Effect of curvature on dual solutions of flow through a curved circular tube

Dual solutions, i.e., two-vortex and four-vortex solutions, of the flow through a curved circular tube are numerically obtained by the spectral method for 0 ≤ δ ≤ 0.8 and 500 ≤ Dn ≤ 10000, where δ is the non-dimensional curvature of the tube and Dn the Dean number. It is found that the critical Dean number above which the four-vortex solution exists takes the lowest value 956 at δ = 0 and increases with δ. In terms of the Reynolds number of the flow, however, the critical Reynolds number decreases from infinity as δ increases from zero, takes the minimum value of about 250 at δ ≈ 0.52, and then increases again.