  Sommerfeld paradox (turbulence paradox) roughly says that mathematically the
Couette linear shear flow is linearly stable for all values of the Reynolds
number, but experimentally transition from the linear shear to turbulence
occurs under perturbations of any size when the Reynolds number is large
enough. In [Li, Lin 2011], we offered a resolution of this paradox. The aim of
this paper is to provide a numerical implementation of the resolution. The main
idea of the resolution is that even though the linear shear is linearly stable,
slow orbits (also called quasi-steady states) in arbitrarily small
neighborhoods of the linear shear can be linearly unstable. The key is that in
infinite dimensions, smallness in one norm does not mean smallness in all
norms. Our study here focuses upon a sequence of 2D oscillatory shears which
are the Couette linear shear plus small amplitude and high frequency sinusoidal
shear perturbations. In the fluid velocity variable, the sequence approaches
the Couette linear shear (e.g. in $L^2$ norm of velocity), thus it can be
viewed as Couette linear shear plus small noises; while in the fluid vorticity
variable, the sequence does not approaches the Couette linear shear (e.g. in
$H^1$ norm of velocity). Unlike the Couette linear shear, the sequence of
oscillatory shears has inviscid linear instability; furthermore, with the
sequence of oscillatory shears as potentials, the Orr-Sommerfeld operator has
unstable eigenvalues when the Reynolds number is large enough, this should lead
to transient nonlinear growth which manifests as transient turbulence as
observed in experiments. The main result of this paper verifies this transient
growth.
