  An investigation of the effect of a destabilizing cross-stream temperature
gradient on the transient growth phenomenon of plane Poiseuille flow and plane
Couette flow is presented. Only the streamwise-uniform and nearly
streamwise-uniform disturbances are highly influenced by the Rayleigh number Ra
and Prandtl number Pr. The maximum optimal transient growth G max of
streamwise-uniform disturbances increases slowly with increasing Ra and
decreasing Pr. For all Ra and Pr, at moderately large Reynolds numbersRe, the
supremum of G max is always attained for streamwise-uniform perturbations (or
nearly streamwise-uniform perturbations, in the case of plane Couette flow)
which produce large streamwise streaks and Rayleigh-B\'enard convection rolls
(RB). The optimal growth curves retain the same large-Reynolds-number scaling
as in pure shear flow. A 3D vector model of the governing equations
demonstrates that the short-time behavior is governed by the classical lift-up
mechanism and that the influence of Ra on this mechanism is secondary and
negligible. The optimal input for the largest long-time response is given by
the adjoint of the dominant eigenmode with respect to the energy scalar
product: the RBeigenmode without its streamwise velocity component. These
short-time and long-time responses depict, to leading order, the optimal
transient growth G(t). At moderately large Ra (or small Pr at a fixed Ra), the
dominant adjoint mode is a good approximation to the optimal initial condition
for all time. Over a general class of norms that can be considered as growth
functions, the results remain qualitatively similar, for example, the dominant
adjoint eigenmode still approximates the maximum optimal response.
