Planar limits of three-dimensional incompressible flows with helical symmetry

TL;DR

This paper investigates how three-dimensional incompressible flows with helical symmetry in a circular pipe simplify to planar flows as the helical step increases, linking three-dimensional and two-dimensional flow models.

Contribution

It establishes the limiting behavior of helical flows, demonstrating convergence to planar flow governed by two-and-half Navier-Stokes and Euler equations as the step approaches infinity.

Findings

Helical flows tend to planar flows as the step size increases.

The limiting flows are described by two-and-half Navier-Stokes and Euler equations.

The results connect three-dimensional helical symmetry with two-dimensional flow models.

Abstract

Helical symmetry is invariance under a one-dimensional group of rigid motions generated by a simultaneous rotation around a fixed axis and translation along the same axis. The key parameter in helical symmetry is the step or pitch, the magnitude of the translation after rotating one full turn around the symmetry axis. In this article we study the limits of three-dimensional helical viscous and inviscid incompressible flows in an infinite circular pipe, with respectively no-slip and no-penetration boundary conditions, as the step approaches infinity. We show that, as the step becomes large, the three-dimensional helical flow approaches a planar flow, which is governed by the so-called two-and-half Navier-Stokes and Euler equations, respectively.