The hydrodynamic stability of ablation flows is a key issue in laser-driven inertial confinement fusion (ICF) where a sufficiently symmetric implosion of a spherical pellet is expected to achieve thermonuclear burn. Such flows which originate from exposing the pellet outer shell to a growing incident heat flux, present the radial structure of an inward-propagating deflagration, or ‘ablation', wave where a shock wave precedes a subsonic heat front that coindices with the leading edge of the heated material expansion wave. Inherently unsteady, these flows are compressible, strongly accelerated and highly nonuniform with a steep heat front, owing to the strong nonlinearity of the heat transport and the intense incident heating. These features, in addition to non-trivial boundary conditions at the shell external surface and shock front, are sources of non-modal thermo-acoustics effects [1, 2]. However non-modal instability growth in ablation flows relevant to ICF has never been studied so far. The development of instabilities leading to non-linear phenomena in ablation flows could result into the loss of symmetry of the implosion and could finally inhibit ignition. Transition mechanisms in ablation flows are therefore of primary importance to ICF ignition.

Here we investigate non-modal effects in planar radiative ablation waves by using self-similar ablation solutions to the Euler equations with nonlinear heat conduction without further approximation as model base flows representative of the early stage of an ICF pellet implosion [3]. Pseudo-spectra of the local approximation of the perturbation evolution operator reveal a potential for strong transient growth. Because of the base flow unsteadiness, our non-modal linear stability analysis relies on a direct-adjoint method. The flow boundary deformations, at the material external surface and shock front, as well as their adjoint variable counterparts enter this method formulation. Both optimal initial conditions and receptivity to perturbations of the incident heat flux and external surface pressure are considered. Different definitions of objective functionals are investigated, some in relation with experimentally measurable quantities. Optimal response computations are carried out for terminal times and perturbation transverse wavelengths which are determined on the basis of pellet implosion features. Computed optimal responses are physically analysed in terms of diffusion and propagation, with the help of a decomposition into linear hyperbolic waves — corresponding to acoustic, entropy, vorticity and radiation-conductivity waves [3] — for a nonuniform heat-conducting flow.

[1] K. Wieczorek et al., Phys. Fluids. 23, (2011)
[2] F. Nicoud et al., AIAA J. 45, (2007)
[3] J. M. Clarisse et al., J. Fluid Mech. 848, (2018)