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

Abstract

Flows through a curved duct of square cross-section are numerically studied by using the spectral method, and covering a wide range of curvature of the duct (0 < 0.5) and the Dean number Dn (0 < Dn 8000), where 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 c ≈ 0.279645, while it remains almost unchanged for < c or > 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.

Traveling-wave solutions of the flow in a curved-square duct

“…3. This fact verifies the conjecture issued by Mondal et al, 5 which argued that time-periodic solution obtained by 2D calculation may correspond to a traveling-wave solution in the real 3D flow. This is also consistent with the results of Wang and Yang.…”

“…Although all the solution branches in Fig. 2 have been already found by, for example, Winters, 4 Wang and Yang, 9 and Mondal et al, 5 they are newly calculated for the present study.…”

“…Numerical and experimental investigations 4,5 have shown that many types of steady and unsteady solutions appear as the Dean number, Dn, increases for the curved-rectangular-duct flow, where Dn is proportional to the Reynolds number multiplied by the square root of the duct curvature. If Dn is very small, a steady solution, which has two secondary vortices ͑two-vortex solution͒ in a cross section, is realized in not only the numerical simulations but the experimental observations.…”

“…Numerical studies of a curved-square-duct flow 5 showed that if Dn is further increased, periodic solutions become quasiperiodic and then the solutions turn into chaos. Therefore, the curved-square-duct flow may exhibit a typical change in flow structures from a steady solution to chaos on the route to turbulence.…”

Non-isothermal Flow through a Curved Channel with Strong Curvature

“…An early co mplete b ifurcation study of two-dimensional (2-D) flow through a curved duct of square cross section was conducted by Winters (1987). Very recently, Mondal et al (2007a) perfo rmed comprehensive numerical study on fully developed bifurcation structure and stability of two-dimensional (2D) flo w through a curved duct with square cross section and found a close relationship between the unsteady solutions and the bifurcat ion diagram of steady solutions. The flow through a curved duct with differentially heated vertical sidewalls has another aspect because secondary flows promote fluid mixing and heat transfer in the fluid (Yanase et al, 2005).…”