[Paper Review] On The Critical Reynolds Number For Transition From Laminar To Turbulent Flow

Khanh Tuoc Trịnh|arXiv (Cornell University)|Jul 6, 2010

Fluid Dynamics and Turbulent FlowsEngineering71 references22 citations

TL;DR

This paper proposes a novel method to determine the critical Reynolds number for transition from laminar to turbulent flow in pipes by analyzing the normalized thickness of the viscous sublayer. The study identifies the critical Reynolds number as the asymptotic value of this normalized thickness, with physical thickness decreasing exponentially post-transition, offering a direct, visualization-based criterion for transition onset in both Newtonian and non-Newtonian flows.

ABSTRACT

In this visualisation, the transition from laminar to turbulent flow is characterised by the intermittent ejection of wall fluid into the outer stream. The normalised thickness of the viscous flow layer reaches an asymptotic value but the physical thickness drops exponentially after transition. The critical transition pipe Reynolds number can be obtained simply by equating it with the asymptotic value of the normalised thickness of viscous flow layer. Key words: Transition, critical stability Reynolds number, critical transition Reynolds number, non-Newtonian pipe flow

Motivation & Objective

• To establish a physically grounded criterion for the onset of turbulence in pipe flows.
• To resolve ambiguity in defining the critical Reynolds number due to intermittent transition behavior.
• To link the transition process to measurable flow layer characteristics, particularly the viscous sublayer thickness.
• To extend the method to non-Newtonian fluid flows, broadening applicability.
• To provide a quantitative, visualization-based approach for identifying the transition threshold.

Proposed method

• The study uses direct flow visualization to observe the intermittent ejection of wall fluid into the outer stream during transition.
• The normalized thickness of the viscous flow layer is tracked as a function of Reynolds number.
• The critical Reynolds number is identified as the asymptotic value of the normalized viscous layer thickness at transition.
• The physical thickness of the viscous layer is modeled as decaying exponentially after transition.
• The method relies on empirical observation and scaling analysis rather than linear stability theory.
• The approach is applied to both Newtonian and non-Newtonian pipe flows to validate generality.

Experimental results

Research questions

• RQ1 What physical parameter can reliably mark the onset of turbulent transition in pipe flows?
• RQ2 How does the viscous sublayer thickness evolve during the transition from laminar to turbulent flow?
• RQ3 Can the critical Reynolds number be defined as the asymptotic value of the normalized viscous layer thickness?
• RQ4 How does the proposed criterion compare with traditional stability-based definitions?
• RQ5 To what extent is the method applicable to non-Newtonian fluid flows?

Key findings

• The normalized thickness of the viscous flow layer reaches a stable asymptotic value at the transition point, defining the critical Reynolds number.
• The physical thickness of the viscous layer decreases exponentially after the transition, indicating a sharp change in flow structure.
• The critical Reynolds number is directly obtained by equating it to the asymptotic value of the normalized viscous layer thickness.
• The method provides a consistent and observable criterion for transition, independent of linear stability analysis.
• The approach is applicable to non-Newtonian pipe flows, suggesting broader utility beyond Newtonian fluids.
• The visualization reveals intermittent ejection of wall fluid into the outer stream as a key signature of transition.