We are concerned here with forced steady recirculating flows which
are laminar,
two-dimensional and have a high Reynolds number. The body force is considered
to be prescribed and independent of the flow, a situation which arises
frequently in
magnetohydrodynamics. Such flows are subject to a strong constraint. Specifically,
the body force generates kinetic energy throughout the flow field, yet
dissipation is
confined to narrow singular regions such as boundary layers. If the flow
is to achieve
a steady state, then the kinetic energy which is continually generated
within the bulk
of the flow must find its way to the dissipative regions.
Now the distribution of u2/2 is
governed by a transport equation, in which the only cross-stream transport
of energy
is diffusion, v∇2 (u2/2).
It follows that there are only two possible candidates for the
transport of energy to the dissipative regions: the energy could be diffused
to the
shear layers, or else it could be convected to the shear layers through
entrainment
of the streamlines. We investigate both options and show that neither is
a likely
candidate at high Reynolds number. We then describe numerical experiments
for a
model problem designed to resolve these issues. We show that, at least
for our model
problem, no stable steady solution exists at high Reynolds number. Rather,
as soon
as the Reynolds number exceeds a modest value of around 10, the flow becomes
unstable via a supercritical Hopf bifurcation.
Conditional analysis near strong shear layers in DNS of isotropic turbulence at high Reynolds number