Transition to turbulence in bent pipes

The flow through curved pipes has received increasing attention in the past decades, but several phenomena still miss an exhaustive explanation. Bent pipes are fundamental components in various industrial devices, and are also studied in the medical field, being an integral part of vascular and respiratory systems [2,3].
The transition to turbulence of these flows has received a considerable amount of interest in the past decades: the recent works by Canton et al. [4] and Kühnen et al. [5] determined that the flow is linearly unstable, and undergoes a Hopf bifurcation for any curvature greater than zero. This behaviour is in contrast to the flow in a straight pipe which is linearly stable and undergoes subcritical transition [1].
Low pipe curvatures present a different scenario: here the flow becomes turbulent before onset of the linear instability, and no clear transition boundary has been observed. For low curvatures a bent pipe appears to behave similarly to a straight pipe, i.e. the flow undergoes transition to turbulence despite being linearly stable to infinitesimal perturbations [5,6].

[1] D. Barkley. Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech., 803:P1, 2016.
[2] S. A. Berger, L. Talbot, and L.-S. Yao. Flow in curved pipes. Annu. Rev. Fluid Mech., 15:461–512, 1983.
[3] K. V. Bulusu, S. Hussain, and M. W. Plesniak. Determination of secondary flow morphologies by wavelet analysis in a curved artery model with physiological inflow. Exp. Fluids, 55:1832, 2014.
[4] J. Canton, P. Schlatter, and R. Örlü. Modal instability of the flow in a toroidal pipe. J. Fluid Mech., 792:894–909, 2016.
[5] J. Kühnen, P. Braunshier, M. Schwegel, H. C. Kuhlmann, and B. Hof. Subcritical versus supercritical transition to turbulence in curved pipes. J.  Fluid Mech., 770:R3, 2015.
[6] K. R. Sreenivasan and P. J. Strykowski. Stabilization effects in flow through helically coiled pipes. Exp. Fluids, 1:31–36, 1983.