Blade-Resolved WMLES: Modeling & Simulation

• Blade-resolved WMLES is a simulation approach that models turbulent flows around blades using spatially filtered Navier–Stokes equations with explicit wall-stress closures.
• It leverages hybrid numerical algorithms, adaptive grid strategies, and ML-enhanced closures to balance accuracy and computational efficiency in complex, moving geometries.
• The approach accurately predicts integral forces and wake dynamics, as validated against experiments and scalability tests, enabling cost-effective high-Reynolds-number simulations.

Blade-resolved wall-modeled large-eddy simulation (WMLES) constitutes a class of computational methods in which large-scale turbulent flow structures around dynamically resolved blades—such as wind turbines, compressors, and airfoils—are simulated with explicit wall stress closure models, while the boundary layer and near-wall turbulence are under-resolved or modeled. These approaches provide a tractable route to accurate, high-Reynolds-number simulations in configurations where wall-resolved LES or DNS would be computationally prohibitive, enabling analysis of integral forces, wake dynamics, and fluid–structure interaction (FSI) in complex, moving geometries.

1. Mathematical Formulation and Wall Modeling Approach

The governing equations underpinning blade-resolved WMLES are spatially filtered Navier–Stokes equations incorporating subgrid-scale (SGS) stress closures and explicit wall-stress models. In the Lattice Boltzmann blade-resolved WMLES framework, the approach starts from a filtered Brinkman–Navier–Stokes system: $$\begin{cases} \nabla\cdot\bar{\bm u} = 0, \ \frac{\partial \bar{\bm u}}{\partial t} + \bar{\bm u}\cdot\nabla\bar{\bm u} = -\frac{1}{\rho}\nabla\bar p + \nu_{\rm mo}\nabla^2\bar{\bm u} + \frac{\nu_{\rm mo}}{K(\bm x,t)}\bar{\bm u} - \nabla\cdot\mathbf{T}_{\rm sgs}, \end{cases}$$ where $K(\bm x,t)$ denotes the local permeability (encoding blade geometry via homogenization), and $\mathbf{T}_{\rm sgs}$ is the subgrid stress tensor (Kummerländer et al., 15 Oct 2025).

Wall stress is imposed via a wall model that provides the shear (or friction velocity $u_\tau$) at the first off-wall cell. A common strategy uses the Spalding–law equilibrium log-layer relation: $$y^+ = u^+ + \frac{1}{E}\left(e^{\kappa u^+} - 1 - \kappa u^+ - \frac{1}{2}(\kappa u^+)^2 - \frac{1}{6}(\kappa u^+)^3\right), \qquad \kappa\approx0.41,\, E\approx9.8,$$ with $u^+ = u/u_\tau$ and $y^+ = y u_\tau/\nu$, solved implicitly at each wall cell via Newton–Raphson iteration (Kummerländer et al., 15 Oct 2025, Balakumar et al., 11 Feb 2026).

For rough surfaces or complex wall physics, wall models may instead use algebraic, semi-empirical, or neural network (NN) closures based on local flow and geometric features, e.g., mapping non-dimensional velocity, distance, and roughness statistics to wall shear (Ma et al., 2024, Ma et al., 28 Jan 2026). Machine learning-based models further include information-theoretic input selection and confidence estimation via Bayesian methods.

2. Homogenized Blade-Resolved Framework and Numerical Algorithms

In the efficient blade-resolved WMLES framework, moving solid surfaces are embedded in a fixed Eulerian lattice (e.g., D3Q19 LBM) as porous regions, with local permeability $K$ governing the fluid–solid interaction. The key steps of the homogenized hybrid regularized recursive LBM (HHRRLBM) are:

• Blending of fluid and prescribed solid velocity fields in partially solid cells: $\widehat{\bm u} = (1-d)\bm u + d\bm u^B$, where $d$ is the local porosity;
• Implementation of the wall model by overwriting $K(\bm x,t)$0 at wall-modeled cells, with stabilization by setting $K(\bm x,t)$1 in collision;
• Momentum-exchange force computed at immersed boundaries to yield integral blade forces;
• Dynamic update of porosity and structure tag fields to track moving blades and rotating machinery in strong-coupling FSI setups (Kummerländer et al., 15 Oct 2025).

Simulation proceeds via sequential update steps: $$y^+ = u^+ + \frac{1}{E}\left(e^{\kappa u^+} - 1 - \kappa u^+ - \frac{1}{2}(\kappa u^+)^2 - \frac{1}{6}(\kappa u^+)^3\right), \qquad \kappa\approx0.41,\, E\approx9.8,$$9 The entire algorithm is deployed in OpenLB, exploiting C++ meta-programming for flexible cellwise kinetics and enabling multi-rotor, wind-farm–scale computations across distributed-memory GPU clusters.

3. Grid Generation, Resolution, and Transition Handling

Successful blade-resolved WMLES requires tailored grid strategies:

• Turbulent regions: The first off-wall point (used for wall model exchange) must lie within the logarithmic layer ($K(\bm x,t)$2–$K(\bm x,t)$3, or for LBM, up to $K(\bm x,t)$4 when modeling at extreme scale) (Kummerländer et al., 15 Oct 2025, Balakumar et al., 11 Feb 2026).
• Laminar and transitional BL: Thin laminar regions necessitate many points per boundary layer thickness (typically $K(\bm x,t)$5–$K(\bm x,t)$6), and the wall model must be deactivated or supplemented by the no-slip condition upstream of transition (Balakumar et al., 11 Feb 2026).
• Transition location: Precursor RANS and linear-stability (N-factor/or Orr–Sommerfeld) analysis are used to estimate $K(\bm x,t)$7. Synthetic disturbances at appropriate frequency/amplitude can be imposed to trigger Tollmien–Schlichting (TS) waves and enforce transition in agreement with stability theory (Balakumar et al., 11 Feb 2026).

Hybrid grid adaptation based on local $K(\bm x,t)$8, extracted from precursor RANS, allows grids to simultaneously resolve laminar regions and provide proper wall-model placement in turbulence. The resultant non-uniform grids minimize total cell count (e.g., $K(\bm x,t)$9 vs. $\mathbf{T}_{\rm sgs}$0 for full wall-resolved LES at $\mathbf{T}_{\rm sgs}$1).

4. Model Validation and Accuracy

Validation of blade-resolved WMLES spans canonical blade flows, representative turbines, and realistic wind-farm or compressor configurations:

• For a canonical three-blade rotor at $\mathbf{T}_{\rm sgs}$2, the LBM-based WMLES achieves thrust-coefficient errors within $\mathbf{T}_{\rm sgs}$3 of reference, with experimental order of convergence (EOC) of $\mathbf{T}_{\rm sgs}$4 (formal order $\mathbf{T}_{\rm sgs}$5 at finest grids) (Kummerländer et al., 15 Oct 2025).
• Wake velocity profiles, phase-averaged velocity fields, and Q-criterion vortex visualizations indicate the capture of tip-vortex structures and leapfrogging phenomena matching both experiment and wall-resolved LBM (Kummerländer et al., 15 Oct 2025).
• Application to high-pressure turbine blades with roughness using ML-based wall models results in wall-shear prediction within $\mathbf{T}_{\rm sgs}$6 and mean-velocity deficit within $\mathbf{T}_{\rm sgs}$7 except in shock-dominated regions (Ma et al., 2024, Ma et al., 28 Jan 2026).
• Grid and model adaptations enable accurate skin-friction and transition location for airfoil flows under both laminar and turbulent boundary layer conditions (Balakumar et al., 11 Feb 2026).

A summary view of computational validation is given below:

Case	Key Metric	Error/Agreement
Three-blade rotor	$\mathbf{T}_{\rm sgs}$8 error	$\mathbf{T}_{\rm sgs}$9 (vs. ref.)
Wind farm scaling	Weak scaling	$u_\tau$0 to 64 GPUs
HPT blade (rough)	$u_\tau$1 vs. DNS	$u_\tau$2 over blade
Airfoil flow	$u_\tau$3 (lam/turb)	Agreement within model error

Significance: These results indicate that blade-resolved WMLES, with proper grid placement and wall-stress modeling, can achieve near-grid-converged integral force prediction and resolve relevant turbulent structures for engineering applications (Kummerländer et al., 15 Oct 2025, Ma et al., 2024, Balakumar et al., 11 Feb 2026).

5. Machine Learning and Building-Block Model Extensions

Recent advances have extended wall modeling beyond classical empirical laws to data-driven closures using artificial neural networks (ANNs), trained on databases from DNS and wall-resolved LES over canonical and application-specific geometries:

• Building-block flow model (BFM/BFWM): Decomposes flows into superpositions of canonical units (e.g., ZPG, APG, FPG, separation, unsteady 3D) with classifier and predictor ANNs; inputs are local two-point invariants constructed for Galilean/covariance invariance; a unified wall/SGS closure allows seamless application to arbitrary geometries, including rotating blades (Arranz et al., 2024, Lozano-Durán et al., 2022).
• Roughness-aware ML wall models: Input features include roughness statistics (e.g., $u_\tau$4, $u_\tau$5, effective slope $u_\tau$6, skewness $u_\tau$7, kurtosis $u_\tau$8) as well as local velocity and temperature. Information-theoretic feature selection and GP-based uncertainty quantification yield robust predictions and runtime confidence diagnostics (Ma et al., 2024, Ma et al., 28 Jan 2026).
• Validation: For rough-wall WMLES, a-priori error on channel-flow DNS is typically $u_\tau$9%, and a-posteriori predictive accuracy in turbine-blade applications remains within $$y^+ = u^+ + \frac{1}{E}\left(e^{\kappa u^+} - 1 - \kappa u^+ - \frac{1}{2}(\kappa u^+)^2 - \frac{1}{6}(\kappa u^+)^3\right), \qquad \kappa\approx0.41,\, E\approx9.8,$$0% for wall shear and $$y^+ = u^+ + \frac{1}{E}\left(e^{\kappa u^+} - 1 - \kappa u^+ - \frac{1}{2}(\kappa u^+)^2 - \frac{1}{6}(\kappa u^+)^3\right), \qquad \kappa\approx0.41,\, E\approx9.8,$$1% for heat flux, with confidence scores flagging low-trust regions (e.g., stagnation, strong shock or curvature) (Ma et al., 28 Jan 2026).

A plausible implication is that systematic expansion of the building-block database to include rotational and curvature-dominated flows will further enhance the fidelity and transferability of machine-learning wall models in blade-resolved WMLES (Lozano-Durán et al., 2022).

6. Computational Performance and Scalability

Efficient realization of blade-resolved WMLES hinges on architecture-optimized solvers and data locality:

• The HHRRLBM approach achieves $$y^+ = u^+ + \frac{1}{E}\left(e^{\kappa u^+} - 1 - \kappa u^+ - \frac{1}{2}(\kappa u^+)^2 - \frac{1}{6}(\kappa u^+)^3\right), \qquad \kappa\approx0.41,\, E\approx9.8,$$2 MLUPs/s single-GPU throughput (NVIDIA A100); kernel arithmetic intensity is $$y^+ = u^+ + \frac{1}{E}\left(e^{\kappa u^+} - 1 - \kappa u^+ - \frac{1}{2}(\kappa u^+)^2 - \frac{1}{6}(\kappa u^+)^3\right), \qquad \kappa\approx0.41,\, E\approx9.8,$$3 FLOP/B post-common-subexpression elimination, supporting bandwidth-bound execution (Kummerländer et al., 15 Oct 2025).
• Weak scaling demonstrates near-ideal efficiency ($$y^+ = u^+ + \frac{1}{E}\left(e^{\kappa u^+} - 1 - \kappa u^+ - \frac{1}{2}(\kappa u^+)^2 - \frac{1}{6}(\kappa u^+)^3\right), \qquad \kappa\approx0.41,\, E\approx9.8,$$4 up to 64 GPUs, $$y^+ = u^+ + \frac{1}{E}\left(e^{\kappa u^+} - 1 - \kappa u^+ - \frac{1}{2}(\kappa u^+)^2 - \frac{1}{6}(\kappa u^+)^3\right), \qquad \kappa\approx0.41,\, E\approx9.8,$$5 at 384 GPUs) for multi-turbine, wind-farm–scale simulations (up to $$y^+ = u^+ + \frac{1}{E}\left(e^{\kappa u^+} - 1 - \kappa u^+ - \frac{1}{2}(\kappa u^+)^2 - \frac{1}{6}(\kappa u^+)^3\right), \qquad \kappa\approx0.41,\, E\approx9.8,$$6 billion lattice cells), with bottlenecks arising only at the force-integration stage due to global reductions and noncontiguous message passing at extreme node counts (Kummerländer et al., 15 Oct 2025).
• OpenLB’s DSL-based modularity allows dynamic adaptation of “dynamics” tuples for different subdomains (wall vs. bulk), supporting platform-transparent GPU/CPU/HPC deployment (Kummerländer et al., 15 Oct 2025).

This level of scalability enables routine execution of blade-resolved WMLES for entire wind farms, with the potential for future elastic FSI and local grid refinement to further reduce cost and error.

7. Current Limitations and Future Directions

Despite substantial progress, several challenges persist:

• Present validation is bounded to moderate Reynolds number ($$y^+ = u^+ + \frac{1}{E}\left(e^{\kappa u^+} - 1 - \kappa u^+ - \frac{1}{2}(\kappa u^+)^2 - \frac{1}{6}(\kappa u^+)^3\right), \qquad \kappa\approx0.41,\, E\approx9.8,$$7–$$y^+ = u^+ + \frac{1}{E}\left(e^{\kappa u^+} - 1 - \kappa u^+ - \frac{1}{2}(\kappa u^+)^2 - \frac{1}{6}(\kappa u^+)^3\right), \qquad \kappa\approx0.41,\, E\approx9.8,$$8); full-scale, multi-MW turbines, high-pressure compressors, and severe stall regimes require further extensions and validation (Kummerländer et al., 15 Oct 2025).
• Wall-model errors are fundamentally limited by their empirical or data-driven training; transfer to strong separation, shock–BL interaction, and blade–tip vortex regimes requires either new models or careful retraining with physics-rich databases (Arranz et al., 2024, Ma et al., 28 Jan 2026).
• Scaling at extreme node count may be compromised by global communication overheads, particularly in force aggregation for multi-rotor FSI (Kummerländer et al., 15 Oct 2025).
• For roughness and multi-physics coupling (e.g., heat transfer, compressibility), building-block and ML wall models must be extended to include additional topographies, anisotropy, curved surfaces, and strong compressible effects (Ma et al., 28 Jan 2026).

Ongoing work focuses on expanding canonical databases, leveraging active learning for coverage, implementing local refinement, and modularizing wall/SGS model components to support rapid adaptation to new geometries and physics. Reliable uncertainty quantification via ML model confidence scores is increasingly standard for robust deployment and error monitoring in production-grade WMLES workflows (Ma et al., 28 Jan 2026, Lozano-Durán et al., 2022).