Self-similar profile of probability density functions in zero-pressure gradient turbulent boundary layers

The probability density function (pdf) of a streamwise velocity component is studied in zero-pressure gradient boundary layers. From analyzing the data up to Rθ≃13,000, it is found that pdfs have self-similar profiles ranging from y+≃180 to 0.049Δ+, where Δ is Rotta–Clauser boundary layer thickness. Pdf profiles asymptote to the universal shape very close to the Gaussian, but are positively skewed at the core region, indicating smaller values in the tail parts. Based on this experimental fact, the mean velocity profile is reconsidered from the standpoint of pdf equation. The log-law profile is expected as the mean velocity distribution. The Kármán constant is evaluated to be 0.38, and the log-region starts at ys+ = 180 and ends at yr+ = 0.021Δ++96.5 for Rθ > 4000. The end point locates in 0.15δ+ ≤ yr+ ≤ 0.2δ+. The relation to the new scaling law derived from Lie-group theory is also discussed.