Asymmetric travelling waves in a square duct

Abstract

Two types of asymmetric solutions are found numerically in square-duct flow. They emerge through a symmetry-breaking bifurcation from the mirror-symmetric solutions discovered by Okino et al. (J. Fluid Mech., vol. 657, 2010, pp. 413–429). One of them is characterized by a pair of streamwise vortices and a low-speed streak localized near one of the sidewalls and retains the shift-and-reflect symmetry. The bifurcation nature as well as the flow structure of the solution show striking resemblance to those of the asymmetric solution in pipe flow found by Pringle & Kerswell (Phys. Rev. Lett., vol. 99, 2007, A074502), despite the geometrical difference between their cross-sections. The solution seems to be embedded in the edge state of square-duct flow identified by Biau & Bottaro (Phil. Trans. R. Soc. Lond. A, vol. 367, 2009, pp. 529–544). The other solution deviates slightly from the mirror-symmetric solution from which it bifurcates: the shift-and-rotate symmetry is retained but the mirror symmetry is broken.