The paper reviews some of the most important challenges and recent progress in modeling and simulations of flow, turbulence and heat transfer of the electrically conductive fluids interacting with electromagnetic fields (EMF). Despite the differences in flow geometries and huge disparity of the length- and time-scales, ranging from astrophysical and geophysical to laboratory-scale applications, it is demonstrated that magneto-fluid-dynamics (MFD) phenomena, in their essence, are characterized by two fundamental features — a time-dependent spiraling flow patterns and non-homogeneous distributions of the magnetic fields. Some representative examples of both two- and one-way coupled MFD phenomena are considered: the turbulent magnetic dynamo under realistic experimental conditions (the Riga-dynamo experimental setup), and the turbulent Rayleigh-Bénard convection in a differentially heated enclosure subjected to the localized Lorentz force originating from combining electrodes and array of permanent magnets located beneath the lower thermally active enclosure wall. In both considered cases, a good agreement between available experimental results and numerical simulations is obtained, proving accuracy and potentials of the multi-scale modeling approach in simulating complex MFD phenomena.

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