Logarithmic and nonlogarithmic scaling laws of two-point statistics in wall turbulence

Wall turbulence has a sublayer where one-point statistics, e.g., the mean velocity and the variances of some velocity fluctuations, vary logarithmically with the distance from the wall. This logarithmic scaling is found here for two-point statistics or specifically two-point cumulants of those fluctuations by means of experiments in a wind tunnel. As for corresponding statistics of the rate of the energy dissipation, the scaling is found to be not logarithmic. We reproduce these scaling laws with some mathematics and also with a model of energy-containing eddies that are attached to the wall.

Figure 1

One-point statistics (a) $U\left(z\right)/u_{\tau }$, (b) ${\langle -uw\left(z\right)\rangle }_c/u_{\tau }^2=\langle -uw\left(z\right)\rangle /u_{\tau }^2$, (c) ${\langle u^2\left(z\right)\rangle }_c/u_{\tau }^2=\langle u^2\left(z\right)\rangle /u_{\tau }^2$, (d) ${\langle u^4\left(z\right)\rangle }_c/u_{\tau }^4=[\langle u^4\left(z\right)\rangle -3{\langle u^2\left(z\right)\rangle }^2]/u_{\tau }^4$, and (e) ${\langle \varepsilon \left(z\right)\rangle }_c/(u_{\tau }^3/z)=\langle \varepsilon \left(z\right)\rangle /(u_{\tau }^3/z)$ against $z/{\delta }_{99}$ for $U_{\infty }=6\mathrm{m}{\mathrm{s}}^{-1}$ (circles) and $12\mathrm{m}{\mathrm{s}}^{-1}$ (squares shifted horizontally by two units). The filled symbols lie in the constant-flux sublayer. To these, solid lines are regression fits of Eq. (1a), (1b), or (3a). We provide $\pm 2\sigma$ errors, albeit not including those for $u_{\tau }$ and $z-z_{\mathrm{wt}}$.

Figure 2

Two-point velocity cumulant in the constant-flux sublayer ${\langle u(x,z)u(x+r,z)\rangle }_c/u_{\tau }^2=\langle u(x,z)u(x+r,z)\rangle /u_{\tau }^2$ against $r/z$ for (a) $U_{\infty }=6\mathrm{m}{\mathrm{s}}^{-1}$ and (b) $12\mathrm{m}{\mathrm{s}}^{-1}$. The solid arrow indicates an increase in ${\delta }_{99}/z$. The gray areas are examples of $\pm 2\sigma$ errors, albeit not including those for $u_{\tau }$ and $z-z_{\mathrm{wt}}$. The panel (c) shows $C_{uu}(r/z)$ and $D_{uu}(r/z)$ of Eq. (3a). For $U_{\infty }=12\mathrm{m}{\mathrm{s}}^{-1}$, we provide $\pm 2\sigma$ errors including those for $u_{\tau }$ but not for $z-z_{\mathrm{wt}}$.

Figure 3

Two-point velocity cumulant ${\langle u(x,z)u(x+r,z)\rangle }_c/u_{\tau }^2=\langle u(x,z)u(x+r,z)\rangle /u_{\tau }^2$ against $ln({\delta }_{99}/z)$ at $r/z=0.1$, 0.3, 1.0, 3.0, and 10.0 for (a) $U_{\infty }=6\mathrm{m}{\mathrm{s}}^{-1}$ and (b) $12\mathrm{m}{\mathrm{s}}^{-1}$. The filled symbols lie in the constant-flux sublayer. To these, solid lines are regression fits of Eq. (3a). Although we provide $\pm 2\sigma$ errors on all the data in the same manner as in Figs. 2 and 2, none of them are discernible.

Figure 4

Same as in Fig. 2 but for ${\langle u^2(x,z)u^2(x+r,z)\rangle }_c/u_{\tau }^4=[\langle u^2(x,z)u^2(x+r,z)\rangle -{\langle u{\left(z\right)}^2\rangle }^2-2{\langle u(x,z)u(x+r,z)\rangle }^2]/u_{\tau }^4$.

Figure 5

Same as in Fig. 3 but for ${\langle u^2(x,z)u^2(x+r,z)\rangle }_c/u_{\tau }^4=[\langle u^2(x,z)u^2(x+r,z)\rangle -{\langle u{\left(z\right)}^2\rangle }^2-2{\langle u(x,z)u(x+r,z)\rangle }^2]/u_{\tau }^4$.

Figure 6

Two-point cumulant of the rate of the energy dissipation in the constant-flux sublayer ${\langle \varepsilon (x,z)\varepsilon (x+r,z)\rangle }_c/{(u_{\tau }^3/z)}^2=[\langle \varepsilon (x,z)\varepsilon (x+r,z)\rangle -{\langle \varepsilon \left(z\right)\rangle }^2]/{(u_{\tau }^3/z)}^2$ against $r/z$ for (a) $U_{\infty }=6\mathrm{m}{\mathrm{s}}^{-1}$ and (b) $12\mathrm{m}{\mathrm{s}}^{-1}$. The solid arrow indicates an increase in ${\delta }_{99}/z$. The panel (c) shows $C_{\varepsilon \varepsilon }(r/z)$ of Eq. (3b). We provide examples of $\pm 2\sigma$ errors as in Figs. 2 and 4. The circles and squares indicate the Kolmogorov length $\eta$. The insets are for large values of $r/z$.

Figure 7

Schematics of (a) attached eddies, (b) $I_{uu}(r/h_e,z/h_e)$, (c) $I_{u^2{u}^2}(r/h_e,z/h_e)$, and (d) $I_{\varepsilon \varepsilon }(r/h_e,z/h_e)$. The solid arrow indicates an increase in $r/h_e$. The dotted lines are examples of ${\widehat{I}}_{u^n{u}^n}^{\left(0\right)}(r/z)\propto D_{u^n{u}^n}(r/z)$ in Eq. (12c). They reproduce $D_{uu}\ge 0$ and $D_{u^2{u}^2}\le 0$ in Figs. 2 and 4. As for integrations of $I_{u^n{u}^n}-{\widehat{I}}_{u^n{u}^n}^{\left(0\right)}$ in Eq. (12b) and of $I_{\varepsilon \varepsilon }$ in Eq. (14b), the gray areas are examples to reproduce $C_{uu}(r/z)$, $C_{u^2{u}^2}(r/z)$, and $C_{\varepsilon \varepsilon }(r/z)$ in Figs. 2, 4, and 6.

Figure 8

Same as in Fig. 1 but for (a) ${\langle u^3\left(z\right)\rangle }_c=\langle u^3\left(z\right)\rangle$, (b) ${\langle u^5\left(z\right)\rangle }_c=\langle u^5\left(z\right)\rangle -10\langle u^3\left(z\right)\rangle \langle u^2\left(z\right)\rangle$, (c) ${\langle u^6\left(z\right)\rangle }_c=\langle u^6\left(z\right)\rangle -15\langle u^4\left(z\right)\rangle \langle u^2\left(z\right)\rangle -10{\langle u^3\left(z\right)\rangle }^2+30{\langle u^2\left(z\right)\rangle }^3$, (d) ${\langle u^7\left(z\right)\rangle }_c=\langle u^7\left(z\right)\rangle -21\langle u^5\left(z\right)\rangle \langle u^2\left(z\right)\rangle -35\langle u^4\left(z\right)\rangle \langle u^3\left(z\right)\rangle +210\langle u^3\left(z\right)\rangle {\langle u^2\left(z\right)\rangle }^2$, and (e) ${\langle u^8\left(z\right)\rangle }_c=\langle u^8\left(z\right)\rangle -28\langle u^6\left(z\right)\rangle \langle u^2\left(z\right)\rangle -56\langle u^5\left(z\right)\rangle \langle u^3\left(z\right)\rangle -35{\langle u^4\left(z\right)\rangle }^2+420\langle u^4\left(z\right)\rangle {\langle u^2\left(z\right)\rangle }^2+560{\langle u^3\left(z\right)\rangle }^2\langle u^2\left(z\right)\rangle -630{\langle u^2\left(z\right)\rangle }^4$ nondimensionalized with use of $u_{\tau }$. The solid lines are regression fits of Eq. (3a).