In this study, mathematical and numerical methodologies are developed, and an induction-based incompressible magnetohydrodynamic (MHD) flow solver was created, to study moderate Rem liquid metal (LM) MHD flows for fusion blanket design. Most LM MHD flow numerical studies in fusion blanket design have traditionally assumed that the magnetic Reynolds number (Rem) is much less than unity. The Rem, a dimensionless parameter in the magnetic induction equation, is a measure of the ratio of convection to diffusion of the magnetic field. The low Rem approximation, also known as the inductionless or quasi-static approximation, assumes that the applied magnetic field is quasi-static and that the ratio of induced to applied magnetic field strength is much less than unity. This assumption is not valid under certain conditions, however. For example, during unsteady plasma events, such as major disruptions, the applied magnetic field changes on the order of Tesla per milliseconds. The strongly unsteady applied magnetic field requires the use of the magnetic induction formulation. Furthermore, these conditions may lead to high velocities such that Rem is greater than unity.

The objectives of this study are to (1) study the effects of moderate Rem on steady MHD flows, (2) compare approximate magnetic boundary conditions (BCs) with proper far-field magnetic BCs for moderate Rem steady MHD flows and our main objective (3) study flow physics of a flow induced from a strongly unsteady applied magnetic field, similar to conditions expected in the LM during a fusion plasma disruption. We limit our scope of the first two objectives to a simple lid-driven cavity (LDC) flow with a transversely applied magnetic field. For the third objective, we consider a long three-dimensional rectangular enclosure (akin to a blanket module) with no-slip conducting walls on all sides and an unsteady applied magnetic field computed from a plasma code, used for the international thermonuclear experimental reactor (ITER).

The Rem effect on MHD flows was analyzed by considering a LDC MHD flow with a transversely applied magnetic field using proper far-field magnetic BCs. Results show that the flow is mostly two-dimensional (except for the Hartmann layers) when Rem < 100, but becomes more three-dimensional as Rem increases. The integral kinetic energy and velocity distributions indicate that the Rem effect on the flow is negligible for Rem. 100 at steady state, suggesting that approximate magnetic BCs are valid in this range under steady-state conditions. While the flow is unsteady, however, the integral kinetic energy deviated significantly with respect to changes in Rem, indicating that the approximate magnetic BCs are likely invalid during flow unsteadiness. For 0 ≤ Rem ≤ 100, the induced magnetic field magnitude increases linearly with Rem while its distribution remains qualitatively unchanged. The induced magnetic field energy in the flow domain is higher than the applied one for Rem values higher than Rem ~ 850. First results of linear and non-linear dynamo tests were performed for the LDC flow problem and while both tests were ultimately inconclusive, the results were analyzed and time-local induced magnetic field generation was observed in the non-linear dynamo test.