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In this paper, the decay and growth of localized wave packet (LWP) intwo-dimensional plane-Poiseuille flow are studied numerically andtheoretically. When the Reynolds number ($Re$) is less than a critical value$Re_c$, the disturbance kinetic energy $E_k$ of LWP decreases monotonicallywith time and experiences three decay periods, i.e. the initial and the finalsteep descent periods, and the middle plateau period. Higher initial $E_k$ of adecaying LWP corresponds to longer lifetime. According to the simulations, thelifetime scales as $(Re_c-Re)^{-1/2}$, indicating a divergence of lifetime as$Re$ approaches $Re_c$, a phenomenon known as "critical slowing-down". Byproposing a pattern preservation approximation, i.e. the integral kinematicproperties (e.g. the disturbance enstrophy) of an evolving LWP are independentof $Re$ and single valued functions of $E_k$, the disturbance kinetic energyequation can be transformed into the classical differential equation forsaddle-node bifurcation, by which the lifetimes of decaying LWPs can bederived, supporting the $-1/2$ scaling law. Furthermore, by applying thepattern preservation approximation and the integral kinematic propertiesobtained as $Re<Re_c$, the Reynolds number and the corresponding $E_k$ of thewhole lower branch, the turning point, and the upper-branch LWPs with$E_k<0.15$ are predicted successfully with the disturbance kinetic energyequation, indicating that the pattern preservation is an intrinsic feature ofthis localized transitional structure.

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