Probability Density Distribution of Velocity Differences at Very High Reynolds Numbers

Abstract

The probability density distributions (PDD) of the velocity differences $\Delta u\left(r\right)=u\left(x\right)-u(x+r)\mathrm{}$ for separations $r$ within the inertial range are studied in a large wind tunnel and in the atmospheric surface layer at $R_{\lambda }\approx \mathrm{}(2.0-12.7)\mathrm{}\times {10}^3$. For large velocity differences, the PDD behave as $P\left(\Delta u\right)\propto \exp \mathrm{}[-b(r\left)\right|\frac{\Delta u}{{\sigma }_{\Delta u}}\mathrm{}\left|\right]\mathrm{}$. The logarithmic decrement $b\left(r\right)\mathrm{}$ is found to scale as $b\left(r\right)=\alpha r^{\beta }$. The value of $\alpha$ is found to be 0.51 ± 0.02 for the negative ($\Delta u<\mathrm{}0\mathrm{}$) tails of the PDD and 0.58 ± 0.02 for the positive ($\Delta u>\mathrm{}0\mathrm{}$) tails. In both cases, $\beta =0.17\mathrm{}\pm 0.01$.