High-order moments of streamwise fluctuations in a turbulent channel flow with spanwise rotation

It is well known that the spanwise rotation in turbulent channel flow alters the mean velocity distribution to a linear law. In the present work, we have studied the higher-order moments of the streamwise fluctuations in a turbulent channel flow with spanwise rotation. Our results show that in a significant part of the channel the $2p$-order moments, raised by the power $1/p$ with $p=1,2,...,6$, also follow linear behavior according to ${\langle {\left(u^{\prime +}\right)}^{2p}\rangle }^{1/p}=a_p(y/h)+b_p$. Here, $u^{\prime +}$ is the streamwise velocity fluctuation normalized by the global friction velocity, $h$ is the channel half width, and $b_p$ and $a_p$ are the intercept and the slope, respectively, which vary with Reynolds and rotation numbers. The linear regions can be extended by introducing a self-similar scaling, that is, $2p$-order moments as a function of $2q$-order moments. The slopes in the self-similar scaling $a_p/a_1$ do not reveal sub-Gaussian behavior as in nonrotating wall-bounded flows, but rather Gaussian or super-Gaussian behaviors.

Figure 1

Profiles of $\langle {\left(u^{\prime +}\right)}^2\rangle$. (a) results at $R{e}_{\tau }=180$; (b) results at ${\text{Re}}_b=10000$; (c) results at ${\text{Re}}_b=20000$, and (d) results at ${\text{Re}}_b=31600$. The data are from Ref. [42] (a) and Ref. [46] (b)–(d). The thin solid lines show linear profiles with different slopes.

Figure 2

Slopes $a_1$ for the linear fitting of $\langle {\left(u^{\prime +}\right)}^2\rangle$ as a function of (a) $\text{Ro}$ and (b) ${\text{Ro}}_{\tau }$. “X180” refers to the data from Ref. [42] at ${\text{Re}}_{\tau }=180$, while “B5k,” “B10k,” “B20k,” and “B31k” refer to the data from Ref. [46] at ${\text{Re}}_b=5000,10000,20000$, and 31 600, respectively.

Figure 3

(a) Profiles of ${\langle {\left(u^{\prime +}\right)}^{2p}\rangle }^{1/p}$ and their linear fittings ${\langle {\left(u^{\prime +}\right)}^{2p}\rangle }^{1/p}=a_p(y/h)+b_p$ for $p=1$ to 6. $U^{+}$ is also plotted to show the linear region for mean velocity (black solid line). (b) Relative error of the linear fitting $1-[a_p(y/h)+b_p]/{\langle {\left(u^{\prime +}\right)}^{2p}\rangle }^{1/p}$ for $p=1$ to 6. The data is from Ref. [42] at ${\text{Ro}}_{\tau }=22$ and fitting coefficients are listed in Table 1. Inset in (b) shows the zoomed-in behavior for $p=1$ to 4.

Figure 4

(a) Profiles of ${\langle {\left(u^{\prime +}\right)}^{2p}\rangle }^{1/p}$ and their linear fittings ${\langle {\left(u^{\prime +}\right)}^{2p}\rangle }^{1/p}=a_p(y/h)+b_p$ for $p=1$ to 6. $U^{+}$ is also plotted to show the linear region for mean velocity (black solid line). (b) Relative error of the linear fitting $1-[a_p(y/h)+b_p]/{\langle {\left(u^{\prime +}\right)}^{2p}\rangle }^{1/p}$ for $p=1$ to 6. The data is from Ref. [46] at ${\text{Re}}_b=31600$ and $\text{Ro}=1.2$. Inset in (b) shows the zoomed-in behavior for $p=1$ to 4.

Figure 5

(a) Plot of ${\langle {\left(u^{\prime +}\right)}^{2p}\rangle }^{1/p}$ against $\langle {\left(u^{\prime +}\right)}^2\rangle$ for $p=1$ to 6. The solid lines indicate the slopes $a_p/a_1$. (b) Plot of ${\langle {\left(u^{\prime +}\right)}^{2p}\rangle }^{1/p}$ against ${\langle {\left(u^{\prime +}\right)}^6\rangle }^{1/3}$ for $p=1$ to 6. The solid lines indicate the slopes $a_p/a_3$, while the vertical dash-dotted lines show the bounds of the linear regions depicted in Fig. 4 with $-0.6984\le y/h\le -0.0123$. The data is from Ref. [46] at ${\text{Re}}_b=31600$ and $\text{Ro}=1.2$.

Figure 6

Slopes of the self-scaling $a_p/a_1$ at different Reynolds number and rotation number. (a) ${\text{Re}}_b=20000$ and (b) ${\text{Re}}_b=31600$. The Gaussian values are also shown for comparison.