Abstract
Streak breakdown caused by a spanwise inflectional instability is
one phase of the
following transition scenarios, which occur in plane Poiseuille and Couette
flow. The
streamwise vortex scenario is described by formula here The oblique wave scenario is described by formula here The purpose of this paper is to investigate the streak breakdown phase
of the above
scenarios by a linear stability analysis and compare threshold energies
for transition
for the above scenarios with those for transition initiated by Tollmien-Schlichting
waves (TS), two-dimensional optimals (2DOPT), and random noise (N) at subcritical
Reynolds numbers. We find that if instability occurs, it is confined to disturbances with
streamwise
wavenumbers α0 satisfying
0<αmin<
[mid ]α0[mid ]<αmax. In these
unstable cases, the
least stable mode is located near the centre of the channel with a phase
velocity
approximately equal to the centreline velocity of the mean flow. Growth
rates for
instability increase with streak amplitude. For Couette flow streak breakdown
is
inhibited by mean shear. Using the linear stability analysis, we determine
lower
bounds on threshold amplitude for transition for scenario (SV) that are
consistent
with thresholds determined by direct numerical simulations. In the final part of the paper we show that the threshold energies for
transition
in Poiseuille flow at subcritical Reynolds numbers for scenarios (SV) and
(OW)
are two orders of magnitude lower than the threshold for transition initiated
by
Tollmien–Schlichting waves (TS) and an order of magnitude lower than
that for
(2DOPT). Scenarios (SV) and (OW) occur on a viscous time scale. However,
even
when transition times are taken into account, the threshold energy required
for
transition at a given time for (SV) and (OW) is lower than that for the
(TS) and
(2DOPT) scenarios at Reynolds number 1500.