ablation flows [23][24][25]. Indeed these self-similar flows present the essential characteristics of an ablation wave as illustrated on Fig.…”

“…[21], with respect to wave propagation and the analysis of the flow conduction region. Indeed, a local analysis of ablation waves in terms of linear waves propagating in the longitudinal direction reveals that temperature and density stratification in the conduction region, in conjunction with the heat-conductivity nonlinearity, lead to the existence of a family of supersonic forward-propagating waves in this region [25]. Classical description of ablation waves, because of the isothermal assumption, are unable to render such phenomenon, hence the interest of self-similar solutions.…”

“…We focus on a self-similar ablation flow with a fast expansion of its conduction region (i.e. presenting an isothermal Chapman-Jouguet point) as it presents the main characteristics of the shock transit phase of an ICF pellet implosion [25].…”