LOCALIZED DISTURBANCES IN PARALLEL SHEAR FLOWS

The development of localized disturbances in parallel shear flows is r eviewed. The inviscid case is considered, first for a general velocity profile and then in the special case of plane Couette flow so as to b ring out the key asymptotic results in an explicit form. In this conte xt, the distinctive differences between the wave-packet associated wit h the asymptotic behavior of eigenmodes and the non-dispersive (invisc id) continuous spectrum is highlighted. The largest growth is found fo r three-dimensional disturbances and occurs in the normal vorticity co mponent. It is due to an algebraic instability associated with the lif t-up effect. Comparison is also made between the analytical results an d some numerical calculations. Next the viscous case is treated, where the complete solution to the initial value problem is presented for b ounded flows using eigenfunction expansions. The asymptotic, wave-pack et type behaviour is analyzed using the method of steepest descent and kinematic wave theory. For short times, on the other hand, transient growth can be large, particularly for three-dimensional disturbances. This growth is associated with cancelation of non-orthogonal modes and is the viscous equivalent of the algebraic instability. The maximum t ransient growth possible to obtain from this mechanism is also present ed, the so called optimal growth. Lastly the application of the dynami cs of three dimensional disturbances in modeling of coherent structure s in turbulent flows is discussed.