• Reformulate the incompressible magneto-hydro-dynamics in terms of differential forms • Underline how the fluid velocity shows up in the magnetic compartment thanks to the Lie’s derivative • Underline how the magnetic induction acts on the fluid compartment thanks to the Laplace’s force Magnetohydrodynamic offers examples of non-scalar advection-diffusion problems which are relevant for applications. We consider its formulation in terms of differential forms, with the presence of operators such as the exterior derivative and Lie’s derivative, being aware of the underneath analogy between electromagnetic dynamic and incompressible fluid dynamic. We analyze the intrinsic structure of the magnetic and fluid coupling, with a special attention to the Laplace’s force. Taking the cue from [11] we focus on the density of virtual power associated with each of the involved forces.

• Research highlight 1: Knowledge-guided initialization enhances training efficiency: By embedding pre-trained models from previous single- or multi-case training into the initial population, the convergence rate of multi-case CFD-driven training is significantly improved. • Research highlight 2: Real-time transformer-based training acceleration: When no prior knowledge is available, integrating the transformer directly into the optimization loop enables adaptive guidance throughout the training process. Recent studies have demonstrated the effectiveness of applying the computational fluid dynamics (CFD)-driven symbolic machine learning (ML) frameworks to assist in the development of explicit physical models within Reynolds-averaged Navier-Stokes (RANS), particularly for modeling transition, turbulence, and heat flux. These approaches can yield improved flow predictions with marginal increase in computational cost compared to baseline models. Nevertheless, a key limitation lies in the substantial computational expense during the training phase, which often requires thousands of RANS evaluations. This challenge becomes severe in training models for complex industrial applications, where each RANS run is computationally intensive, and is further exacerbated when attempting to develop more generalizable and coupled multiple models across multiple product designs. Take the development of general transition and turbulence model corrections for both low- and high-pressure turbines as the study case, this work introduces two transformer-assisted strategies to accelerate model training. In the first, previously trained models are stored and used as inputs to the transformer, which generates new models informed by prior knowledge to partially replace randomly initialized models at the first training iteration. Results show that leveraging prior knowledge trained from different turbine configurations all effectively guide the search toward more promising regions of the solution space, thereby accelerating the training process. In the second scenario, when no prior knowledge is available, the transformer is integrated into the training loop to dynamically generate candidate models based on the small error models from the last training iteration and discarding high-error models. Results indicate that more frequent transformer updates, such as after every training iteration, further enhance the acceleration effect.

The present contribution focuses on an extended modelling of elongated, non-spherical particle transport in wall-bounded, particle-laden flows. A turbulent channel flow is considered using a point-particle Euler/Lagrange framework, with the fluid phase computed by a numerically filtered, scale-resolving approach. The developed method for inertial fibres was implemented in OpenFOAM Ⓡ , neglecting two-way coupling. Particle tracking with respect to translation and rotation is conducted in different frames of reference which are transformed through the use of quaternions, so that the fibre centroid position and orientation are known along their trajectory. Aerodynamic resistance coefficients for drag, lift, and torque are taken from correlations dependent on fibre orientation at an aspect ratio of five. Wall collisions of fibres are treated with an extended hard-body collision model that includes fibre orientation and the actual contact point. By solving the impulse equations with the parameters for restitution ratio and Coulomb friction coefficient the momentum loss was modelled. The flow validation was carried out against DNS data for a turbulent channel. Particular consideration was focused on the fibre-wall interactions, comparing the extended model with reduced approaches, such as centre-of-gravity specular reflection and spherical particle wall collision for the same equivalent diameter. The results highlight the important role of realistic wall-collision modelling. Accounting for the actual fibre-wall contact point leads to significantly different predictions of near-wall mean concentration, particle flux, and orientation profiles. In particular, fibre tilting during wall interactions enhances wall contact, increasing collision rates and modifying rebound angles compared to simplified models.

• A Hybrid High-Order method for incompressible Navier-Stokes is presented. • The method is Reynolds semi-robust and pressure-robust by design. • Conservation of volume is enforced cell-wise up to machine precision. • ESDIRK time stepping enables local high-order accurate time integration. • Validated on 2D/3D cases including the Taylor-Green vortex at Re = 1600. We propose a Hybrid High-Order (HHO) formulation of the incompressible Navier–Stokes equations, that is well suited to be employed for the simulation of turbulent flows. The spatial discretization relies on hybrid velocity and pressure spaces and the temporal discretization is based on Explicit Singly Diagonal Implicit Runge-Kutta (ESDIRK) methods. The formulation possesses some attractive features that can be fruitfully exploited when high-fidelity computations are required, namely: pressure-robustness, conservation of volume enforced cell-by-cell up to machine precision, robustness in the inviscid limit, implicit high-order accurate time stepping with local time step adaptation, reduced memory footprint thanks to static condensation of both velocity and pressure, possibility to exploit inherited p -multilevel solution strategies to improve performance of iterative solvers. After demonstrating the relevant properties of the scheme in practice, performing challenging 2D and 3D test cases, we consider the simulation of the Taylor–Green Vortex flow problem at Reynolds 1 600.