Numerical evidence of logarithmic regions in channel flow at $\mathrm{R}{\mathrm{e}}_{\tau }=8000$

Direct numerical simulations of channel flow up to $\mathrm{R}{\mathrm{e}}_{\tau }=8000$ have been performed to determine the existence of a logarithmic region in channel flow at high-Reynolds number. It is notable that the logarithmic variations both in the mean velocity, $U^{+}=\left(1/\phantom{1\kappa }\kappa \right)ln\left(y^{+}\right)+B$ and the streamwise Reynolds stress, $\overline{u^{+}{u}^{+}}=B_1-A_{1}ln(y/h)$ are first confirmed for $\mathrm{R}{\mathrm{e}}_{\tau }=8000$, with $\kappa =0.387,B=4.21$ and $A_1=1.65,B_1=1.23$, where $y$ is the distance from the wall and $h$ is the channel half-height. The logarithmic region of $U^{+}$ locates between $300<y^{+}<1100$ ($y/h=0.14$) and that of $\overline{u^{+}{u}^{+}}$ locates between $1200\left(y/h=0.15\right)<y^{+}<2000(y/h=0.25)$. Therefore, these logarithmic regions do not overlap with each other. Furthermore, the region of ${k_x}^{-1}$ law is studied without using the frozen turbulence hypothesis in logarithmic regions of the mean velocity and the streamwise Reynolds stress.

Figure 1

Verification of statistical convergence; total shear balance (YT8000C), and residual of total shear stress for all cases.

Figure 2

Logarithmic variations of mean velocity and streamwise Reynolds stress. Darker-shaded regions denote the logarithmic regions in YT8000C. (a) Mean velocity: indicator parameter $\beta (=1/\kappa )$ and (b) streamwise Reynolds stress: test function $\mathrm{\Xi }(=-A_1)$.

Figure 3

Comparison with fitting equation of logarithmic variation in streamwise Reynolds stress and the current DNS in YT8000C and the experiments of pipe flow [30, 31]. Darker-shaded region denotes the logarithmic region in YT8000C.

Figure 4

Streamwise premultiplied spectrum of $u$ at overlap region: $90<y^{+}<150$. Darker-shaded regions denote the ${k_x}^{-1}$ regions. (a) ${\mathrm{Re}}_{\tau }=4000$ (YT4000) and (b) ${\mathrm{Re}}_{\tau }=8000$ (YT8000C).

Figure 5

Contour lines of streamwise premultiplied spectra of $u$ in the $y^{+}-{{\lambda }_x}^{+}$ plane with profiles of turbulent intensity of $u$. Contour maps are lined from 0.1 (blue) to 1.95 (red); spacing in 0.05. Darker-shaded regions denote the ${k_x}^{-1}$ regions. Thin broken lines show ${k_x}^{-1}$ limits ${\lambda }_x=15.7y$, from $y^{+}>100$. Blue dotted lines denote ${\lambda }_x=100y$ [41]. Red dotted lines denote $y^{+}=3.9\mathrm{R}{{\mathrm{e}}_{\tau }}^{0.5}$ [39, 40]. (a) ${\mathrm{Re}}_{\tau }=1000$, (b) ${\mathrm{Re}}_{\tau }=2000$, (c) ${\mathrm{Re}}_{\tau }=4000$, and (d) ${\mathrm{Re}}_{\tau }=8000$.