Multiple solution of a flow in a curved rectangular tube (1st report, symmetric steady solutions and their stability)

The flow through a curved rectangular duct with the curvature = 0.1 is investigated numerically at the Dean number Dn = 100 for various aspect ratio of the duct cross section by use of the spectral method of the polynomial function expansion. The flow is assumed to be uniform in the direction of the duct axis. Steady solutions which are symmetric with respect to the mid-horizontal plane of the duct cross section are obtained by the Newton-Raphson's method and their linear stability is studied. The bifurcation diagram of the solution shows that there exist three solution branches, the main vortex solution, the multi-vortex solution and the singular vortex solution. Some of the secondary flow patterns found by Akiyama et al.⁽¹⁾ are allotted to the solutions in the bifurcation diagram.