This paper is concerned with the behaviour of Tollmien-Schlichting (T-S) waves experiencing
small localised distortions within an incompressible boundary layer developing
over a flat-plate. In particular, the distortion is produced by an isolated roughness element
located at Rexc = 440 000. We considered the amplification of an incoming T-S wave
governed by the two-dimensional linearised Navier-Stokes equations, where the base flow
is obtained from the two-dimensional non-linear Navier-Stokes equations. We compare
these solutions with asymptotic analyses which assume a linearised triple-deck theory for
the base flow and determine the validity of this theory in terms of the height of the smallscale
humps/indentations taken into account. The height of the humps/indentations is
denoted by h which is considered to be less than or equal to xcRe−5/8
xc
(corresponding
to h/δ99 < 6% for our choice of Rexc
). The rescaled width ˆd(≡ d/(xcRe−3/8
xc
)) of
the distortion is of the order O(1) and the width d is shorter than the T-S wavelength
(λT S = 11.3 δ99).
We observe that for distortions which are smaller than 0.1 of the inner deck height
(h/δ99 < 0.4%) the numerical simulations confirm the asymptotic theory in the vicinity
of the distortion. For larger distortions which are still within the inner deck (0.4% <
h/δ99 < 5.5%) and where the flow is still attached the numerical solutions show that
both humps and indentations are destabilising and deviate from the linear theory even
in the vicinity of the distortion.
We numerically determine the transmission coefficient which provides the relative amplification
of the T-S wave over the distortion as compared to the flat-plate. We observe
that for small distortions, h/δ99 < 5.5%, where the width of the distortion is of order
of the boundary layer a maximum amplification of only 2% is achieved. This amplification
can however be increased as the width of the distortion is increased or if multiple
distortions are present. Increasing the height of the distortion so that the flow separates
(7.2% < h/δ99 < 12.8%) leads to a substantial increase in the transmission coefficient of
the hump up to 350 %.