On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow

TLDR

The study examines non‑linear, two‑dimensional Navier‑Stokes solutions with small disturbance amplification rates, noting that in subcritical cases the presence of an unstable equilibrium implies unstable disturbances. The authors formulate two problem types—tracking growth of unstable infinitesimal disturbances (supercritical) or decay of finite disturbances from unstable equilibria (subcritical)—and plan numerical calculations to decide which behavior occurs in plane Poiseuille flow. The authors suggest that their method, and its generalization in Part 2, remains valid across a broad range of Reynolds numbers and wave‑numbers both inside and outside the neutral‑stability curve.

Abstract

This paper considers the nature of a non-linear, two-dimensional solution of the Navier-Stokes equations when the rate of amplification of the disturbance, at a given wave-number and Reynolds number, is sufficiently small. Two types of problem arise: (i) to follow the growth of an unstable, infinitesimal disturbance (supercritical problem), possibly to a state of stable equilibrium; (ii) for values of the wave-number and Reynolds number for which no unstable infinitesimal disturbance exists, to follow the decay of a finite disturbance from a possible state of unstable equilibrium down to zero amplitude (subcritical problem). In case (ii) the existence of a state of unstable equilibrium implies the existence of unstable disturbances. Numerical calculations, which are not yet completed, are required to determine which of the two possible behaviours arises in plane Poiseuille flow, in a given range of wave-number and Reynolds number. It is suggested that the method of this paper (and of the generalization described by Part 2 by J. Watson) is valid for a wide range of Reynolds numbers and wave-numbers inside and outside the curve of neutral stability.