Rayleigh-B\'{e}nard-Poiseuille flow: Optimal growth of streamwise-uniform disturbances

An investigation of the dominant transient growth mechanisms in plane Poiseuille flow subjected to a destabilizing cross-stream temperature gradient is presented. It was pointed out by the same authors in DFD meeting 2009 that only the streamwise-uniform and nearly-streamwise-uniform disturbances are highly influenced by the Rayleigh number \textbf{\textit{Ra}} and Prandtl number \textbf{\textit{Pr}}. Here, it is demonstrated that the \textit{short-time} behavior is governed by the classical inviscid lift-up mechanism and the optimal input for the largest \textit{long-time} response is given by the adjoint of the dominant eigenmode with respect to the energy scalar product: the Rayleigh-B\'{e}nard eigenmode without its streamwise velocity component. These short and long-time responses are then shown to depict, up to leading order, the optimal transient growth \textbf{\textit{G(t)}}. It is thereby brought out that, at moderately large \textbf{\textit{Ra}} (or small \textbf{\textit{Pr}} at a fixed \textbf{\textit{Ra}}), the dominant adjoint mode is a good approximation to the optimal initial condition for all time. The results remain qualitatively similar over a general class of norms that can be considered as growth functions. For instance, the dominant adjoint eigenmode still approximates the maximum optimal response.