Bifurcation diagram for two-dimensional steady flow and unsteady solutions in a curved square duct

Flows through a curved duct of square cross-section are numerically studied by using the spectral method, and covering a wide range of curvature δ of the duct (0 < δ ≤ 0.5) and the Dean number Dn (0 < Dn ≤ 8000), where δ is non-dimensionalized by the half width of the square cross-section. The main concern is the relationship between the unsteady solutions, such as periodic, multi-periodic and chaotic solutions, and the bifurcation diagram of the steady solutions. It is found that the bifurcation diagram topologically changes if the curvature is increased and exceeds the critical value δ_c ≈ 0.279645, while it remains almost unchanged for δ < δ_c or δ > δ_c. A periodic solution is found to appear if the Dean number exceeds the bifurcation point, whether it is pitchfork or Hopf bifurcation, where no steady solution is stable. It is found that the bifurcation diagram and its topological change crucially affect the realizability of the steady and periodic solutions. Time evolution calculations as well as their spectral analysis show that the periodic solution turns to a chaotic solution if the Dn is further increased no matter what the curvature is. It is interesting that the chaotic solution is weak for smaller Dn, where the solution drifts among the steady solution branches, for larger Dn, on the other hand, the chaotic solution becomes strong, where the solution tends to get away from the steady solution branches.