Three-dimensional coherent structures in a curved pipe flow

TL;DR

This paper extends Dean's approximation to study three-dimensional travelling waves in curved pipe flow, identifying two bifurcation types and exploring their structures and connections to straight pipe solutions.

Contribution

It introduces a new analysis of three-dimensional travelling waves in curved pipes, revealing bifurcation types and their relation to straight pipe solutions, expanding understanding of flow structures.

Findings

Two bifurcation types of solutions identified

Vortex-wave interaction solutions computed at high Dean numbers

Some solution branches remain disconnected from classical Dean vortex solutions

Abstract

Dean's approximation for curved pipe flow, valid under loose coiling and high Reynolds numbers, is extended to study three-dimensional travelling waves. Two distinct types of solutions bifurcate from the Dean's classic two-vortex solution. The first type arises through a supercritical bifurcation from inviscid linear instability, and the corresponding self-consistent asymptotic structure aligns with the vortex-wave interaction theory. The second type emerges from a subcritical bifurcation by curvature-induced instabilities and satisfies the boundary region equations. Despite the subcritical nature of the second type of solutions, it is not possible to connect their solution branches to the zero-curvature limit of the pipe. However, by continuing from known self-sustained exact coherent structures in the straight pipe flow problem, another family of three-dimensional travelling waves can be shown to exist across all Dean numbers. The self-sustained solutions also possess the two high-Reynolds-number limits. While the vortex-wave interaction type of solutions can be computed at large Dean numbers, their branch remains unconnected to the Dean vortex solution branch.