A primitive variable finite-volume method for incompressible viscous magnetohydrodynamic flows

A recently developed numerical method is employed for computing the numerical solutions of the incompressible Navier-Stokes equations in the presence of a magnetic field. The method is based on the pressure correction approach, but employs a regular grid finite-volume variable arrangement instead of the usual staggered grid arrangement. The pressure equation is derived such that effects which promote the well-known checkerboard instability are not present. A relevant compatibility constraint on pressure is satisfied by Neumann boundary conditions obtained using a vector identity. The transport equations of the magnetic field with solenoidal condition are solved with a similar approach. The unified computational framework is thus developed for the solution of incompressible viscous magnetohydrodynamic flows. Implemented in a second-order-accurate finite-volume code, the algorithm is used to compute the magnetohydrodynamic flow in a pipe. Numerical solution is compared with existing analytical and computational results for various Hartmann numbers.