The Developments and the Applications of the Numerical Algorithms in Simulating the Incompressible Magnetohydrodynamics with Complex Boundaries and Free Surfaces

Throughout the present thesis, magnetohydrodynamics (MHD, hereafter) deal with flows of liquid metal which is subjected to the external magnetic fields (MFs, hereafter), and therefore, the fluid flow is treated as incompressible. MHD is relatively young in natural science and engineering starting with the pioneering work of Hartmann (Hg-Dynamics-I, Math-Fys. MLA, 1937, [34]) in liquid metal duct flow under the influence of a strong external MF. Under such circumstances, the motion of the electrically conducting fluids generate the electric current and the Lorentz force, and as a consequence, the flow field is coupled and varied with the induced electromagnetic field. Today the research in MHD maybe subdivided into two purposes, there are on the one hand fluid mechanics and applied mathematics because of its complex flow characteristics, such as the MHD instability and the MHD turbulence, which contain particular dynamic behaviors due to the coupling of the flow field and the electromagnetic field. On the other hand, it is also very important in industrial production processes and technology applications, whose working principle is based on the MHD effects, which may be effective in optimizing and controlling the liquid flows in metallurgical and casting industries. Furthermore, the development of MHD pumping devices and the electromagnetic voltmeter also depends on the research in MHD flows. In addition, MHD has been advanced significantly during the last three decades by efforts to take the massive energy out of magnetic-confinement fusion device, which is thought as the most potential energy in future, and therefore, the flow characteristics and the heat transfer in liquid metal flows such as lithium or lithium lead under the influence of MHD effect deserves more attention. The design and construction of a liquid metal fusion blanket or fusion divertor requires detailed knowledge of MHD duct flows and MHD free surface flows.

The set of equations governing the viscous incompressible fluid motion under the influence of an external MF are the momentum and mass conservation equations, given by where \(\mathbb {S}\) is the viscous stress tensor written as with density \(\rho \), pressure p and dynamics viscosity \(\mu \). \({\varvec{F}}_{s}\) stands for the surface tension which just acts on the interface and is formulated as a volume force based on the CSF technique.

In Gerris, an approximate projection method making use of a staggered-in-time discretization is employed to discretize the governing equations, and the numerical schemes are second-order accurate.

In this chapter, detailed validations are presented to investigate the accuracy and the efficiency of the Gerris-MHD solver in simulating the MHD flows, respectively with complex solid boundaries or free surfaces.

As introduced in Chap. 1, the bubbly driven flows are encountered frequently in the metallurgical engineering and the casting engineering, where the bubbles are injected into the molten metal to stir and homogenize the liquid metal (Bai and Thomas, Metall Mater Trans B 32(6):1143–1159, 2001, [1]). For such type of two-phase flows, the MFs are used to control the bubble motion through a contactless way. In addition, in the fusion reactor, the MHD effect will greatly suppress the turbulence of the flow in the energy transfer section, and a possible remedy is to inject gas bubbles into the coolant to enhance the flow disturbances. From this perspective, the bubble behaviors in the fluids subjected to the external MF has became a topic of considerable interest.

After a previous investigation of the rising motion of a gas bubble in the liquid GaInSn under the influence of a vertical MF, this chapter focuses on the case of a uniform horizontal MF. Similarly, in order to allow some benchmark, most of the parameters selected for this new investigation are the same as in an experiment recently performed in Dresden, Germany. The Ar bubble diameter is either 4.3 mm or 6.4 mm, and the liquid metal is still GaInSn, resulting in Reynolds numbers in the parameter space of \(2000<Re<4000\), which is typically larger than that rising in water. The research is carried out in a similar manner, by investigating the influence of the horizontal MF on the rising velocity, the rising path, the vortex structures and the shape of the bubble, and finally, we give a brief comparison of the numerical results between the vertical and horizontal MFs.

In the previous two chapters (Chaps. 5 and 6), we confirm that the bubble motion is greatly influenced by imposing an external MF, which modifies the bubble shapes, the vortex structures, and also the rising paths. Meanwhile, the relations between the wake evolution and the path transition are more clear by investigating the transition process under the influence of the MF. However, the physical mechanics causing the path instability are still not so clear. In this chapter, we try to study a particular case of the vortex evolutions when the single bubble transits from zigzag to spiral.

In order to improve the performance of the numerical methodologies in studying the incompressible MHD flows, a new solver is developed and implemented into the open source Gerris code. The new Gerris-MHD solver, which is based on the Cartesian grids and the AMR technique, is able to simulate the single-phase MHD flows respectively of being bounded by electrically insulating or conducting walls with a cut-cell approach. In addition, by using the VOF method, the solver is also able to solve the multi-phase MHD flows and the discontinuities of the physical properties across the fluid-fluid interface can be handled very well. After that, the single bubble motion with or without the influence of the MF is investigated numerically, and the physical mechanisms are discussed in details.