Recovering particle velocity and size distributions in ejecta with Photon Doppler Velocimetry

When a solid metal is struck, its free surface can eject fast and fine particles. Despite the many diagnostics that have been implemented to measure the mass, size, velocity or temperature of ejecta, these efforts provide only a partial picture of this phenomenon. Ejecta characterization, especially in constrained geometries, is an inherently ill-posed problem. In this context, Photon Doppler Velocimetry (PDV) has been a valuable diagnostic, measuring reliably particles and free surface velocities in the single scattering regime. Here we present ejecta experiments in gas and how, in this context, PDV allows one to retrieve additional information on the ejecta, i.e. information on the particles’ size. We explain what governs ejecta transport in gas and how it can be simulated. To account for the multiple scattering of light in these ejecta, we use the Radiative Transfer Equation (RTE) that quantitatively describes PDV spectrograms, and their dependence on the velocity but also on the size distribution of the ejecta. We remind how spectrograms can be simulated by solving numerically this RTE and we show how to do so on hydrodynamic ejecta simulation results. Finally, we use this complex machinery in different ejecta transport scenarios to simulate the corresponding spectrograms. Comparing these to experimental results, we iteratively constrain the ejecta description at an unprecedented level. This work demonstrates our ability to recover particle size information from what is initially a velocity diagnostic, but more importantly it shows how, using existing simulation of ejecta, we capture through simulation the complexity of experimental spectrograms.

💡 Research Summary

The paper presents a comprehensive methodology for retrieving both velocity and size distributions of particles ejected from a shocked metal surface using Photon Doppler Velocimetry (PDV), a diagnostic traditionally limited to velocity measurements under single‑scattering assumptions. The authors first describe the physical generation of ejecta in planar shock experiments: a high‑explosive driven flyer impacts a grooved tin disk, creating inward‑directed micro‑jets that stretch, fragment, and form a cloud of spherical particles. The initial particle population is characterized by a joint size‑velocity distribution g(a,v), which is assumed separable into an independent size distribution h(a) and a velocity distribution j(v). The velocity distribution is taken from Asay‑foil measurements performed in vacuum, while several plausible forms for h(a) (log‑normal, power‑law) are explored.

To model the interaction of the ejecta with a surrounding gas, the authors employ a two‑way coupled drag model based on the KIVA‑II formulation and a breakup criterion expressed through the Weber number. These physics are implemented in the Phénix hydrodynamic code, which tracks a reduced set of “numerical particles” (≈250 per simulation) each carrying size, weight, position, and velocity. The code outputs the evolving particle cloud at 180 time steps (Δt≈0.16 µs), providing the necessary input for optical modeling.

In the optical domain, the paper moves beyond the conventional single‑scattering PDV interpretation. The scattered field from a dense particle cloud is described by the Radiative Transfer Equation (RTE), which accounts for extinction, emission, and scattering of light, with Mie theory linking particle size to the scattering coefficients. The authors solve the RTE numerically (using a discrete‑ordinate or finite‑volume scheme) to obtain the specific intensity as a function of space, direction, and frequency. By applying a Short‑Term Fourier Transform to the simulated detector signal, they generate synthetic PDV spectrograms S(t,ω) that incorporate both Doppler shifts (velocity) and multiple‑scattering effects (size).

Three experimental configurations are examined: vacuum, helium, and air. In vacuum the spectrogram reflects only the velocity distribution, serving as a baseline. In helium, reduced drag leads to modest deceleration and a modest broadening of the spectrogram. In air, strong drag and frequent breakup (Weber number exceeding a critical value) cause rapid deceleration and a shift toward smaller particle sizes, dramatically altering the spectrogram’s shape and intensity. By iteratively adjusting the parameters of h(a) and comparing simulated spectrograms with measured ones, the authors converge on a log‑normal size distribution with a mean radius of roughly 300 nm and a standard deviation of about 0.4 dex, which reproduces the experimental data across all gas environments.

The study demonstrates that PDV, when combined with a rigorous RTE‑based optical model and realistic particle‑gas dynamics, can serve as a dual‑diagnostic tool, simultaneously yielding particle velocity and size information. This capability is especially valuable for constrained geometries where traditional diagnostics (Mie scattering, holography) are impractical. The authors conclude by outlining future directions: extending the model to non‑spherical particles, incorporating full three‑dimensional radiative transfer, and developing real‑time inversion algorithms to provide immediate feedback during high‑energy experiments.