Fractality and the law of the wall

Fluid motions in the inertial range of isotropic turbulence are fractal, with their space-filling capacity slightly below regular three-dimensional objects, which is a consequence of the energy cascade. Besides the energy cascade, the other often encountered cascading process is the momentum cascade in wall-bounded flows. Despite the long-existing analogy between the two processes, many of the thoroughly investigated aspects of the energy cascade have so far received little attention in studies of the momentum counterpart, e.g., the possibility of the momentum-transferring scales in the logarithmic region being fractal has not been considered. In this work, this possibility is pursued, and we discuss one of its implications. Following the same dimensional arguments that lead to the $D=2.33$ fractal dimension of wrinkled surfaces in isotropic turbulence, we show that the large-scale momentum-carrying eddies may also be fractal and non-space-filling, which then leads to the power-law scaling of the mean velocity profile. The logarithmic law of the wall, on the other hand, corresponds to space-filling eddies, as suggested by Townsend [The Structure of Turbulent Shear Flow (Cambridge University Press, Cambridge, 1980)]. Because the space-filling capacity is an integral geometric quantity, the analysis presented in this work provides us with a low-order quantity, with which, one would be able to distinguish between the logarithmic law and the power law.

Figure 1

A sketch of a coarse-grained wrinkled interface, across which a transported quantity $\varphi$ has a finite difference $\mathrm{\Delta }\varphi$. The interface is self-similar across a range of scales $l_S<l<l_L$, where $l_L$ and $l_S$ are two coarse-graining length scales. Flow structures are statistically similar at the two coarse-graining length scales.

Figure 2

(a) A sketch of the hypothesized wall-bounded turbulence at high Reynolds numbers. Three hierarchies of wall-attached eddies are sketched I, II, III. An eddy affects the flow within. The eddies are wall-attached, and are inclined towards the flow direction. The sizes of the eddies scale with their distance from the wall, and therefore the number of the eddies double as the size halves. Here, there are one hierarchy-I eddy, two hierarchy-II eddies, and four hierarchy-III eddies. A, B, and C are at three wall normal heights. (b) Filtered streamwise velocity at $y^{+}=100$ as a function of the variance of the streamwise velocity fluctuation. $\langle {u_l^{\prime }}^2\rangle$ is a function of the filtering length scale $l$, and $\langle {u^{\prime }}^2\rangle$ is a function of the wall-normal distance $y$. By relating $y$ and $l$, i.e., $z=ltan\left(\theta \right)$, we can relate $\langle {u_l^{\prime }}^2\rangle$ to $\langle {u^{\prime }}^2\rangle$, which leads to the plot shown here. Here, $\theta ={12}^{\circ }$ is the inclination angle of a typical wall-attached eddy [33]. We have used the top-hat filter in Fourier space. While not shown, using a top-hat filter in physical space leads to similar results. The two thin solid lines are at ${\left.\langle {u^{\prime }}^2\rangle \right|}_{y=0.3\delta }$ and ${\left.\langle {u^{\prime }}^2\rangle \right|}_{y^{+}=100}$. The bold solid line indicates the proportionality between $\langle {u_l^{\prime }}^2\rangle$ and $\langle {u^{\prime }}^2\rangle$.

Figure 3

(a) The measured $\mathcal{S}\left(y\right)$ as a function of the wall-normal distance on a log-log scale. We have normalized $\mathcal{S}\left(y\right)$ using the length of the signal. The two solid lines indicate $\sim y^0$ and $\sim y^{0.13}$, respectively. The dashed lines are at wall normal distances $y^{+}=C_y\sqrt{{\mathrm{Re}}_{\tau }},y/\delta =0.1$ and $y/\delta =0.3$, respectively, where $C_y=4.4$. (b) The mean velocity (bold line) as a function of the wall-normal distance on a semi-log scale. The constants in the log scaling and in the power-law scaling are $\kappa =0.39,B=4.4$, and $C=9.7,\alpha =0.13$.

Figure 4

(a) Measured $\mathcal{S}\left(y\right)$ as functions of the wall-normal distance. $c=1.5$ and $c=2$ are used. $\mathrm{\Delta }x$ is the filtering length scale. The resolution of the hot-wire data is $\mathrm{\Delta }{x}^{+}\approx 10$ (assuming a constant convective velocity $U_c=0.8U_{\infty }$, and invoking the Taylor's frozen turbulence hypothesis. $U_{\infty }$ is the free-stream velocity). (b) $\mathcal{S}\left(y\right)$ in a DNS channel at ${\mathrm{Re}}_{\tau }=5200$ [35]. $c=2$ is used.

Figure 5

A sketch of the wall layer.

Figure 6

(a) Power-law exponents and (b) prefactor as functions of the friction Reynolds number. Symbols are measurements, and are best fits of power-law scalings to $U$ in the wall-normal distance range $0.1<y/\delta <0.3$, within which region $\mathcal{S}\left(y\right)$ follows a power law. The bold line, thin line and the dashed line correspond to Eq. (9) for boundary layer, pipe, and channel, respectively. The thin red line in (a) corresponds to the empirical correlation by [52]: $\alpha =1.085/log\left(\mathrm{Re}\right)+6.535/{(log\left(\mathrm{Re}\right))}^2$, where $\mathrm{Re}$ is the bulk Reynolds number, and is defined based on the pipe diameter and the flow rate. The bulk Reynolds number $\mathrm{Re}$ can be related to the friction Reynolds number ${\mathrm{Re}}_{\tau }$ using the Karman-Prandtl resistance relation, i.e., $1/\sqrt{f}=-1.930log(1.90/\left(\mathrm{Re}\sqrt{f}\right))$, where $f=8{\tau }_w/\left(\rho {\overline{U}}^2\right)$, and $\overline{U}$ is the mean flow rate. The thin red line in (b) corresponds to $C=0.7053log\left(\mathrm{Re}\right)+0.3055$ [52].

Figure 7

Mean velocities as functions of the wall-normal distance for pipe flow at (a) ${\mathrm{Re}}_{\tau }\approx 3300$ and (b) ${\mathrm{Re}}_{\tau }\approx 452000$ on a semilog scale. Legends are the same as in Fig. 3.

Figure 8

Mean velocities as functions of the wall-normal distance for channel flow at (a) ${\mathrm{Re}}_{\tau }\approx 1000$ and (b) ${\mathrm{Re}}_{\tau }\approx 5200$ on a log-linear scale. Legends are the same as in Fig. 3.

Figure 9

Mean velocity as functions of the wall-normal distance for boundary layer flow at ${\mathrm{Re}}_{\tau }\approx 4800$ on a semi-log scale.