Edge States and Phase Space of Channel Flow with Spanwise Magnetic Field

The edge state between laminar and turbulent flows is crucial for understanding the transition process. This boundary usually exhibits a highly convoluted and fractal structure. In this study, we employ a tracking method to investigate the phase space of the laminar-turbulent edge state in plane Poiseuille flow subjected to a spanwise magnetic field. Notably, the presence of the magnetic field allows for the sustained perturbations along the field direction, while eliminating the counterparts in the vertical plane. By conducting high resolution direct numerical simulations (DNS), we accurately compute the exact coherent structures (ECS) and connecting orbits in the channel flow, considering a constant and uniform magnetic field aligned in the spanwise direction. Using the bisection method, we search for the exact coherent states within the channel. Our findings reveal that at low Reynolds numbers, the edge state in the channel flow comprises relative periodic orbits characterized by streamwise streaks and vortices on both sides of the streaks. The phase diagram demonstrates that the cycle period of the ECS increases while the amplitude of velocity perturbation decreases with the Reynolds number. The applied magnetic field elongates the exact coherent structures along its direction and compresses them in the remaining two directions, exhibiting distinct characteristics compared to the non-magnetic case. Furthermore, the cycle period of the exact coherent states exhibits a decreasing trend as the magnetic field strength intensifies. At high strengths of the spanwise magnetic field, the flow field decay to laminar flow.