Bifurcation study of three-dimensional solutions of the curved square-duct flow

Steady, periodic and traveling-wave solutions of the flow through a curved square duct are obtained numerically, and their linear stability to two- and three-dimensional disturbances is investigated. It is found that a steady traveling-wave (STW) solution bifurcates from the symmetric steady solution branch and is linearly stable in the intermediate range between low and high pressure gradient regions. Two stable periodic traveling-wave (PTW) solutions are also found, which have oscillating components with different symmetry. One keeps the same symmetry as the STW solution and the other breaks it. They seem to bifurcate from the STW solution branch via Hopf bifurcation although their solution branches are not obtained exactly. It is strongly suggested that a two-dimensional time-periodic (2DP) solution is created via heteroclinic bifurcation, not from local bifurcation. 2DP solution is always linearly unstable against threedimensional disturbances though it has linearly stable region to two-dimensional disturbances.