Particle Spray Simulations

• Particle spray simulations are numerical models that predict the behavior of particulate flows in multiphase systems by coupling fluid dynamics with discrete particle tracking.
• These simulations employ various methods, such as Eulerian–Lagrangian, DNS/VOF, and PBM, to capture phenomena like atomization, breakup, and thermal processes.
• Applications range from combustion and materials deposition to astrophysics and propulsion, enabling improved design and optimization in complex environments.

Particle spray simulations refer to the broad class of numerical and mathematical models used to study, predict, and analyze the behavior of particulate or droplet-laden flows—often in multiphase environments such as combustion, material deposition (thermal or cold spray), propulsion, and astrophysical contexts. The domain encompasses both high-fidelity interface-resolving approaches and a variety of reduced-order or hybrid Lagrangian–Eulerian methods that simulate the transport, interaction, and fate of discrete particles or droplets under external forcing, turbulence, and interfacial phenomena.

1. Governing Mathematical Frameworks

Particle spray simulations typically build upon the coupled solution of the carrier-phase (gas or liquid) conservation equations and a particle-phase description. The prevailing formulations include:

• Eulerian–Lagrangian Approach: The fluid is resolved using Eulerian conservation equations, while particles/droplets are advanced as discrete Lagrangian entities governed by ODEs for position, momentum, and potentially internal state (temperature, phase) (Haghshenas et al., 2021, Farrokhpanah et al., 2016, Pakseresht et al., 2020).
• Direct Numerical Simulation (DNS) with Volume-of-Fluid (VOF): Resolves all interfaces, surface tension effects, and the full multiphase Navier–Stokes dynamics (Ling et al., 2015, Jeske et al., 2021).
• Population Balance Models (PBM): Statistical treatment for number, size, and distribution evolution of droplets or particles, often coupled to Lagrangian or Eulerian transport (Ren et al., 2020).
• Mean-Field Kinetic PDEs/Vlasov Equations: E.g., for collective dynamics of a dispersed phase in an ideal fluid, sometimes accounting for sophisticated coupling such as gyroscopic or feedback effects (Moussa et al., 2011).

Key governing equations include:

• Mass, momentum, and energy conservation for the carrier phase, incorporating source terms from particle–fluid interaction;
• Particle trajectory ODEs, with drag, gravity, lift, thermophoresis, and possibly Coulombic interactions (Liu et al., 4 May 2025);
• Subgrid-scale turbulence, energy partitioning, and mixture-fraction evolution in LES or RANS-based frameworks (Li et al., 2020).

2. Physical Regimes and Applicability

The design of a spray simulation, including model selection and numerical technique, is dictated by flow regimes, particulate size and density, carrier phase properties, and the physical objectives of the simulation:

• Primary Atomization and Breakup: Resolved in full 3D DNS for fundamental investigations (Kelvin–Helmholtz instability, ligament and rim formation, Rayleigh–Plateau breakup) (Ling et al., 2015).
• Dense, Two-Way or Volumetric Coupling: At significant droplet volume fractions (α_d ≳ 5%), volumetric displacement effects modify both the continuity and momentum equations of the carrier phase, introducing non-divergence-free corrections and strong back-coupling (Pakseresht et al., 2020, Pakseresht et al., 2019).
• Sparse (Dilute) Sprays: Standard two-way coupling with point-particle models is generally sufficient for α_d ≪ 1%, as volumetric effects are negligible (Pakseresht et al., 2020).
• Thermal/Chemical Processes: Particle heating, melting, evaporation, solidification, and chemical conversion (e.g., in flame spray pyrolysis or suspension plasma spraying) require internal ODE/PDE solvers per particle (Farrokhpanah et al., 2016, Ren et al., 2020).
• Charged Particle Interactions: For electrospray, space propulsion and other charged droplet phenomena, Coulomb N-body simulations with direct field computations, e.g., using the Boris pusher, are essential for plume divergence and focusing studies (Liu et al., 4 May 2025).

3. Numerical Methods and Convergence Criteria

The complexity of spray simulations spans from high-resolution, interface-fitted DNS to stochastic parcel-based methods and hybrid algorithms:

• Finite-Volume and VOF: Multi-phase DNS/VOF methods employ sharp interface tracking (VOF, height-function curvature, continuum surface force) and require mesh sizes sufficient to resolve the thinnest sheets and ligament radii (O(10 μm)) (Ling et al., 2015).
• SPH and Meshless Methods: Smoothed Particle Hydrodynamics (SPH) frameworks are utilized for free-surface and solidification dynamics, notably in thermal spraying and impact studies (Jeske et al., 2021).
• Stochastic Lagrangian/Eulerian Schemes: Stochastic parcel methods implemented in industrial and research LES codes demand careful scaling of parcel-per-cell count: constant n_pc yields only c=1/2 convergence; linear convergence (c=1) requires doubling n_pc as Δx is halved, quadratic convergence (c=2) requires 8× per halving (Schmidt et al., 2018).
• LES SGS Modeling: Subgrid-scale energy dissipation modeling, such as the dynamic structure family, employs Leonard-type terms and local test filters to maintain mesh independence and physical fidelity of penetration rates and spray statistics (Li et al., 2020).
• Pressure-based Solvers: Dense spray flows with high particle loading introduce variable-density, zero-Mach number pressure correction equations, often requiring an auxiliary volumetric source term to ensure correct continuity (Pakseresht et al., 2020, Pakseresht et al., 2019).

4. Model Coupling, Breakup, Collision, and Sub-models

Spray simulations require robust sub-modeling for the physical processes that govern droplet/particle evolution:

• Breakup and Coalescence: Kelvin–Helmholtz/Rayleigh–Taylor instabilities govern secondary breakup (KH–RT model), while collision–coalescence is typically modeled via stochastic NTC algorithms (Farrokhpanah et al., 2016, Haghshenas et al., 2021).
• Thermal/Solid-State Response: Simulation of rapid heating, phase change (melting/solidification), and recoil in droplet/substrate impact mandates coupled energy and momentum conservation, along with enthalpy-porosity methods or detailed atomistic models for viscoplasticity and interdiffusion (Ahmed et al., 10 Oct 2025, Jeske et al., 2021).
• Chemical Conversion and PBM: Multicomponent single-droplet models with population balance equations allow rigorous prediction of nanoparticle yield and size from precursor vaporization, in both “gas-to-particle” and “droplet-to-particle” regimes (Ren et al., 2020).
• Turbulence–Particle Interactions: Particle dispersion, settling, and turbulence modulation are captured using point-particle feedback and advanced coupling closures (Pakseresht et al., 2019, Li et al., 2020).
• Mixing-Limited and Capsule Models: For high-pressure diesel/IC sprays, the ELMO model enforces entrainment-limited vaporization and spreading ([mixing-limited] vs. [interface-limited])—superseding classical interface-tracking approaches in certain regimes (Haghshenas et al., 2021).

5. Validation, Mesh Sensitivity, and Best Practices

Physical and numerical validation is central in particle spray simulation practice:

• Mesh and Parcel Convergence: Quantitative guidelines specify parcel count and mesh refinement necessary for desired convergence rates. For 3D transient sprays, halving Δx with constant n_pc gives only 0.5 order convergence; first order demands n_pc ∝ Δx^{-1}, and second order ∝ Δx^{-3} (Schmidt et al., 2018).
• Experimental Comparison: Metrics such as velocity, temperature, and droplet size distributions at impact are systematically compared against experimental data (e.g., laser phase Doppler anemometry, schlieren imaging) (Farrokhpanah et al., 2016, Pakseresht et al., 2019).
• Statistical and Thermodynamic Consistency: Models must be validated against known limiting behaviors: dilute vs. dense regimes, correct turbulence energy partitioning, and void-fraction effects up to α_d ≈ 40% (Pakseresht et al., 2019, Pakseresht et al., 2020).
• Algorithmic Recommendations: For dense sprays, volumetric displacement of the carrier phase is required at α_d ≳ 5%; for accurate particle–substrate impact, nanocell mesh and fully coupled boundary ODEs are mandated (Pakseresht et al., 2020, Ahmed et al., 10 Oct 2025).

6. Specialized Applications and Future Directions

Particle spray simulations continue to diversify into high-impact domains:

• Materials Processing: High-fidelity cold spray and plasma spray simulations resolve turbulence, shock structure, phase change, and detailed particle trajectories, guiding process parameter optimization and new nozzle design (Bouthier et al., 2020, Farrokhpanah et al., 2016).
• Additive Manufacturing: Atomistic MD studies link control parameters (impact velocity, ultrasound) to diffusivity and plastic deformation, enabling in situ alloying and overcoming refractory metal limitations (Ahmed et al., 10 Oct 2025).
• Electrospray Propulsion and Plume Modeling: N-body particle simulations, with direct calculation of full Coulomb interaction and Boris-pusher-based integration, enable predictive design for space thrusters and advanced diagnostics for plume divergence (Liu et al., 4 May 2025).
• Astrophysical Systems: Particle spray codes (“corespray”) simulate the ejection of stars via three-body interactions for understanding galactic halo formation and globular cluster dynamics (Grondin et al., 2022).

Continuing challenges include mesh-independent convergence at scale, variance reduction for statistical error, and robust two-way coupling for transient, dense, or chemically active sprays across all relevant parameter regimes.