Investigation of electro-vortex flow between planes using different computational approaches

Electrovortex flows arise when an electric current of varying density passes through a well-conducting fluid (e.g., acid or metal melt). In such a case, the electric current generates a magnetic field, which leads to the Lorentz force causing swirling currents of the medium. There are different methods of theoretical study of such currents. As a rule, to avoid the necessity to find the pressure dependence on coordinates, the variables "vector potential of velocity - swirl" ("scalar current function – swirl" in the case of axisymmetric flows) are used. In such a case, it is quite effective to use automodel variables, which allow to reduce the dimensionality of the problem. In this case, the solution for the introduced function can be sought in the form of an expansion by the electro-vortex flow parameter proportional to the square of the magnetic Reynolds number. Also, this solution can be obtained numerically, for example, using finite-difference methods. Nowadays, more and more often the solutions are investigated by means of direct numerical modeling methods, when no automodel approximations are made, which reduce the accuracy of the solution. Nevertheless, in such a case the number of computations can be quite large and requires the use of supercomputer resources. A separate difficulty is presented by the boundary conditions: for example, for the velocity vector potential we obtain a fourth-order equation, which imposes significant restrictions on the time steps in the evolution equation. The problem can be avoided by using approximate boundary conditions, but this again reduces the accuracy of the solution. In this paper, using the example of electro-vortex flow between planes, the solutions that can be obtained using the various computational approaches mentioned above are examined. The results obtained are compared, and they are also compared with analytical approximations.