Transition from Steady to Chaotic States of Isothermal and Non-isothermal Flows through a Curved Rectangular Duct

Flows through a curved rectangular duct for the aspect ratio l =2 are numerically studied by use of the spectral method with and without a temperature difference between the vertical outer and inner sidewalls. We find three branches of steady solutions for the case without the temperature difference (isothermal case). Then we investigate linear stability of the steady solutions and find that only a portion of one steady solution branch is linearly stable but other branches are unstable. We obtain five branches of steady solutions for the case with the temperature difference (non-isothermal case) for the Grashof number Gr =100. Linear stability shows that only a portion of one of them is linearly stable while other branches are unstable like the isothermal case. The change of the flow state, as the Dean number Dn is increased, obtained by time evolution calculations, is found to be similar for both the isothermal and non-isothermal cases. When there is no stable steady solution, the time evolution calculations show that typical transition occurs from a steady flow to a chaotic state through a periodic flow when Dn is increased whether the system is isothermal or non-isothermal.