Plane viscous channel flows are perturbed and the ensuing initial-value
problems are
investigated in detail. Unlike traditional methods where travelling wave
normal modes
are assumed as solutions, this work offers a means whereby arbitrary initial
input can
be specified without having to resort to eigenfunction expansions. The full temporal
behaviour, including both early-time transients and the long-time asymptotics, can
be determined for any initial small-amplitude three-dimensional disturbance. The
bases for the theoretical analysis are: (a) linearization of the governing
equations; (b)
Fourier decomposition in the spanwise and streamwise directions of the flow;
and (c)
direct numerical integration of the resulting partial differential equations.
All of the
stability criteria that are known for such flows can be reproduced. Also,
optimal initial
conditions measured in terms of the normalized energy growth can be determined
in a straightforward manner and such optimal conditions clearly reflect transient
growth data that are easily determined by a rational choice of a
basis for the initial
conditions. Although there can be significant transient growth for subcritical values
of the Reynolds number, it does not appear possible that arbitrary initial conditions
will lead to the exceptionally large transient amplitudes that
have been determined by
optimization of normal modes when used without regard to a particular initial-value
problem. The approach is general and can be applied to other classes of problems
where only a finite discrete spectrum exists (e.g. the Blasius boundary layer).
Finally, results from the temporal theory are compared with the equivalent
transient test case
in the spatially evolving problem with the spatial results having been obtained using
both a temporally and spatially accurate direct numerical simulation code.
The onset of transition in a boundary layer is dependent on the
initialization and
interaction of disturbances in a laminar flow. Here, theory and full Navier–Stokes
simulations focus on the transient period just after disturbances enter
the boundary
layer. The temporal evolution of disturbances within a boundary layer is
investigated
by examining a series of initial value problems. In each instance, the
complete
spectra (i.e. the discrete and the continuum) are included so that the
solutions can
be completely arbitrary. Both numerical and analytical solutions of the
linearized
Navier–Stokes equations subject to the arbitrary initial conditions
are presented. The
temporal evolution of disturbances during the transient period are compared
with
the spatial evolution of the same disturbances and a strong correlation
between the
two approaches is demonstrated indicating that the theory may be used for
the
transient period of disturbance evolution. The theory and simulations demonstrate
that strong amplification of the disturbances can occur as a result of
the inclusion
of the continuum in the prediction of disturbance evolution. The results
further
show that any approach proposed for use in bypass boundary layer transition
must
include the transient growth that results from the continuum. Finally,
although a
connection between temporal and spatial evolution in the transient period
has been
demonstrated, a theoretical basis as an explanation for this connection
remains the
focus of additional study.
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