The flow through curved pipes has received increasing atten
tion in the past decades\, but several phenomena still miss an exhaustive
explanation. Bent pipes are fundamental components in various industrial d
evices\, and are also studied in the medical field\, being an integral par
t of vascular and respiratory systems [2\,3].

\nThe transition to turbu
lence 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 bi
furcation for any curvature greater than zero. This behaviour is in contra
st to the flow in a straight pipe which is linearly stable and undergoes s
ubcritical transition [1].

\nLow pipe curvatures present a different sc
enario: here the flow becomes turbulent before onset of the linear instabi
lity\, and no clear transition boundary has been observed. For low curvatu
res a bent pipe appears to behave similarly to a straight pipe\, i.e. the
flow undergoes transition to turbulence despite being linearly stable to i
nfinitesimal perturbations [5\,6].

\n

\n[1] D. Barkley. Theoretical
perspective on the route to turbulence in a pipe. J. Fluid Mech.\, 803:P1\
, 2016.

\n[2] S. A. Berger\, L. Talbot\, and L.-S. Yao. Flow in curved
pipes. Annu. Rev. Fluid Mech.\, 15:461–512\, 1983.

\n[3] K. V. Bulusu
\, S. Hussain\, and M. W. Plesniak. Determination of secondary flow morpho
logies by wavelet analysis in a curved artery model with physiological inf
low. Exp. Fluids\, 55:1832\, 2014.

\n[4] J. Canton\, P. Schlatter\, and
R. Örlü. Modal instability of the flow in a toroidal pipe. J. Fluid M
ech.\, 792:894–909\, 2016.

\n[5] J. Kühnen\, P. Braunshier\, M. Sch
wegel\, H. C. Kuhlmann\, and B. Hof. Subcritical versus supercritical tran
sition to turbulence in curved pipes. J. Fluid Mech.\, 770:R3\, 2015.

\n[6] K. R. Sreenivasan and P. J. Strykowski. Stabilization effects in fl
ow through helically coiled pipes. Exp. Fluids\, 1:31–36\, 1983.

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