Dual solutions of the flow through a curved tube

Dual solutions, i.e. two-vortex and four-vortex solutions, and their stability of flow through a slightly curved circular tube are numerically investigated by the spectral method in the range 96 ⩽ Dn ⩽ 10000, where Dn is the Dean number. It is found that the two-vortex solution is stable in response to any small disturbances, while the four-vortex solution is unstable to asymmetric disturbances. Time evolution of the unsteady four-vortex flow is also studied by a numerical simulation of the Navier-Stokes equation when Dn = 1000. The four-vortex flow eventually turns into a two-vortex flow.