This article focuses on the basic turbulent flow, and the thermal, mass-transfer, and hydrodynamic phenomena for use in modeling physical processes during induction melting. It provides a discussion on transport phenomena equations that includes the approximation of convective terms in the transport equation and computational schemes for the fluid dynamics equation. The aspects of computational algorithms for specific magnetohydrodynamic problems with mutual influence of the magnetic field and melt flow due to the changing shape of the free surface are also considered. The article illustrates the application of the basic equations and approaches formulated for electromagnetic field and melt turbulent flow for the numerical study of an induction crucible furnace.