The film flow down an inclined plane has several features that make it an interesting
prototype for studying transition in a shear flow: the basic parallel state is an exact
explicit solution of the Navier–Stokes equations; the experimentally observed transition
of this flow shows many properties in common with boundary-layer transition;
and it has a free surface, leading to more than one class of modes. In this paper,
unstable wavepackets – associated with the full Navier–Stokes equations with viscous
free-surface boundary conditions – are analysed by using the formalism of absolute
and convective instabilities based on the exact Briggs collision criterion for multiple
k-roots of D(k, ω) = 0; where k is a wavenumber,
ω is a frequency and D(k, ω) is the
dispersion relation function.The main results of this paper are threefold. First, we work with the full
Navier–Stokes equations with viscous free-surface boundary conditions, rather than a
model partial differential equation, and, guided by experiments, explore a large region of the
parameter space to see if absolute instability – as predicted by some model equations
– is possible. Secondly, our numerical results find only convective instability, in
complete agreement with experiments. Thirdly, we find a curious saddle-point bifurcation which
affects dramatically the interpretation of the convective instability. This is the first finding
of this type of bifurcation in a fluids problem and it may have implications for the
analysis of wavepackets in other flows, in particular for three-dimensional instabilities.
The numerical results of the wavepacket analysis compare well with the available experimental
data, confirming the importance of convective instability for this problem.The numerical results on the position of a dominant saddle point obtained by
using the exact collision criterion are also compared to the results based on a
steepest-descent method coupled with a continuation procedure for tracking convective
instability that until now was considered as reliable. While for two-dimensional
instabilities a numerical implementation of the collision criterion is readily available,
the only existing numerical procedure for studying three-dimensional wavepackets
is based on the tracking technique. For the present flow, the comparison shows a
failure of the tracking treatment to recover a subinterval of the interval of unstable
ray velocities V whose length constitutes 29% of the length of the entire unstable
interval of V. The failure occurs due to a bifurcation of the saddle point, where V
is a bifurcation parameter. We argue that this bifurcation of unstable ray velocities
should be observable in experiments because of the abrupt increase by a factor of
about 5.3 of the wavelength across the wavepacket associated with the appearance of
the bifurcating branch. Further implications for experiments including the effect on
spatial amplification rate are also discussed.
Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain