Particle-Resolved DNS (pr-DNS)

• Particle-Resolved DNS is a computational approach that solves the full Navier–Stokes equations around each particle to capture detailed micro-scale hydrodynamics.
• It employs high-fidelity numerical methods, such as Immersed Boundary and Lattice Boltzmann methods, to accurately simulate drag, lift, collisions, and lubrication forces.
• It provides model-free datasets that underpin the development of closure laws and hybrid physics–machine learning models for advanced multiphase flow predictions.

Particle-Resolved Direct Numerical Simulations (pr-DNS) refer to computational strategies that resolve the full Navier–Stokes equations around each individual particle in a multiphase, particle-laden flow, capturing all relevant micro-scale hydrodynamic interactions without relying on closure models for interphase momentum exchange or subgrid particle dynamics. Unlike lower-fidelity approaches (e.g. Euler–Euler or point-particle Euler–Lagrange), pr-DNS provide “model-free” datasets and physical insight into the underlying mechanisms of dispersion, clustering, breakup, mass/momentum transfer, and nonlinear collective dynamics across a range of flow regimes and applications.

1. Fundamental Methodologies and Formulations

The core of pr-DNS is the simultaneous numerical solution of the incompressible Navier–Stokes equations for the fluid,

$$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho_f}\nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{f}_{\text{IBM}}, \quad \nabla \cdot \mathbf{u} = 0,$$

with explicit enforcement of no-slip and no-penetration boundary conditions at the surface of each rigid (or deformable) particle. Particles are advanced via the Newton–Euler equations,

$$m_p \frac{d \mathbf{V}_p}{dt} = \int_{S_p} \mathbf{\sigma}_f \cdot \mathbf{n} \,dS + \mathbf{F}_{\text{coll}} + \mathbf{F}_{\text{body}},\quad I_p \frac{d \boldsymbol{\Omega}_p}{dt} = \int_{S_p} (\mathbf{r}\times\mathbf{\sigma}_f\cdot\mathbf{n})\,dS + \mathbf{T}_{\text{coll}},$$

where the fluid–particle stress integration and particle–particle contact or cohesion models (e.g. Hertzian, lubrication, van der Waals) are fully coupled.

Several classes of numerical techniques are prevalent:

• Immersed Boundary Methods (IBM): Apply a forcing term (distributed on Lagrangian markers and interpolated to the Eulerian mesh) to impose no-slip at arbitrarily located particle surfaces within a fixed Cartesian grid (Uhlmann et al., 10 Dec 2024, Costa et al., 2019, Yao et al., 2021).
• Fictitious Domain or DLM Methods: Enforce rigid-body motion within particle regions using distributed Lagrange multipliers, often with a penalization strategy (Uhlmann et al., 10 Dec 2024, Münster et al., 28 Jun 2025).
• Lattice Boltzmann (LBM): Utilize a mesoscopic kinetic framework for the fluid, integrating momentum exchange at the boundary via bounce-back or more advanced interfaces (Rettinger et al., 2020, Uhlmann et al., 10 Dec 2024).
• Volume-Filtering Immersed Boundary (VFIB): Introduce mathematically rigorous filtering of the transport equations over a compact kernel, rendering the particle–fluid interface “thickness” δf/d_p a resolution/fidelity parameter (Kasbaoui et al., 29 Apr 2024).

All such methods must couple hydrodynamics with robust short-range (lubrication) and contact models to resolve near-contact and collision physics accurately, particularly under four-way coupling regimes.

2. Microphysics, Coupling, and Closure Strategies

Pr-DNS explicitly resolves all force/moment exchanges between fluid and particles, including:

• Hydrodynamic Forces: Direct numerical integration over the resolved stress field on particle surfaces—no empirical closure needed for drag, lift, or added mass.
• Collisional Interactions: Soft-sphere discrete element models (DEM) or hard-sphere event-driven approaches, incorporating elasticity, friction, rolling resistance, and in some systems cohesion (e.g., parabolic or cutoff van der Waals models for sediment) (Vowinckel et al., 2018, Khalifa et al., 24 May 2025).
• Lubrication Forces: Singularity-regularized analytic expressions for short-range viscous damping are critical to accurately capture normal and tangential interactions as interparticle gaps vanish (Rettinger et al., 2020, Münster et al., 28 Jun 2025).
• Four-Way Coupling: Hydrodynamics, interparticle collisions, and fluid-mediated interaction all fully coupled to address dense suspensions or bed–fluidization phenomena.

Boundary conditions are typically periodic within the computational domain or employ wall models for canonical flows (e.g., turbulent channels, sediment beds, viscometers). High-order finite-volume or LBM solvers often deliver ~10–30 grid points per particle diameter to resolve thin boundary layers and wake structures, with uniform or adaptively refined meshes.

3. Applications: Fundamental and Complex Particulate Flows

Pr-DNS has enabled progress in several key areas:

• Static Particle Assemblies: Benchmarking quasi-steady drag, lift, and heat/mass transfer correlations, providing closure laws as direct functions of volume fraction, local arrangement, and Reynolds number (Kasbaoui et al., 29 Apr 2024, Chouippe et al., 7 Jan 2025, Fernandes et al., 2021).
• Freely Evolving Suspensions: Quantification of mean drag in dynamically evolving particle distributions, dissecting the role of clustering, velocity fluctuations, and particle mobility (Tavanashad et al., 2020). Drag force can increase or decrease relative to fixed beds depending on the interplay between structure and dynamics.
• Dense Suspensions and Rheology: High-fidelity data for effective viscosity incorporating wall force and energy dissipation analyses, leading to polynomial or piecewise-smooth closures suitable for CSP or other industrially relevant flows (Münster et al., 28 Jun 2025).
• Cohesive and Breakup Dynamics: Modeling of cohesive force “smearing” and scale-dependent cohesion number for sedimentation/flocculation (Vowinckel et al., 2018). High-resolution turbulence-induced breakage of agglomerates, including quantification of erosion-dominated breakup, fragment size distributions, and ejection directionality (Khalifa et al., 24 May 2025).
• Turbulent Multiphase Flows (Channel, Homogeneous, Isotropic): Resolution of near-wall effects, including the accurate prediction of turbophoretic accumulation and the need for shear-induced lift forces (e.g., Saffman) to match interface-resolved and point-particle predictions (Costa et al., 2019).
• Wall-bounded Sediment Transport: Erosion, bedload formation, and roughness evolution, linked to the Shields number and resolved particle–fluid and particle–wall interactions (Chouippe et al., 7 Jan 2025).

Pr-DNS provides unrivaled access to spatially and temporally resolved statistics, Lagrangian and Eulerian fields, which underpin model development for lower-fidelity simulation strategies.

4. Impact on Model Development and Data-Driven Closures

Model-free datasets from pr-DNS form the empirical foundation for:

• Drag, Lift, and Torque Correlations: New closure laws parameterized by local volume fraction, Reynolds number, particle arrangements, and velocity fluctuations, replacing or augmenting classical pointwise models (Tavanashad et al., 2020, Kasbaoui et al., 29 Apr 2024, Wachem et al., 8 May 2025).
• Pairwise Interaction and Machine Learning Models: Hybrid or surrogate closures using axisymmetric wake superposition, generalized Faxén representations, and neural network regression to predict hydrodynamic forces as a function of local environment encoded via position or volume-fraction representations (Balachandar et al., 2020, Metelkin et al., 28 Jul 2025). Recent studies demonstrate FCNN models with volume-fraction grid inputs provide improved accuracy and flexibility, especially for polydisperse systems and wall-bounded flows.
• Stochastic Force Closures: Ornstein–Uhlenbeck-process-based models for unresolved drag fluctuations in Euler–Lagrange frameworks, with parameters calibrated to pr-DNS data, improving velocity variance and dispersion predictions (Lattanzi et al., 2021).
• Rheology and Effective Viscosity: High-density suspension data yields polynomial or piecewise-smooth closures for effective viscosity as functions of particle concentration and shear rate, validated via wall force and energy dissipation (Münster et al., 28 Jun 2025).

5. Current Limitations, Computational Challenges, and Grid Convergence

Despite its accuracy, pr-DNS remains computationally intensive, with O(10^9–10^11) grid points and O(10^5–10^6) particles being the upper bounds for high-fidelity studies at present (Chouippe et al., 7 Jan 2025). Main challenges include:

• Grid Resolution: To ensure converged force and transport estimates, 10–24 grid points per particle diameter are typically required. The interface thickness parameter (e.g., δf/d_p for VFIB methods) must be small (≲ 1/8) to recover full fidelity (Kasbaoui et al., 29 Apr 2024).
• Collision/Contact Handling: Near-contact regularization, multi-direct forcing, and iteration in IBM/DLM schemes (Uhlmann et al., 10 Dec 2024); proper calibration of lubrication, friction, and restitution; stability constraints especially for low-density-contrast systems.
• Non-sphericity, Polydispersity, and Shape Effects: Current frameworks predominantly address spheres, though recent advances allow for irregular shapes via grid markers or volume-fraction-based neural network input features (Metelkin et al., 28 Jul 2025).
• Scalability and Parallelism: Efficient use of resources via fixed grid (non-grid-conforming) approaches, as enabled by IBM/LBM paradigms (Uhlmann et al., 10 Dec 2024, Rettinger et al., 2020), is key for large particle ensembles.

6. Emerging Trends: Hybrid Physics–Machine Learning and Multiscale Coupling

Promising directions include:

• Hybrid Surrogates: Neural operator learning and FNO architectures accelerate Navier–Stokes solvers in pr-DNS, targeting velocity/vorticity field prediction in aerosol–cloud–turbulence applications, with up to 70–90% speedup relative to conventional solvers (Atif et al., 2023). This hybrid workflow preserves accuracy while drastically reducing cost.
• Volume Filtering and Point-Particle Bridging: Volume-filtered formulations (VFIB) provide a mathematically rigorous link between pr-DNS and EL/EE models through filter-width-dependent closures, facilitating transition across fidelity scales and eliminating the ambiguous force-point tuning in classic IBM (Kasbaoui et al., 29 Apr 2024, Wachem et al., 8 May 2025).
• Physics-Informed Data-Driven Closures: Integration of model structure (e.g., generalized Faxén decomposition, pairwise maps) with machine learning leverages large flow fields from pr-DNS as training sets, yielding closures with demonstrably superior accuracy, flexibility, and the ability to handle size, shape, and wall effects (Balachandar et al., 2020, Metelkin et al., 28 Jul 2025).
• Rheology and Dense Flow Modeling: Systematic DNS–derived closure tables for effective viscosity and other emergent macroscopic properties permit reliable upscaling to the Euler–Euler level for CSP and other industrial suspensions (Münster et al., 28 Jun 2025).

7. Broader Significance and Future Outlook

Pr-DNS has evolved into an indispensable, “ground-truth” tool for quantifying fundamental multiphase phenomena—clustering, pattern formation, pseudo-turbulence, transport, and non-Newtonian behavior—across canonical and application-driven regimes (Subramaniam, 2020, Chouippe et al., 7 Jan 2025). Its datasets enable development, calibration, and validation of closure models used in Euler–Lagrange and Euler–Euler simulations, and guide analytical theoretical advances.

Challenges remain: extending pr-DNS to non-spherical particles, refining force and transport closures in highly inertial and turbulent regimes, automating extraction of physics-informed features for data-driven models, and bridging scales across DNS, filtered, and macroscopic formulations. Multiscale and hybrid approaches—including neural operators and advanced feature engineering—are poised to mitigate computational burdens while preserving fidelity.

As computational resources scale and methodological advances mature, pr-DNS will underpin increasingly predictive, robust, and generalizable models for industrial, environmental, and geophysical particulate flows, driving progress in simulation-informed design and control across a spectrum of engineering disciplines.