Math 2.0

Fluid mechanics Definition

Laminar–turbulent transition

Builds on TurbulenceReynolds numberHydrodynamic stabilityTollmien–Schlichting waveNavier–Stokes equations

Definition

Consider a family of flows governed by the Navier–Stokes equations that depends on a control parameter, usually the Reynolds number Re\mathrm{Re}. A flow is laminar when its velocity field is smooth and regular, steady or periodic, with fluid layers moving in an orderly way. Laminar–turbulent transition is the change of the flow from a laminar state to a turbulent state as Re\mathrm{Re} increases. In a spatially developing flow, such as a boundary layer, the change can also occur along the direction of the flow.

Remarks

The definition is descriptive, since there is no single accepted mathematical definition of turbulence. Two main scenarios are distinguished.

In supercritical, or natural, transition, the laminar flow loses linear stability at a critical Reynolds number. Small disturbances such as Tollmien–Schlichting waves then grow, secondary instabilities follow, and the flow breaks down into turbulence. In subcritical, or bypass, transition, turbulence appears at Reynolds numbers where the laminar flow is linearly stable, triggered by disturbances of finite size.

Example: in Reynolds' pipe experiments, transition occurred near Re≈2000\mathrm{Re} \approx 2000 based on mean speed and diameter. Yet Hagen–Poiseuille flow is linearly stable at all Reynolds numbers for which it has been computed, so this is subcritical transition. Non-example: the onset of steady Taylor vortices between rotating cylinders is a laminar-to-laminar bifurcation, not a transition to turbulence, since the new flow is still steady and regular.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Hydrodynamic stability, concept, no evidence levelHydrodynamic stabilityconceptNavier–Stokes equations, concept, no evidence levelNavier–Stokes equatio…conceptReynolds number, concept, no evidence levelReynolds numberconceptTollmien–Schlichting wave, concept, no evidence levelTollmien–Schlichting …conceptTurbulence, concept, no evidence levelTurbulenceconceptLaminar–turbulent transition, definition, no evidence levelLaminar–turbulent tra…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Hydrodynamic stability concept
  2. Navier–Stokes equations concept
  3. Reynolds number concept
  4. Tollmien–Schlichting wave concept
  5. Turbulence concept
  6. Laminar–turbulent transition definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Laminar–turbulent_transition
About this page
TagP229CC7D05
URIm2://P229CC7D05
Kinddefinition
WritingW0 stub
MSC76F06
DefinesLaminar–turbulent transition
UsesTurbulence, Reynolds number, Hydrodynamic stability, Tollmien–Schlichting wave, Navier–Stokes equations
Changed3 November 2026

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