The stability of an incompressible swept attachment-line
boundary layer flow is studied
numerically, within the Görtler–Hämmerlin
framework, in both the linear and nonlinear two-dimensional
regimes in a self-consistent manner. The initial-boundary-value
problem resulting from substitution of small-amplitude excitation
into the incompressible Navier–Stokes equations
and linearization about the generalized Hiemenz
profile is solved. A comprehensive comparison of all linear
approaches utilized to
date is presented and it is demonstrated that the linear
initial-boundary-value problem
formulation delivers results in excellent agreement with
those obtained by solution of
either the temporal or the spatial linear stability theory
eigenvalue problem for both
zero suction and a layer in which blowing is applied. In
the latter boundary layer
recent experiments have documented the growth of instability
waves with frequencies
in a range encompassed by that of the unstable
Görtler–Hämmerlin linear modes
found in our simulations. In order to enable further
comparisons with experiment
and, thus, assess the validity of the
Görtler–Hämmerlin theoretical model, we make
available the spatial structure of the eigenfunctions at
maximum growth conditions.The condition on smallness of the imposed excitation is
subsequently relaxed and
the resulting nonlinear initial-boundary-value problem is
solved. Extensive numerical
experimentation has been performed which has verified
theoretical predictions on
the way in which the solution is expected to bifurcate from
the linear neutral loop.
However, it is demonstrated that the two-dimensional
model equations considered do
not deliver subcritical instability of this flow; this
strengthens the conjecture that three-dimensionality is,
at least partly, responsible for the observed discrepancy
between the linear theory critical Reynolds number and the
subcritical turbulence observed either
experimentally or in three-dimensional numerical simulations.
Further, the present
nonlinear computations demonstrate that the unstable
flow has its line of maximum
amplification in the neighbourhood of the experimentally
observed instability waves,
in a manner analogous to the Blasius boundary layer. In
line with previous eigenvalue
problem and direct simulation work, suction is observed to
be a powerful stabilization
mechanism for naturally occurring instabilities of small amplitude.
On vanishing limits of the shear viscosity and Hall coefficients for the planar compressible Hall-MHD system
