Vorticity evolution in a rigid pipe of circular cross-section

TL;DR

This paper models the evolution of vorticity in long circular pipes using Navier-Stokes equations in cylindrical coordinates, revealing insights into turbulence development and laminar-turbulent transition.

Contribution

It provides a detailed analytical framework for vorticity dynamics in pipe flows, accounting for complex interactions and initial conditions affecting turbulence onset.

Findings

Vorticity solutions define complex flow fields with many degrees of freedom.

Increased Reynolds number leads to vorticity proliferation and turbulence.

Flow remains globally regular without bifurcation, with transition driven by non-linearity.

Abstract

In this paper, we show that the spatio-temporal evolution of incompressible flows in a long circular pipe can be described by vorticity dynamics. The principal techniques to obtain solutions are similar to those used for flows in the whole space. As the consideration of the Navier-Stokes equations is given in a cylindrical co-ordinates system, two aspects of complication arise. One is the interaction of the velocity components in the radial and azimuthal directions, due to the fictitious centrifugal force in the equations of motion. The rate of the vorticity production at the pipe wall depends on the initial data at entry, and hence is unknown a priori; it must be determined as part of the solution. The vorticity solution obtained defines an intricate flow-field of multitudinous degrees of freedom. As the Reynolds number increases, the analytical solution predicts vorticity-scale proliferations in succession. For sufficiently large initial data, pipe flows are of a turbulent nature. The solution of the governing equations is globally regular and does not bifurcate in space or in time. It is asserted that laminar-turbulent transition is a dynamic process inbred in the non-linearity. The presence of exogenous disturbances, due to imperfect test environments or purpose-made artificial forcing, distorts the course of the intrinsic transition. The flow structures observed by Reynolds (1883) and others can be synthesised and elucidated in light of the current theory.