Law of the wall

The law of the wall describes how mean fluid velocity varies with distance from a solid boundary in a turbulent flow. In the inner region, velocity scaled by friction velocity depends logarithmically on distance scaled by viscous length.

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Logarithmic velocity profile: A velocity distribution in which velocity varies logarithmically with distance from a reference boundary. The law of the wall’s central prediction is a logarithmic mean-velocity profile.
Turbulent boundary layer: A near-surface flow region in which turbulence strongly influences momentum transport and velocity. The law describes mean velocity within this type of boundary layer.
Skin friction coefficient: A dimensionless measure of the wall-shear force relative to a characteristic flow’s dynamic pressure. The law helps relate measured outer-flow conditions to wall friction.
Ludwig Prandtl: A German fluid dynamicist who established foundational theories of boundary layers and turbulent flow. His mixing-length theory provided an early route to the logarithmic wall profile.
Rough-wall turbulence: Turbulent flow near a rough surface, where surface texture changes momentum transfer and mean velocity. Roughness can shift or modify the smooth-wall logarithmic profile.
Friction velocity: A velocity scale defined as the square root of wall shear stress divided by fluid density. It normalizes velocity and sets the scale of turbulent shear near the wall.
Wall shear stress: The tangential force per unit area exerted by a fluid on a boundary. It determines friction velocity, the velocity scale used by the law.
Pipe flow: Fluid motion through a conduit, with wall friction shaping its velocity and pressure distributions. The law’s near-wall profile informs turbulent velocity and friction estimates in pipes.
Theodore von Kármán: A Hungarian-American engineer whose work shaped modern theories of turbulence and fluid mechanics. The logarithmic law is commonly written with the von Kármán constant bearing his name.
Viscous sublayer: The near-wall region of a turbulent flow where viscous stresses dominate turbulent stresses. Its approximately linear velocity profile contrasts with the logarithmic region.