Linear Exponential Instability of the Hagen-Poiseuille Flow with Respect to Synchronous Bi-Periodic Disturbances

TL;DR

This paper demonstrates that Hagen-Poiseuille flow exhibits linear exponential instability under bi-periodic disturbances, with the critical Reynolds number linked to the disturbance periods, shedding light on transition to turbulence.

Contribution

It introduces the concept that bi-periodic axial disturbances can induce exponential instability in Hagen-Poiseuille flow, identifying conditions for transition to turbulence.

Findings

Exponential instability occurs only with bi-periodic disturbances.

The threshold Reynolds number depends on the ratio of disturbance periods.

Minimum threshold aligns with observed transition conditions.

Abstract

For Gagen-Poiseuille flow, we show that exponential instability (to extremely small, axially symmetric disturbances represented by Galerkin's approximation) is possible only if there exists bi-periodic variability of the disturbances along the pipe axis when the threshold Reynolds number depends on the ratio of two longitudinal periods. Absolute minimum (for) is obtained that corresponds to the observed conditions of transition from the laminar resistance law to the turbulent one and Tollmien-Schlichting waves exciting in the boundary layer.