Math 2.0

Fluid mechanics Definition

Tollmien–Schlichting wave

Builds on Orr–Sommerfeld equationBoundary layerBlasius boundary layerReynolds number

Definition

Consider a parallel or nearly parallel shear flow U(y)U(y) of a viscous incompressible fluid along a wall y=0y = 0, at Reynolds number Re\mathrm{Re}. Examples are channel flows and boundary layers. Assume that the profile UU has no inflection point inside the flow. A Tollmien–Schlichting wave is a two-dimensional disturbance with stream function φ(y)eiα(x−ct)\varphi(y) e^{i \alpha (x - c t)} of the following kind. The pair (φ,c)(\varphi, c) is the eigenmode of the Orr–Sommerfeld equation that becomes unstable, Im⁡c>0\operatorname{Im} c > 0, as Re\mathrm{Re} passes its critical value.

Remarks

Such a profile is stable in inviscid theory by Rayleigh's inflection point criterion, so the instability of the Tollmien–Schlichting mode is a viscous effect. The critical layer is where U(y)U(y) equals the real part of cc. Viscosity near the wall and near the critical layer shifts the phase of the Reynolds stress, so that the wave extracts energy from the mean shear. The waves are slow, with phase speed a fraction of the free-stream speed, and long compared with the boundary layer thickness.

Example: for the Blasius boundary layer on a flat plate, the parallel-flow approximation gives a critical Reynolds number of about 520520 based on the displacement thickness. For plane Poiseuille flow the critical mode, at Re≈5772\mathrm{Re} \approx 5772, is also a Tollmien–Schlichting wave. Non-example: the growing waves of a free mixing layer U(y)=tanh⁡yU(y) = \tanh y are not Tollmien–Schlichting waves. That profile has an inflection point, and its instability persists in inviscid theory as a Kelvin–Helmholtz type instability.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Boundary layer, concept, no evidence levelBoundary layerconceptDisplacement thickness, concept, no evidence levelDisplacement thicknessconceptOrr–Sommerfeld equation, concept, no evidence levelOrr–Sommerfeld equati…conceptPrandtl boundary layer equations, concept, no evidence levelPrandtl boundary laye…conceptReynolds number, concept, no evidence levelReynolds numberconceptSimilarity solution, concept, no evidence levelSimilarity solutionconceptBlasius boundary layer, example, no evidence levelBlasius boundary layerexampleTollmien–Schlichting wave, definition, no evidence levelTollmien–Schlichting …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. Boundary layer concept
  2. Displacement thickness concept
  3. Orr–Sommerfeld equation concept
  4. Prandtl boundary layer equations concept
  5. Reynolds number concept
  6. Similarity solution concept
  7. Blasius boundary layer example
  8. Tollmien–Schlichting wave definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Tollmien–Schlichting_wave
About this page
TagVED29VB4ZS
URIm2://VED29VB4ZS
Kinddefinition
WritingW0 stub
MSC76E05, 76D10, 76F06
DefinesTollmien–Schlichting wave
UsesOrr–Sommerfeld equation, Boundary layer, Blasius boundary layer, Reynolds number
Changed3 November 2026

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