## Laminar-turbulent transition in pipe flow: wall effects and critical Reynolds number

### Abstract

This article describes possible causes of natural laminar–turbulent transition in circular pipe flow. Our starting points are the observations that under natural disturbance conditions, transition appears to take place only in the developing entrance region, as observed in Reynolds' color-band experiments, and that the critical Reynolds number $R_c$ has a minimum value of about 2000 when using a sharp-edged uniform radius pipe, as observed in our earlier color-band experiments. The entrance region is defined as the region from the pipe inlet to the point where the inlet flow fully develops into Hagen-Poiseuille flow for a sharp-edged entrance pipe. In the case of a bell-mouth entrance pipe, the entrance region includes the bell-mouth entrance region. We derive for the entrance region a new ratio of the increase in kinetic energy flux (${\varDelta \text{KE flux}}$) to a wall effect, where the wall effect is the radial wall power (${\text{R-Wall-Power}}$) exerted on the wall by the radial component of the viscous term in the Navier-Stokes equations. In dimensionless form, ${\varDelta \text{KE flux}}$ is a constant, although ${\text{R-Wall-Power}}$ decreases as the Reynolds number Re increases. Our previous calculations for the case of a sharp-edged entrance pipe indicate that ${\varDelta \text{KE flux}} \approx$ total ${\text{R-Wall-Power}} ~({\text{T-R-Wall-Power}})$ at Re $\approx$ 2000. Accordingly, our hypothesis is that $R_c$ can be derived from the relation between ${\varDelta \text{KE flux}}$ and ${\text{T-R-Wall-Power}}$. We discuss, moreover, whether or not this criterion can apply to different entrance geometries such as the bell-mouth entrances considered by Reynolds.

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### Key words

hydrodynamic stability, mesh refinement, thermodynamics

### AMS subject classifications

76E05, 65M50, 80A05

### Links to the cited ETNA articles

 [25] Vol. 30 (2008), pp. 10-25 K. Shimomukai and H. Kanda: Numerical study of normal pressure distribution in entrance pipe flow