Co-Evolving Fluid-Particle Framework

• The framework synchronizes fluid and particulate phases via two-way coupling to accurately represent complex multiphysics interactions.
• It employs Eulerian–Lagrangian and dynamic closure models, resolving both resolved and subgrid-scale phenomena with high fidelity.
• The approach enhances simulation accuracy in turbulent flows, high-speed aerothermodynamics, and biomedical as well as environmental systems.

A co-evolving fluid-particle framework denotes a class of computational models and numerical methods in which the fluid and particulate phases are advanced in time in a tightly coupled, interdependent manner, with fully two-way feedbacks. Unlike decoupled or weakly coupled approaches, the co-evolution paradigm treats fluid and particle subsystems with equal fidelity, consistently resolves their dynamic interplay at all relevant spatial and temporal scales, and includes advanced closure models for unresolved interactions. Applications span turbulent multiphase flow, environmental and biomedical systems, high-speed aerothermodynamics, plasma physics, and beyond.

1. Fundamental Principles and Coupling Strategies

Co-evolving frameworks are typically based on the Eulerian–Lagrangian or Eulerian–Eulerian formalisms, with the fluid phase solved on an Eulerian grid (finite-volume, DG/FEM, or LBM), and the discrete particulate phase tracked in the Lagrangian frame, potentially as rigid, elastic, or even metaball-represented (non-spherical) entities. The defining characteristic is a synchronized or tightly coupled operator-splitting time integration where the fluid equations are advanced subject to instantaneous feedback from the particle system, and the particle dynamics incorporate instantaneous or temporally averaged fluid forces.

The canonical structure involves:

• Two-way coupling: Particles both feel and exert forces on the fluid at each step, often involving volumetric, localized, or surface-based interaction terms.
• Closure modeling: For unresolved subgrid-scale (SGS) fluctuations, volume-filtering, or drag/lift/heat-exchange correlations, physically consistent and sometimes dynamic closure models are incorporated, as in velocity-enriching LES and volume-filtered IBM approaches (Hausmann et al., 2023, Hausmann et al., 2024).
• Collocated evolution: Both phases are subcycled or co-advanced on a common or harmonized timeline, using efficient data exchange and parallelization approaches (e.g., MPI+MPI shared-memory for four-way coupled systems (Schwarz et al., 9 Jan 2025)).

2. Governing Equations and Physical Closures

The governing system consists of the Navier-Stokes (or MHD, Boltzmann, etc.) equations for the fluid phase, extended by coupling source terms:

$$\frac{\partial (\rho_f \bm{u})}{\partial t} + \nabla \cdot (\rho_f \bm{u} \bm{u}) = -\nabla p + \mu_f \nabla^2 \bm{u} + \bm{S}(\bm{x}, t)$$

where the source $\bm{S}$ originates from particle drag, lift, momentum-exchange, and heat-transfer. Particle equations instantiate Newton–Euler dynamics:

$$m_p \frac{d \bm{v}_p}{dt} = \bm{F}_{\text{fluid-particle}} + \bm{F}_{\text{collisions}} + m_p \bm{g}$$

$$I_p \frac{d \bm{\omega}_p}{dt} = \bm{T}_{\text{fluid-particle}} + \bm{T}_{\text{collisions}}$$

Coupling terms may be integrated over smeared delta functions (immersed boundary), assigned by compact kernels (Particle-In-Cell, IDW), or, for volume-filtered methodologies, as analytical integrals over convolution kernels that define porosity/volume-fraction fields (Hausmann et al., 2024, Malipeddi et al., 2023, Robinson et al., 2013, Schwarz et al., 9 Jan 2025).

Subgrid- and volume-filter closures accommodate the disparity between grid and particle scales:

• LES velocity enrichment: $u = \tilde{u} + u'$, with $u'$ reconstructed via truncated Fourier modes to supply unresolved fluctuations to particle drag, and modulation from particles fed back into SGS energy (Hausmann et al., 2023).
• Physically Consistent IBM: Systematic closure of SGS stresses, viscous slip, and transpiration terms via Gaussian convolution filtering and dynamic gradient models (Hausmann et al., 2024).

3. Numerical Schemes, Algorithms, and Implementation

The co-evolving paradigm demands synchronizing the fluid and particle solvers, often via globally implicit or explicit–partitioned time stepping. Key features include:

• Data exchange: Conservative transfer of particle feedback (forces, torques, heat) onto the fluid grid using mesh-aware kernels compatible with FV, DG, or LBM discretizations (Schwarz et al., 9 Jan 2025, Zhang et al., 2022).
• Time integration: Subcycling of particle advance (especially for high-inertia or Stiff drag/collision regimes), with fluid fields held fixed or interpolated as appropriate (Nam et al., 6 Dec 2025, Malipeddi et al., 2023).
• Parallelization: Space-filling curve domain decomposition for fluid, local storage and fast neighbor search (linked cell/Morton hashing) for particles, and hybrid shared/distributed memory for collision pairing and force deposition (Schwarz et al., 9 Jan 2025, Malipeddi et al., 2023).
• Coupling algorithms: Varied strategies, including explicit operator splitting (no sub-iterations), one-pass BDF2 with coupling at each time step, or monolithic Newton–Raphson for block-coupled implicit integrators (Nam et al., 6 Dec 2025, Hausmann et al., 2024, Malipeddi et al., 2023).

Specialized models are required for compressible flow (shock-capturing hybrid DG–FV), non-spherical shapes (metaballs and sharp-interface LBM coupling (Zhang et al., 2022)), and high-fidelity treatments of SGS turbulence, especially in LES contexts (Hausmann et al., 2023, Hausmann et al., 2024).

4. Applications, Advanced Features, and Validation

Co-evolving frameworks now underpin a wide array of large-scale, multiphysics simulations:

Application Domain	Frameworks/Features	Validation Approach
Turbulent particle-laden LES (SGS feedback)	Velocity enrichment, dynamic SGS models	DNS comparison for pair dispersion, clustering (Hausmann et al., 2023)
High-speed and hypersonic flows	Modular Euler–Lagrangian, surface coupling	Supersonic nozzle flow, surface erosion (Nam et al., 6 Dec 2025)
Dense four-way coupled multiphase	Hard-sphere collision, parallel IDW kernel	Strong/weak scaling to >10k cores, shock–particle interaction (Schwarz et al., 9 Jan 2025)
Complex geometries (patient-specific, biomedical)	FEM–DEM with mesh-independent projection	Bifurcation, pipe flow, convergence analysis (Malipeddi et al., 2023)
Non-spherical particles, polydispersity	Metaball DEM–LBM, advanced IBM	Experimental settling, DKT, shape-induced segregation (Zhang et al., 2022)
Multiphysics: phase change, FSI, plasmas	SPH-FEM/LBM, MHD–PIC, hybrid phase field	Gastric bolus, additive manufacturing, reconnection (Liang et al., 30 Dec 2025, Fuchs et al., 2021, Hong et al., 2021)

Common validation strategies include comparison with DNS, experimental trajectories (settling, DKT), drag coefficients (Schiller–Naumann, body-fitted references), energy/decay rates, clustering statistics, and analytic results (e.g., Faxén, RTI growth).

5. Limitations, Stability, and Prospects

The main limiting factors and active areas of research in co-evolving frameworks are:

• Resolution constraints: Particle-resolved approaches demand $d_p/h \approx 6$–20 for resolved forces; volume-filtered/PC-IBM reduces this cost at acceptable error (e.g., drag within 10% at $d_p/\Delta x = 6$ (Hausmann et al., 2024)).
• SGS/closure model generality: Accurate representation across regimes (dense/dilute, high turbulence, compressible shock, multi-relaxation time) requires ongoing closure refinement.
• Load balance: Dense particle packings and heterogeneous distributions challenge parallel performance, especially in four-way coupled models and unstructured meshes (Malipeddi et al., 2023, Schwarz et al., 9 Jan 2025).
• Explicit partitioning: Most frameworks employ explicit or loosely implicit coupling, which can be unstable or inaccurate for very high mass-loading ratios or large time steps (Malipeddi et al., 2023).
• Surface/boundary fidelity: Immersed boundary and meshless approaches deliver broad flexibility but can exhibit force/torque oscillations near contact interfaces, especially for non-spherical or deformable particles (Zhang et al., 2022, Hong et al., 2021).
• Extension to new physics: Incorporation of heat/mass transfer, chemical reaction, fragmentation/coalescence, and magnetized/charged particle interactions remains an active area (Nam et al., 6 Dec 2025, Liang et al., 30 Dec 2025).

This suggests that hybrid and adaptive schemes, including dynamically refined SGS closures, implicit coupling, and machine-learned modeling, are probable avenues for future development.

6. References to Key Frameworks and Surveys

• LES with two-way particle-SGS feedback: "A large eddy simulation model for two-way coupled particle-laden turbulent flows" (Hausmann et al., 2023)
• Modular Euler–Lagrangian coupling for hypersonics: "An Euler-Lagrangian Multiphysics Coupling Framework for Particle-Laden High-Speed Flows" (Nam et al., 6 Dec 2025)
• Physically Consistent IBM for volume-filtered flows: "Physically consistent immersed boundary method: a framework for predicting hydrodynamic forces on particles with coarse meshes" (Hausmann et al., 2024)
• High-performance compressible four-way coupling: "Efficient computation of particle-fluid and particle-particle interactions in compressible flow" (Schwarz et al., 9 Jan 2025)
• Meshless two-way coupled SPH–DEM: "Fluid-particle flow and validation using two-way-coupled mesoscale SPH-DEM" (Robinson et al., 2013)
• Advanced IBM on collocated grids: "Particle-resolved simulations of four-way coupled, polydispersed, particle-laden flows" (Yao et al., 2021)
• Volume-filtered FEM–DEM for unstructured geometries: "Volume filtered FEM-DEM framework for simulating particle-laden flows in complex geometries" (Malipeddi et al., 2023)
• Sharp-interface LBM–MDEM for non-spherical particles: "Coupled Metaball Discrete Element Lattice Boltzmann Method for Fluid-Particle Systems with non-spherical particle shapes" (Zhang et al., 2022)
• SPH-based co-evolving frameworks including phase change: "An SPH framework for fluid-solid and contact interaction problems..." (Fuchs et al., 2021)
• Hybrid Onsager phase-field FSI: "A hybrid phase field method for fluid-structure interactions in viscous fluids" (Hong et al., 2021)

These references collectively illustrate the state-of-the-art in co-evolving fluid-particle frameworks, with extensive validation, practical scalability, and the ability to capture complex multiphysics phenomena across engineering, environmental, and astrophysical applications.