Law of the wall

Builds on Turbulence No-slip condition Viscosity Reynolds decomposition

Statement

Consider a statistically steady turbulent flow of an incompressible fluid along a smooth flat wall, with the no-slip condition at the wall. Let $\rho$ be the density, $\nu$ the kinematic viscosity, $\tau_w$ the mean shear stress at the wall and $u_\tau = \sqrt{\tau_w / \rho}$ the friction velocity. Let $U(y)$ be the mean velocity parallel to the wall at distance $y$ , in the sense of the Reynolds decomposition, and set $u^+ = U / u_\tau$ and $y^+ = y u_\tau / \nu$ . Let $\delta$ be the thickness of the boundary layer, or the half-width of the channel.

In the viscous sublayer, roughly $y^+ < 5$ , one has $u^+ = y^+$ .
In the overlap region, roughly $y^+ > 30$ and $y$ small compared with $\delta$ , one has $u^+ = \frac{1}{\kappa} \ln y^+ + B,$ with constants $\kappa \approx 0.41$ , the von Kármán constant, and $B \approx 5$ .

Remarks

The linear law follows from the fact that very close to the wall the viscous stress $\rho \nu\, dU/dy$ carries all of $\tau_w$ . The logarithmic law comes from an overlap argument. Near the wall $U$ depends on $y$ only through $y^+$ , and far from it the velocity defect depends only on $y/\delta$ . The two forms can agree on a common range only if $dU/dy = u_\tau / (\kappa y)$ there. The buffer layer between the two regions has no simple formula. On a rough wall the logarithmic law survives but with a different additive constant, and in strong pressure gradients or near separation it fails.

Going deeper

For a student

The law of the wall is the basis of wall functions in turbulence models, which avoid resolving the thin viscous sublayer. Integrated across a pipe, it gives the logarithmic friction law for smooth pipes. The values of $\kappa$ and $B$ are empirical, and their exact values and universality are still debated.

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No-slip condition concept
Reynolds decomposition concept
Turbulence concept
Viscosity concept
Law of the wall statement

Where this leads

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Monin–Obukhov similarity theory

Tag	`7AX1EA410X`
URI	`m2://7AX1EA410X`
Kind	statement, fact
Writing	W0 stub
MSC	76F40, 76F10
Status	This copy of the site has no build data. Run `m2 status 7AX1EA410X`.
About	Turbulence
Uses	Turbulence, No-slip condition, Viscosity, Reynolds decomposition
Same as	Wikidata Q6503510 Wikipedia Law of the wall
Changed	3 November 2026

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Monin–Obukhov similarity theory