Velocity probability distribution scaling in wall-bounded flows at high Reynolds numbers

Probability density functions (PDFs) give well-rounded statistical descriptions of stochastic quantities and therefore are fundamental to turbulence. Wall-bounded turbulent flows are of particular interest given their prevalence in a vast array of applications, but for these flows the scaling of velocity probability distribution is still far from being well founded. By exploiting the self-similarity in wall-bounded turbulent flows and modeling velocity fluctuations as results of self-repeating processes, we present a theoretical argument, supported by empirical evidence, for a universal velocity PDF scaling in high-Reynolds-number wall turbulence.

Figure 1

(a) Streamwise velocity PDF in the logarithmic layer, i.e., $3\sqrt{{\text{Re}}_{\tau }}<z^{+},z/\delta <0.2$, where ${\text{Re}}_{\tau }$ is the friction Reynolds number, $z$ is the wall-normal coordinate, and $\delta$ is the boundary layer height. Details of this data set are presented later. Data at different Reynolds numbers are color coded. For data at a given Reynolds number, darker colors are used for velocities closer to the wall. (b) Same as (a) but plotted as a function of $u/u_{\mathrm{rms}}$, where $u_{\mathrm{rms}}$ is the root mean square of the streamwise velocity fluctuation.

Figure 2

(a) Scaled streamwise velocity PDF [the scale factor $N_z\sim ln\langle exp(\pm q_{0}u)\rangle$, where $u<0$ corresponds to $q_0=-1.5$ and $u>0$ corresponds to $q_0=1.5]$. (b) Logarithm of the unscaled streamwise velocity PDF at wall-normal distances from $z^{+}\approx 60$ to $z/\delta \approx 0.4$ in a DNS channel flow at ${\text{Re}}_{\tau }=1000$. Darker colors are for data closer to the wall. The wall-normal distances are logarithmically spaced from $z^{+}\approx 60$, and $z/\delta \approx 0.4$. (c) Logarithm of scaled velocity PDF $P/\langle u^2\rangle$.

Figure 3

(a) Same as Fig. 2 but using a different $q_0=1$. (b) Same as Fig. 2 but using a different $q_0=2$.

Figure 4

(a) Spanwise velocity PDF within $3\sqrt{{\text{Re}}_{\tau }}<z^{+},z<0.2\delta$. Data at ${\text{Re}}_{\tau }\approx 10000$ are shown. Darker colors are used for data closer to the wall. (b) Scaled spanwise velocity PDF [the scale factor $N_z\sim ln\langle exp\left(v\right)\rangle ]$. (c) Unscaled spanwise velocity PDF in a channel at ${\text{Re}}_{\tau }\approx 1000$. The legends are the same as in Fig. 2. (d) Same as (c) but for the scaled velocity PDF.

Figure 5

(a) Measured $\tau \left(q\right)$ as a function of $q$. (b) Plot of $ln\left(P\right)/cln(\delta /z)$ and $c^{\prime }-f_1$. The two constants are $c=1.26$ and $c^{\prime }=0.45$.