Pattern preservation during the decay and growth of localized wave packet in two-dimensional channel flow

Abstract

In this paper, the decay and growth of localized wave packet (LWP) in a two-dimensional plane-Poiseuille flow are studied numerically and theoretically. When the Reynolds number (Re) is less than a critical value Rec, the disturbance kinetic energy Ek of LWP decreases monotonically with time and experiences three decay periods, i.e., the initial and the final steep descent periods and the middle plateau period. Higher initial Ek of a decaying LWP corresponds to longer lifetime. According to the simulations, the lifetime scales as (Rec−Re)−1/2, indicating a divergence of lifetime as Re approaches Rec, a phenomenon known as “critical slowing-down.” By proposing a pattern preservation approximation, i.e., the integral kinematic properties (e.g., the disturbance enstrophy) of an evolving LWP are independent of Re and single valued functions of Ek, the disturbance kinetic energy equation can be transformed into the classical differential equation for saddle-node bifurcation, by which the lifetimes of decaying LWPs can be derived, supporting the −1/2 scaling law. Furthermore, by applying the pattern preservation approximation and the integral kinematic properties obtained as Re<Rec, the Reynolds number and the corresponding Ek of the whole lower branch, the turning point, and the upper-branch LWPs with Ek<0.15 are predicted successfully with the disturbance kinetic energy equation, indicating that the pattern preservation is an intrinsic feature of this localized transitional structure.