Linear stability of flow in rectangular ducts in the vicinity of the critical aspect ratio

We have performed three-dimensional BiGlobal linear stability analyses of flow in rectangular ducts by using a spectral element method. A linear stability theory is applied to a steady laminar solution and an eigenvalue problem is generated in the matrix form. The critical Reynolds numbers are evaluated for some aspect ratios to describe the neutral curves. It is found that the neutral curves consist of two bifurcation branches. One is the first bifurcation branch where the flow loses its stability with respect to three-dimensional disturbances, while the other is the second bifurcation branch where it recovers its stability. The first and second branches connect at the critical point of As=3.19 and Re=2.47×105, where the saddle-node bifurcation occurs. We have revealed that the discontinuous divergence of the critical Reynolds number to infinity in the vicinity of the critical aspect ratio As is caused by the saddle-node bifurcation.