Instability of MHD Flow in the Duct with Electrically Perfectly Conducting Walls

The stability problem of conducting fluid flow in a square duct with perfectly conducting walls is investigated. A homogeneous and constant static magnetic field is applied along the vertical direction of the flow. Nonmodal linear stability analysis is performed on this problem for the first time and the effect of the imposed magnetic field is also taken into account. The amplification and distribution of primary optimal perturbations are obtained by solving iteratively the direct and adjoint governing equations with respect of perturbations. Four modes of perturbations with different symmetries in the space are investigated. Computational results show that, the MHD duct flow is stable at either small or large Hartmann number, but unstable at moderate one. The primary optimal perturbations are in the form of streamwise vortices, which are located inside the thin sidewall layers parallel to the magnetic field. The size of the vortices is decreased with the growing of Hartmann number Ha, meanwhile the amplification of the perturbations is reduced due to the magnetic damping effect. The Hartmann layer perpendicular to the magnetic field seems to be irrelevant to the stability of the MHD duct flow. The most unstable perturbation is in the form of Mode I, which having co-rotating vortices at opposite sidewalls and the vortices tend to enhance each other.