Influences of conservative and non-conservative Lorentz forces on energy conservation properties for incompressible magnetohydrodynamic flows

In the analysis of magnetohydrodynamic (MHD) flow, the Lorentz force significantly affects energy properties because the work generated by the Lorentz force changes the kinetic and magnetic energies. Therefore, the Lorentz force and energy conversion should be predicted accurately. Some energy conservation schemes have been proposed and validated. However, the influences of the Lorentz force discretization on conservation and conversion of energy have not yet been clarified. In this study, a conservative finite difference method is constructed for incompressible MHD flows considering the induced magnetic field. We compare the difference in energy conservation properties among three methods of calculating the Lorentz force. The Lorentz forces are calculated in conservative and non-conservative forms, and both compact and wide-range interpolations of magnetic flux density are used to calculate the non-conservative Lorentz force. The compact interpolation method proposed in this study can perform conversions between conservative and non-conservative forms of the Lorentz force even when using the finite difference method. The present numerical method improves the conservation of transport quantity. Five models were analyzed, and the accuracy and convergence of the present numerical method were verified. From the viewpoint of the conservation of the total energy in an ideal inviscid periodic MHD flow, we consider that the calculation using compact interpolation for the Lorentz force is appropriate. This method preserves the total energy even on non-uniform grids. Moreover, the divergence-free condition of the magnetic flux density is discretely satisfied even without the correction of the magnetic flux density. The present numerical method can capture the Hartmann layer in the propagation of an Alfvén wave and accurately predict the tendency of energy attenuation in the analysis of a Taylor decaying vortex under magnetic fields. Analysis of the Orszag–Tang vortex reveals energy dissipation processes and the generation of high current densities. The present numerical method has excellent energy conservation properties and can accurately predict energy conversion. Therefore, this method can contribute to understanding complex unsteady MHD flows.