Stability Analysis Of Channel Streaky Flow Under The Magnetic Field

The stability of flow has been one of the most important and central problems in fluid mechanics. It becomes more complex when the fluid is electrically conducting and working in the condition with magnetic field. In recent years, the so-called MHD flow instability problem, has received a lot of attention from scientists and engineers.The thesis presents linear stability analysis and direct numerical simulation of MHD channel flows at low Reynolds number. The stability of large-scale streaks in a channel flow with streamwise magnetic field is investigated. The magnetic field is constant and uniform in the orientation and low magnetic Reynolds number assumption is employed in this study. Two different kinds of streaks are considered, steady streaks with only the streamwise velocity component and unsteady streaks with all the three velocity components. The secondary optimal perturbations and the corresponding amplifications of initial perturbation energy are calculated in an iterative procedure by solving the direct and adjoint governing equations, which are linearized around the basic flow state, streaks evolving on top of channel Poiseuille flow.When the large amplitude steady streaks with only the streamwise velocity component are concerned, exponential instability of perturbations occurring at different streamwise wavenumbers is suppressed by the imposed magnetic field. A simple scaling law between the most unstable streamwise wavenumber of exponential instability and the magnetic interaction parameter can be explained by the balance of inertial force and Lorentz force. Transient growth of secondary perturbation is also suppressed by the streamwise magnetic field, when the small amplitude streaks are considered. Similar trend can be found when the real unsteady streaks with full velocity components are considered, but exponential growth behavior of secondary perturbations is absent even for large amplitude streaks. A similar scaling law between the most unstable streamwise wavenumber of transient instability and the magnetic interaction parameter is also valid in some range. The predicted critical Hartmann number, at which all the perturbations are suppressed and the streaky flow is supposed to be stable in the linear frame, increases with the increment of Reynolds number, which is also verified in the relaminarization tests with direct numerical simulation. Those results are in good agreement with experiment on pipe flow with streamwise magnetic field.The suppression effect of streamwise magnetic field on fully developed turbulence in the channel is also investigated with direct numerical simulation. Large elongated streamwise coherent structures are formed when a strong magnetic field is imposed. Further increment of Hartmann number leads to the occurrence of relaminarization, but the flow can not transit to turbulent state by imposing noises in the presence of streamwise magnetic field.Secondary stability analysis is also performed on small-scale streaks evolving on top of Hartmann flow in the channel under wall-normal magnetic field. The results show that, Hartmann flow becomes unstable when the Reynolds number based on the depth of Hartmann layer is great than400, which is qualitatively consistent with the results of previous simulations and experiments.