All Mach Number Schemes for Highly Magnetised MHD flows and Multi-material Modelling

The design of numerical methods that are equally valid and effective across different flow regimes has been a longstanding goal, or "holy grail", for CFD researchers. Building on a rich body of literature, and relying on a flux vector splitting framework, in this thesis we develop novel all Mach number algorithms in two key areas of research and applications: (i) visco-resistive, magnetically dominated magnetohydrodynamic (MHD) flows and (ii) multi-material flows. First, we present a new conservative semi-implicit scheme for the ideal and visco-resistive MHD system, designed to address the multiscale nature of the equations and able to efficiently handle different acoustic and Alfvén Mach number regimes. The scheme is highly attractive for fusion applications, as it can simulate magnetically dominated plasmas that rapidly transition from a near-equilibrium state into a time-dependent compressible regime. The scheme utilises a well-studied flux vector splitting approach to partition the MHD system into two sub-systems: one containing all the advective terms and the other containing the rest of the terms, including not only the gas pressure but also magnetic pressure terms. The explicit treatment of the former via a finite volume method ensures good shock-capturing capabilities, while the implicit treatment of the latter using a nested Picard method renders the scheme particularly well-suited for low sonic Mach number flows and the incompressible limit of the MHD equations. Strongly magnetised flows can be effectively simulated, benefitting from the scheme’s stability under a mild velocity-based CFL condition, which is independent of both the sound and Alfvén speeds. The visco-resistive dissipative terms are treated implicitly to preserve the stability and, consequently, the efficiency, of the scheme even in visco-resistive dominated problems. A cell-centred spatial discretisation on Cartesian meshes is used and adaptations of the constrained transport methodology are proposed to enforce the divergence-free condition of the magnetic field up to machine precision levels. Second, we present a new sharp interface semi-implicit method that extends single-fluid all Mach number schemes for the Euler equations to multi-material scenarios. The proposed algorithm effectively addresses the main challenges encountered by Godunov-type multi- material compressible solvers and is well-suited for applications requiring efficient and accurate handling of multi-material flows with physical interfaces across a wide range of Mach numbers. The scheme employs a level-set method to track the motion of material interfaces within an Eulerian framework, where each material is governed by the compressible Euler equations and is described by its own equation of state. Using a well-known flux vector splitting approach, the governing system is divided into a slow- and a fast-scale component, referred to as the advection and pressure sub-system, respectively. A Riemann-based ghost fluid method is utilised within the explicit solution of the advection component, allowing for the separate update of each material while capturing their nonlinear interactions at the physical interface. For the pressure sub-system, an implicit discretisation of the energy and momentum equations is applied in the bulk of the flow, following the approach typically used in single-material all Mach number solvers, and an ad-hoc procedure is introduced to implicitly enforce the continuity of both pressure and normal velocity across the material interface. Both algorithms are thoroughly validated against a large suite of benchmark test problems, carefully selected to assess the performance and the suitability of the proposed methods across different regimes.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2024

Author dc:creator

• Dematte, Riccardo

Advisor dc:contributor.advisor

• Nikiforakis, Nikolaos

• Semi-implicit conservative finite volume
• All Mach number flow algorithm
• Flux vector splitting
• Magnetohydrodynamic
• Multi-material modelling

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/4fa287ce-d35a-4a79-8a79-95a60f728bd5/download
• http://purl.org/NET/rdflicense/allrightsreserved

Author Identifier

0000-0002-3869-6475

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/385565