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The extreme heat fluxes in the divertor region of tokamaks may require an alternative to solid plasma-facing components, for the extraction of heat and the protection of the surrounding walls. Flowing liquid metals are proposed as an alternative, but raise additional challenges that require investigation and numerical simulations. Free surface designs are desirable for plasma-facing components (PFCs), but steady flow profiles and surface stability must be ensured to limit undesirable interactions with the plasma. Previous studies have mainly used steady-state, 2D, or simplified models for internal flows and have not been able to adequately model free-surface liquid metal (LM) experiments. Therefore, FreeMHD has been recently developed as an open-source magnetohydrodynamics (MHD) solver for free-surface electrically conductive flows subject to a strong external magnetic field. The FreeMHD solver computes incompressible free-surface flows with multi-region coupling for the investigation of MHD phenomena involving fluid and solid domains. The model utilizes the finite-volume OpenFOAM framework under the low magnetic Reynolds number approximation. FreeMHD is validated using analytical solutions for the velocity profiles of closed channel flows with various Hartmann numbers and wall conductance ratios. Next, experimental measurements are then used to verify FreeMHD, through a series of cases involving dam breaking, 3D magnetic fields, and free-surface LM flows. These results demonstrate that FreeMHD is a reliable tool for the design of LM systems under free surface conditions at the reactor scale. Furthermore, it is flexible, computationally inexpensive, and can be used to solve fully 3D transient MHD flows.

A high-order Flux reconstruction implementation of the hyperbolic formulation for the incompressible Navier-Stokes equation is presented. The governing equations employ Chorin's classical artificial compressibility (AC) formulation cast in hyperbolic form. Instead of splitting the second-order conservation law into two equations, one for the solution and another for the gradient, the Navier-Stokes equation is cast into a first-order hyperbolic system of equations. Including the gradients in the AC iterative process results in a significant improvement in accuracy for the pressure, velocity, and its gradients. Furthermore, this treatment allows for taking larger time-steps since the hyperbolic formulation eliminates the restriction due to diffusion. Tests using the method of manufactured solutions show that solving the conventional form of the Navier-Stokes equation lowers the order of accuracy for gradients, while the hyperbolic method is shown to provide equal orders of accuracy for both the velocity and its gradients which may be beneficial in several applications. Two- and three-dimensional benchmark tests demonstrate the superior accuracy and computational efficiency of the developed solver in comparison to the conventional method and other published works. This study shows that the developed high-order hyperbolic solver for incompressible flows is attractive due to its accuracy, stability and efficiency in solving diffusion dominated problems.

This paper presents a simple and highly accurate method for capturing sharp interfaces moving in divergence-free velocity fields using the high-order Flux Reconstruction approach on unstructured grids. A well-known limitation of high-order methods is their susceptibility to the Gibbs phenomenon; the appearance of spurious oscillations in the vicinity of discontinuities and steep gradients makes it difficult to accurately resolve shocks or sharp interfaces. In order to address this issue in the context of sharp interface capturing, a novel, preconditioned and localized phase field method is developed in this work. The numerical accuracy of interface normal vectors is improved by utilizing a preconditioning procedure based on the level set method with localized artificial viscosity stabilization. The developed method was implemented in the framework of the multi-platform Flux Reconstruction open-source code PyFR. Numerical tests in 2D and 3D conducted on different mesh types showed that the preconditioning procedure significantly improves accuracy. The results demonstrate the conservativeness of the proposed method and its ability to capture highly distorted interfaces with superior accuracy when compared to conventional and high-order VOF and level set methods. The high accuracy and locality of the proposed method offer a promising route to carrying out massively-parallel, high accuracy simulations of multi-phase, incompressible phenomena.