Timescales and Statistics of Shock-induced Droplet Breakup

Detonation-based propulsion devices, such as rotating detonation engines (RDEs), must be able to leverage the higher energy densities of liquid fuels in order for them to be utilized in practical contexts. This necessitates a comprehensive understanding of the physical processes and timescales that dictate the shock-induced breakup of liquid droplets. These processes are difficult to probe and quantify experimentally, often limiting measurements to macroscopic properties. Here, fundamental mechanisms in such interactions are elucidated through detailed numerical simulation of Mach 2 and 3 shock waves interacting with 100 $μ$m water droplets. Using a thermodynamically consistent two-phase formulation with adaptive mesh refinement, the simulations capture droplet surface instabilities and atomization into secondary droplets in great detail. The results show that droplet breakup occurs through a coupled multi-stage process, including droplet flattening, formation of surface instabilities and piercing, and the shedding of secondary droplets from the ligaments of the deformed primary droplet. When considering the dimensionless timescale of Ranger and Nicholls ($τ$), these processes occur at similar rates for the different shock strengths. The PDFs for the Sauter mean diameters of secondary droplets are bimodal log-normal distributions at $τ=2$. Modest differences in the degree and rate of liquid mass transfer into droplets less than 5 $μ$m in diameter are hypothesized to partially derive from differences in droplet surface piercing modes. These results are illustrative of the complex multi-scale processes driving droplet breakup and have implications for the ability of shocks to effectively process liquid fuels.

💡 Research Summary

This paper investigates the fundamental mechanisms and timescales governing shock‑induced breakup of liquid droplets, a process that is critical for the practical use of liquid fuels in detonation‑based propulsion systems such as rotating detonation engines (RDEs). Because experimental diagnostics of shock‑droplet interactions are limited by optical access and temporal resolution, the authors employ high‑fidelity three‑dimensional numerical simulations to resolve the full multiscale physics.

The study focuses on Mach 2 (We ≈ 8.2 × 10²) and Mach 3 (We ≈ 3.8 × 10³) planar shock waves impacting isolated 100 µm water droplets initially at rest in ambient air (1 atm, 300 K). A thermodynamically consistent two‑phase formulation is used: the gas phase follows the ideal‑gas equation of state, while the liquid is modeled as a stiffened gas. Volume‑of‑Fluid (VOF) with ρ‑THINC interface reconstruction captures sharp phase boundaries, and adaptive mesh refinement (AMR) via the AMReX framework provides a finest cell size of 0.78 µm (≈ 0.008 d₀). Five AMR levels lead to peak cell counts of roughly 1 billion, ensuring that both the primary droplet surface and the emerging ligaments are well resolved.

Mechanical and thermal equilibrium across the interface is enforced through relaxation operators; phase change is deliberately omitted to keep droplet mass accounting exact during breakup. The governing equations are discretized with a second‑order finite‑volume scheme, HLLC fluxes for convection, central differences for diffusion, and an explicit two‑stage Runge‑Kutta time integrator. Surface tension, viscosity, and thermal conductivity are treated with constant values appropriate for 300 K water and air.

A dimensionless time scale τ, introduced by Ranger and Nicholls, is used to compare the two shock strengths:

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