Analysis based on KL expansion and the Differential Geomer Universal Velocity Scaling in Wall Bounded Shear Flow

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The logarithmic velocity profile is considered in zero-pressure gradient smooth and rough wall turbulent boundary layer. We propose a new definition of the log-law region using the probability profiles of the streamwise velocity fluctuation. The log-law profile is extracted readly from the experimental data and as well as the probability density fuction (PDF) equation. The measure called Kullback Leibler divergence is applied for distinguishing the probability profile. If the logarithmin profile ; UィイD1+ィエD1 = A log y ィイD1+ィエD1 + B is a good representation of the experimental data, our results show that A is independent of the Reynolds number while B depends on it. The ratio of boundary layer thickness δto the upper end of log-law extent, δィイD2LィエD2, is not constant but approaches the value δィイD2LィエD2/δ = 0.2 as the Reynolds number increases. Although this condition is not satisfied in rough wall boundary layers.