In fluid dynamics, the law of the wall (also known as the logarithmic law of the wall) states that the average velocity of a turbulent flow at a certain point is proportional to the logarithm of the distance from that point to the "wall", or the boundary of the fluid region. This law of the wall was first published in 1930 by Hungarian-American mathematician, aerospace engineer, and physicist Theodore von Kármán. It is only technically applicable to parts of the flow that are close to the wall (<20% of the height of the flow), though it is a good approximation for the entire velocity profile of natural streams. General logarithmic formulation The logarithmic law of the wall is a self similar solution for the mean velocity parallel to the wall, and is valid for flows at high Reynolds numbers — in an overlap region with approximately constant shear stress and far enough from the wall for (direct) viscous effects to be negligible: with and where {| border="0" |- ||| is the wall coordinate: the distance y to the wall, made dimensionless with the friction velocity uτ and kinematic viscosity ν, |- ||| is the dimensionless velocity: the velocity u parallel to the wall as a function of y (distance from the wall), divided by the friction velocity uτ, |- ||| is the wall shear stress, |- ||| is the fluid density, |- ||| is called the friction velocity or shear velocity, |- ||| is the Von Kármán constant, |- ||| is a constant, and |- ||| is the natural logarithm. |} From experiments, the von Kármán constant is found to be and for a smooth wall. With dimensions, the logarithmic law of the wall can be written as: where y0 is the distance from the boundary at which the idealized velocity given by the law of the wall goes to zero. This is necessarily nonzero because the turbulent velocity profile defined by the law of the wall does not apply to the laminar sublayer. The distance from the wall at which it reaches zero is determined by comparing the thickness of the laminar sublayer with the roughness of the surface over which it is flowing. For a near-wall laminar sublayer of thickness and a characteristic roughness length-scale , {| border="0" |- | || : hydraulically smooth flow, |- | || : transitional flow, |- | || : hydraulically rough flow.