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

TL;DR

This study investigates the decay and growth of localized wave packets in two-dimensional channel flow, revealing a pattern preservation property and a critical scaling law for their lifetime near a critical Reynolds number.

Contribution

It introduces a pattern preservation approximation and derives a scaling law for wave packet lifetime, advancing understanding of localized structures in transitional flows.

Findings

Lifetime scales as (Re_c - Re)^(-1/2) near critical Re.

Pattern preservation is an intrinsic feature of localized wave packets.

Reynolds number and energy of wave packets can be predicted using the derived energy equation.

Abstract

In this paper, the decay and growth of localized wave packet (LWP) in two-dimensional plane-Poiseuille flow are studied numerically and theoretically. When the Reynolds number ($Re$) is less than a critical value $Re_c$, the disturbance kinetic energy $E_k$ 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 $E_k$ of a decaying LWP corresponds to longer lifetime. According to the simulations, the lifetime scales as $(Re_c-Re)^{-1/2}$, indicating a divergence of lifetime as $Re$ approaches $Re_c$, 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 $E_k$, 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 $Re<Re_c$, the Reynolds number and the corresponding $E_k$ of the whole lower branch, the turning point, and the upper-branch LWPs with $E_k<0.15$ are predicted successfully with the disturbance kinetic energy equation, indicating that the pattern preservation is an intrinsic feature of this localized transitional structure.