The intermittent vibration observed by Reynolds in a cylindrical tube

TL;DR

This paper investigates the intermittent vibration in a cylindrical tube filled with fluid, using a variable parameter method to analyze energy density distribution and its relation to Reynolds' historical observations.

Contribution

It introduces a novel application of the variable parameter method to analyze energy density and intermittent vibrations in fluid-filled tubes, linking spatial sampling to observed phenomena.

Findings

Intermittent vibration depends on spatial sampling interval.

Energy density is proportional to an integer N, tending to infinity as N increases.

Proper sampling intervals are crucial to observe Reynolds' intermittent vibrations.

Abstract

In a cylindrical tube filled with incompressible fluid, the variable parameter method is applied to solve the vorticity motion equation, and the obtained results are substituted into the Poisson equation satisfied by the energy density. The static distribution of energy density in the tube and the energy density of the flowing fluid after a certain spatial sampling interval are obtained. Further numerical calculations are conducted on the energy density of the flowing fluid, the results show that the intermittent vibration observed by Reynolds 140 years ago can only be obtained with the appropriate spatial sampling interval. The energy density in the tube is proportional to to , where N is an integer It tends to infinity with increasing N.