Advanced Wall Modelling for Magnetohydrodynamics Simulations Using Physics-Informed Machine Learning

Magnetohydrodynamics (MHD) is critical in engineering applications such as fusion power plant design and metallurgical processes. However, the computational demands of full-scale 3D MHD simulations are significant, particularly due to the need for extra fine meshing in boundary layers, such as those on Hartmann and side walls, where the behaviour of electric currents must be precisely captured.

The research group (joint academic/industry team) has recently developed a simulation paradigm centred on wall-resolved large-eddy simulation (LES) coupled with MHD. Ideal for studying canonical cases to explore fundamental physics and to build database, however, this is computationally too expensive for parametric design tasks for the next step.

The proposed new research aims to develop efficient wall models for MHD simulations using advanced deep learning techniques, specifically Physics-Informed Neural Networks (PINNs) and Physics-Informed Neural Operators (PINOs). By leveraging these methods, the research seeks to reduce the computational load associated with these meshing requirements, enabling faster and more accessible parametric analysis and design optimisation.

Recent advancements in deep learning, specifically through Physics-Informed Neural Networks (PINNs) and Physics-Informed Neural Operators (PINOs), offer a promising solution to these challenges. These methods embed physical laws into neural networks, enabling the efficient solving of partial differential equations (PDEs) while potentially reducing the need for extremely fine meshing. 

The aim of this research is to leverage PINNs and PINOs to build accurate wall models for MHD simulations and extend this work to further develop surrogate models which would provide an additional layer of efficiency by approximating the behaviour of complex systems with significantly reduced computational costs. The integration of surrogate models with PINNs and PINOs can result in a powerful framework reducing the computational load and enhancing the accessibility of these simulations for practical engineering applications.

The primary objective of this research is to develop a set of wall models for MHD simulations using PINNs/PINOs that can reduce computational costs and enable fast parametric analysis. The specific goals include:

• Developing mathematical models for PINNs and PINOs tailored to MHD wall modelling, particularly for capturing the behaviour of electric currents in boundary layers adjacent to conductive walls.
• Creating wall models that are applicable to both laminar and turbulent MHD flows.
• Conducting a parametric analysis of key variables, (eg., conductivity ratio, characteristic length, wall thickness, surface heat flux, velocity).
• Investigate various configurations, eg.( (non-)isothermal circular and rectangular pipes subjected to (non-)uniform magnetic fields.
• Investigating the impact of these parameters on critical outputs (eg. pressure gradients, Nusselt number and surface temperature).
• Creating surrogate models that can approximate complex MHD system behaviours with reduced computational demands.
• Develop a flexible, expandable and efficient framework that can be used for design purposes.

This research is expected to provide advanced wall models and surrogate models that significantly reduce the computational load imposed by strict meshing requirements in MHD simulations. By leveraging the strengths of PINNs and PINOs, the developed models will enable rapid parametric analysis and design optimisation, making MHD simulations more practical and accessible. The framework could also be adapted for other fluid dynamics problems where computational efficiency is essential.