Logarithmic velocity profile

logarithmic velocity profile

The variation of the mean wind speed with height in the surface boundary layer derived with the following assumptions: 1) the mean motion is one-dimensional; 2) the Coriolis force can be neglected; 3) the shearing stress and pressure gradient are independent of height; 4) the pressure force can be neglected with respect to the viscous force; and 5) the mixing length l depends only on the fluid and the distance from the boundary, l = kz.

Near aerodynamically smooth surfaces, the result is

that is, the logarithmic velocity profile, where u_* is the friction velocity and ν the kinematic viscosity. k ≅ 0.4 and has been called the Kármán constant or von Kármán's constant. The equation fails for a height z sufficiently close to the surface. For aerodynamically rough flow, molecular viscosity becomes negligible. The profile is then

z₀ is a constant related to the average height ε of the surface irregularities by z₀ = ε/30 and is called the aerodynamic roughness length. Another derivation of the logarithmic profile was obtained by Rossby under the assumption that for fully rough flow the roughness affects the mixing length only in the region where z and z₀ are comparable. Then l = k(z + z₀) and

For statically nonneutral conditions, a stability correction factor can be included (
see equation in definition of aerodynamic roughness length).

Haugen, D. A. 1973. Workshop on Micrometeorology. Amer. Meteor. Soc., 392 pp.

Sutton, O. G. 1953. Micrometeorology. sect. 3.9.

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