An Euler-Lagrangian Multiphysics Coupling Framework for Particle-Laden High-Speed Flows

Particle-laden effects in high-speed flows require a coupled Euler and Lagrangian prediction technique with varying fidelity of thermochemical models, depending on the simulation conditions of interest. This requirement makes the development of a conventional monolithic solver challenging to manage the different fidelity of the thermochemical models within a single computational framework. To address this, the present study proposes a multi-solver framework for the coupled Euler-Lagrangian predictions applicable to various particle-laden high-speed flow conditions. Volumetric and surface couplings are established between a particle solver ORACLE (OpenFOAM-based lagRAngian CoupLEr) and a thermochemical nonequilibrium flow solver based on an adaptable data exchange algorithm. The developed framework is then validated by predicting particle-laden supersonic nozzle flows and aerothermal heating around a hypersonic Martian atmospheric entry capsule. Finally, a quasi-1D approximation is proposed in conjunction with a surrogate method to efficiently and accurately predict particle-laden surface erosion, with quantified parametric uncertainty, for hypersonic aerothermal characterization.

💡 Research Summary

The paper presents a modular multi‑solver framework for simulating particle‑laden high‑speed flows, addressing the difficulty of integrating thermochemical models of varying fidelity within a single monolithic code. The authors couple an Eulerian nonequilibrium flow solver (HEGEL) with a Lagrangian particle tracker (ORACLE, built on OpenFOAM) using the preCICE coupling library. Data exchange occurs at user‑defined coupling intervals (Δt_c) and includes gas variables (density, velocity, pressure, translational‑rotational and electron‑vibrational temperatures) from HEGEL to ORACLE, and momentum and energy source terms generated by particles back to HEGEL. The coupling follows a parallel‑explicit scheme: both solvers advance concurrently on independent time steps (Δt_g for HEGEL, Δt_p for ORACLE) and synchronize only at the coupling points, iterating until the flow residual meets a prescribed tolerance.

In the particle domain, particles are treated as spherical point masses. Their translational motion follows a drag‑force balance (Eq. 2) with drag coefficients drawn from Clift et al. for continuum regimes and Henderson for rarefied, nonequilibrium conditions. Heat transfer uses Nusselt correlations (Drake Jr. for equilibrium, Fox et al. for nonequilibrium). Particle vaporization is activated when surface temperature exceeds a material‑specific boiling point, reducing particle radius, but the evaporated mass is not re‑introduced into the gas phase. Particle‑wall interactions consist of (i) collision heating, where the full kinetic energy of an impacting particle is assumed to convert to heat, and (ii) surface recession, modeled as hemispherical craters whose penetration depth follows an empirical law (Eq. 6). To mitigate Monte‑Carlo noise, a moving‑average filter with a stencil width of two neighboring faces is applied to surface heat‑flux data.

The framework is validated against two benchmark problems. First, a supersonic converging‑diverging nozzle (JPL data) demonstrates that the coupled solver correctly predicts shock‑position shifts, temperature fields, and surface heat fluxes when particles are injected. Second, the hypersonic entry of the ExoMars Schiaparelli capsule reproduces measured aerothermal loads and erosion patterns, confirming the necessity of both volumetric and surface coupling for accurate predictions.

Recognizing the prohibitive cost of full 3‑D simulations for early‑stage design, the authors develop a quasi‑1D approximation of the flow field combined with a data‑driven surrogate model for particle‑induced recession. By sampling the high‑fidelity coupled simulations across a range of entry conditions, they train a surrogate that predicts recession rates with quantified uncertainty using Bayesian inference. This surrogate enables rapid assessment of thermal protection system (TPS) thickness requirements, offering confidence intervals that account for variability in particle size distribution, entry angle, and atmospheric composition.

The study concludes that the multi‑solver approach provides flexibility (independent development, verification, and maintenance of each physics module), scalability (parallel execution), and accuracy (validated against experiments). Limitations include the neglect of inter‑particle collisions, omission of vapor‑phase feedback, and simplified material response (no surface roughness or nonlinear erosion behavior). Future work is suggested to incorporate these effects, extend the framework to reactive particles, and explore adaptive coupling strategies for further computational savings.