Math 2.0

Fluid mechanics Definition

Turbulence

Builds on Navier–Stokes equationsReynolds numberVorticityViscosity

Definition

Consider the flow of a viscous fluid governed by the Navier–Stokes equations. The flow is turbulent when its velocity field has the following properties.

  1. It is irregular in space and time, and sensitive to small changes in the initial and boundary data, so that only statistical quantities are reproducible.
  2. It is rotational, with strong three-dimensional fluctuations of the vorticity.
  3. Its fluctuations cover a wide continuous range of length and time scales, which widens as the Reynolds number grows.
  4. It mixes momentum, heat and dissolved substances much faster than molecular diffusion does.
  5. It dissipates kinetic energy into heat through viscosity at the smallest scales, so it needs a continuous supply of energy to persist.

Remarks

This is a descriptive definition; there is no accepted definition of turbulence as a precise mathematical property of solutions of the Navier–Stokes equations. Example: the flow in a pipe at a Reynolds number of 10510^5, based on mean speed and diameter, has all five properties. Non-example: the Kármán vortex street behind a cylinder at a Reynolds number near 100100 is unsteady but periodic and regular, so it fails clause 1 and is laminar. Random but non-rotational motions, such as small-amplitude random water waves, also fail clause 2 and are not called turbulence. Two-dimensional turbulence, studied as a model for large-scale atmospheric flows, keeps most of these properties but not the three-dimensional vortex stretching of clause 2.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Navier–Stokes equations, concept, no evidence levelNavier–Stokes equatio…conceptReynolds number, concept, no evidence levelReynolds numberconceptViscosity, concept, no evidence levelViscosityconceptVorticity, concept, no evidence levelVorticityconceptTurbulence, definition, no evidence levelTurbulencedefinition
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  1. Navier–Stokes equations concept
  2. Reynolds number concept
  3. Viscosity concept
  4. Vorticity concept
  5. Turbulence definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Turbulence
About this page
TagD6XW03W3TN
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Kinddefinition
WritingW0 stub
MSC76F02
DefinesTurbulence
UsesNavier–Stokes equations, Reynolds number, Vorticity, Viscosity
Changed3 November 2026

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