On the mechanism of optimal disturbances: The role of a pair of nearly parallel modes

Abstract

In the present work the role of a pair of nearly parallel least stable modes (having opposite phases and almost identical amplitude distributions), as a key element of optimal transient growth in shear flows, is explored. The general character of this mechanism is demonstrated by four examples. The first two examples are the temporal and spatial growth of optimal disturbances in circular pipe flow. The time and distance, at which the maximum energy amplification of an initial disturbance is achieved, are well predicted analytically by considering only the pair of least stable modes. Furthermore, the dependence of the maximum energy amplification on the Reynolds number matches previous numerical results based on the analysis of many modes. In the temporal case the predicted amplification factor agrees well with these numerical results. The other two examples are concerned with a two-dimensional potential shear layer over a compliant surface, and with a two-dimensional wall-jet. In these examples, a similar growth mechanism takes place near a point where two kinds of two-dimensional modes bifurcate. In the potential shear layer, the maximum optimal growth achieved by two nearly parallel modes is solved analytically for temporal disturbances, whereas in the wall-jet case the distance, at which the maximum amplification is achieved, is well predicted for spatial disturbances. Finally, it appears that the transient growth mechanism based on the interference between two nearly parallel modes is a general case which includes the direct resonance mechanism as a limit for short times.