Three-dimensional disturbances; Considering starting profiles and optimal profiles in Couette and Poiseuille flow

The influence of the velocity profile on transient (algebraic) growth of three-dimensional disturbances in channel flow is investigated. Only streamwise-independent perturbations are considered and the effect of a single forcing Orr-Sommerfeld mode is studied. This restriction is motivated by previous investigations, which show that the largest possible (global optimal) energy growth of a disturbance is generally obtained for streamwise-independent perturbations. Analytic solutions for the energy growth on starting profiles for plane Couette and plane Poiseuille flow are deduced and the findings reveal that the most favorable profiles for transient growth are indeed the fully developed ones. An isoperimetric method to compute optimal profiles is then presented with the purpose to increase the energy growth compared to the fully developed profiles. These profiles are dependent on the spanwise wave number, as well as the perturbation growth time. A numerical investigation show that significant energy magnification can take place on such a profile, even at short growth times of the perturbation. It is further established that these profiles are stable only for low Reynolds number. (C) 1996 American Institute of Physics.