Bifurcation and stability of combined free and forced convection in rotating curved ducts of square cross-section

A numerical study is made on fully developed bifurcation structure and stability of combined free and forced convection in a rotating curved duct of square cross-section. The solution structure is determined as the variation of a parameter indicating the magnitude of buoyancy force. Steady solution structure is very complicated. Flow and temperature fields on various solution branches are identified to be symmetric/asymmetric multi-cell patterns. Dynamic responses of multiple solutions to finite random disturbances are examined by direct transient computation. Five types of physically realizable solutions are identified numerically. They are stable steady 2-cell solution, stable steady multi-cell solution, periodic oscillation, chaotic oscillation and symmetry-breaking oscillation led by sub-harmonic bifurcation (period doubling). Among them, three kinds of stable steady solutions are found to co-exist within a range of parameters. In addition, temporal periodic and chaotic oscillations can also co-exist in another range of parameters. Furthermore, sub-harmonic bifurcation is identified to be another route to chaos. Spectral analysis is used to demonstrate the presence of additional frequencies for the case of sub-harmonic bifurcations. Results show that symmetry-breaking oscillation driven by sub-harmonic bifurcations appear to be identical with the mode observed in Lipps [J. Fluid Mech. 75 (1976) 113], McLaughlin and Orszag [J. Fluid Mech. 122 (1982) 123], and Gollub and Benson [J. Fluid Mech. 100 (1980) 449] for problem of free convection between flat horizontal plates.