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An investigation of the effect of a destabilizing cross-stream temperaturegradient on the transient growth phenomenon of plane Poiseuille flow and planeCouette flow is presented. Only the streamwise-uniform and nearlystreamwise-uniform disturbances are highly influenced by the Rayleigh number Raand Prandtl number Pr. The maximum optimal transient growth G max ofstreamwise-uniform disturbances increases slowly with increasing Ra anddecreasing Pr. For all Ra and Pr, at moderately large Reynolds numbersRe, thesupremum of G max is always attained for streamwise-uniform perturbations (ornearly 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 scalingas in pure shear flow. A 3D vector model of the governing equationsdemonstrates that the short-time behavior is governed by the classical lift-upmechanism and that the influence of Ra on this mechanism is secondary andnegligible. The optimal input for the largest long-time response is given bythe adjoint of the dominant eigenmode with respect to the energy scalarproduct: the RBeigenmode without its streamwise velocity component. Theseshort-time and long-time responses depict, to leading order, the optimaltransient growth G(t). At moderately large Ra (or small Pr at a fixed Ra), thedominant adjoint mode is a good approximation to the optimal initial conditionfor all time. Over a general class of norms that can be considered as growthfunctions, the results remain qualitatively similar, for example, the dominantadjoint eigenmode still approximates the maximum optimal response.

Transient growth in Rayleigh-B\'enard-Poiseuille/Couette convection