Refinement of the logarithmic law of the wall

Available direct numerical simulation of turbulent channel flow at moderately high Reynolds numbers data show that the logarithmic diagnostic function is a linearly decreasing function of the outer-normalized wall distance $\eta =y/\delta$ with a slope proportional to the von Kármán constant, $\kappa =0.4$. The validity of this result for turbulent pipe and boundary layer flows is assessed by comparison with the mean velocity profile from experimental data. The results suggest the existence of a flow-independent logarithmic law ${\overline{U}}^{+}=\overline{U}/u_{\tau }=(1/\kappa )ln(y^{*}/a)$, where $y^{*}=y{U}_S/\nu$ with $U_S=yS\left(y\right)$ being the local shear velocity and the two flow-independent constants $\kappa =0.4$ and $a=0.36$. The range of its validity extends from the inner-normalized wall distance $y^{+}=300$ up to half the outer-length scale $\eta =0.5$ for internal flows, and $\eta =0.2$ for zero-pressure-gradient turbulent boundary layers. Likewise, and within the same range, the mean velocity deficit follows a flow-dependent logarithmic law as a function of a local mean-shear-based coordinate. Furthermore, it is illustrated how the classical friction laws for smooth pipe and zero-pressure-gradient turbulent boundary layer are recovered from this scaling.

Figure 1

Log-law diagnostic function $\mathrm{\Xi }$ (broken lines) and $\mathrm{\Xi }/(1-\beta \eta )$ (solid lines) from DNSs of channel flow. $\mathbf{—}$, $\mathbf{\text{−}} \mathbf{\text{−}} \mathbf{–}$, $R_{\tau }=3990$, $\mathbf{—}$, $\mathbf{\text{−}} \mathbf{\text{−}} \mathbf{–}$, $R_{\tau }=8016$ Yamamoto and Tsuji [28]; $\mathbf{—}$, $\mathbf{\text{−}} \mathbf{–}$, $R_{\tau }=1000$, $\mathbf{—}$, $\mathbf{\text{−}} \mathbf{–}$, $R_{\tau }=5186$ Lee and Moser [29]; $\mathbf{—}$, $\mathbf{\text{−}} \mathbf{\text{−}}$, $R_{\tau }=2003$ Hoyas and Jiménez [30]. (a) $\mathrm{\Xi }$ (broken lines) and $\mathrm{\Xi }/(1-\beta \eta )$ (solid lines) as a function of $\eta$. (b) $\mathrm{\Xi }$ (broken lines) and $\mathrm{\Xi }/(1-\beta \eta )$ (solid lines) plotted against $y^{+}$. Error bars, with an uncertainty of $\pm 1.5\%$, are located at $y^{+}={\delta }^{+}/2$. Horizontal long dashed line, $\mathrm{\Xi }=0.34$. Vertical dashed line, $y^{+}=300$. Bullets denote $\mathrm{\Xi }({\delta }^{+}/2)$. The gray band represents an uncertainty of $\pm 1.5\%$. The thin dashed lines show Eq. (3).

Figure 2

Evolution with Kármán number of $\beta$ obtained from Eq. (4) with ${\eta }_o=0.5$ for the three canonical flows. Channel flow: $◊$, Hoyas and Jiménez [30]; $▿$, Laadhari [31]; $\odot$, Schultz and Flack [32]; $◊$, Bernardini et al. [33]; $⊡$, Lee and Moser [29]; $▵$, Yamamoto and Tsuji [28]. Pipe flow: $⧫$, Zagarola and Smits [8]; $\blacksquare$, Wu and Moin [34]; $⧫$, Hultmark et al. [35]; $▴$, El-Khoury et al. [36]; $•$, Chin et al. [37]; $•$, Ahn et al. [38]; $▾$, Furuichi et al. [39]; $▾$, Bauer et al. [40]. Boundary layer: $\times$, Schlatter and Örlü [20]; $\times$, Örlü and Schlatter [41]; $+$, Sillero et al. [42]; $*$, Bailey et al. [43]; $*$, Vallikivi et al. [44]. $\mathbf{\text{▁}} \mathbf{\text{▁}} \mathbf{\text{▁}}$, ${\beta }_{ch}=0.3$; $\mathbf{\text{−}} \mathbf{—}$, ${\beta }_{pf}=0.6$; $\mathbf{\text{−}} \mathbf{\text{−}} \mathbf{—}$, ${\beta }_{bl}=0.9$. Vertical dashed line: $R_{\tau }=4000$.

Figure 3

Mean velocity profiles ${\overline{U}}^{+}$ as a function of $y^{*}$. (a) Channel flow: large black symbols show mean velocity at $\eta =0.5$. (b) Pipe flow: large black symbols show mean velocity at $\eta =0.5$. (c) Boundary layer: large black symbols show mean velocity at $\eta =0.2$. $\mathbf{—} \mathbf{—}$, Eq. (7a) with $\kappa =0.4$ and $a=0.36$; $....$, $\sqrt{y^{*}}$. Vertical dashed lines: $y^{+}=300$. All other symbols and lines are defined in Table 1.

Figure 4

Mean velocity-defect profiles as a function of ${\eta }^{*}$. (a) Channel flow: dotted line, Eq. (7b) with $\kappa =0.4$ and $b=0.36$; large black symbols, mean velocity deficit at ${\eta }_o=0.5$. (b) Pipe flow: dashed line, Eq. (7b) with $\kappa =0.40$ and $b=0.2$; dashed-double dotted line, Eq. (7b) with $\kappa =0.4$ and $b=0.25$; large black symbols, mean velocity deficit at ${\eta }_o=0.5$. (c) Boundary layer: dashed-dotted line, Eq. (7b) with $\kappa =0.4$ and $b=0.11$; large black symbols, mean velocity deficit at ${\eta }_o=0.2$. Vertical dashed lines: $y^{+}=300$. All other symbols and lines are defined in Table 1.