Scaling of turbulent kinetic energy and dissipation in turbulent wall-bounded flows

A new scaling is developed for the turbulent kinetic energy (TKE) and its dissipation in a turbulent wall-bounded flow. In the traditional dimensional analysis of wall-bounded turbulence, the control parameters for the near-wall region are the kinematic viscosity $\nu$ and the wall shear stress, resulting in the friction velocity $u_{\tau }$ as the characteristic velocity scale and the viscous length scale $\nu /u_{\tau }$ as the characteristic length scale. Although the mean streamwise velocity scales well with the friction velocity, the TKE, and, in particular, the TKE dissipation does not scale with the friction velocity. In the present paper, a new dimensional analysis is performed to identify a proper scaling for the TKE and its dissipation. The control parameters in the near-wall region are the kinematic viscosity and the TKE dissipation at the wall ${\epsilon }_{k,\mathrm{w}}$. The new inner velocity scale for the TKE budget equation is the Kolmogorov wall velocity $u_{\epsilon }\stackrel{def}{=}{\left(\nu {\epsilon }_{k,\mathrm{w}}\right)}^{1/4}$, and the proper length scale is the Kolmogorov wall length $\nu /u_{\epsilon }$. The profiles of the TKE and its dissipation in the near-wall region collapse well under the new scaling, and the TKE profiles in the outer layer also scale well with the new scaling. However, the TKE peak value $k_{\max }$ does not scale with the Kolmogorov wall velocity $u_{\epsilon }$ or the friction velocity $u_{\tau }$ but with a mixed scale. One mixed scale is $u_{\tau }{U}_{\infty }$ first proposed by DeGraaff and Eaton [J. Fluid Mech. 422, 319 (2000)], where $U_{\infty }$ is the mean velocity in the free stream or at the channel centerline. A new mixed scale developed in this paper for $k_{\max }$ is $\nu {\epsilon }_{k,\mathrm{w}}/u_{\tau }^2$, and justification is provided by adding a control parameter to the new dimensional analysis.

Figure 1

Dimensional analysis for the near-wall region in a turbulent channel flow. Left: Traditional inner scaling. Middle: Alternative scaling 1. Right: Alternative scaling 2.

Figure 2

(a) Traditional inner-scaled TKE dissipation. (b) Traditional inner-scaled TKE. (c) TKE dissipation on log-log axes. (d) TKE on log-log axes. DNS data are from Lee and Moser [13]. DNS data from Kawamura and co-workers are also plotted at ${\mathrm{Re}}_{\tau }=180,1020$ (triangles) [14, 15].

Figure 3

(a) Traditional outer-scaled TKE dissipation. The inset is on the log-log axes to better show the near-wall region. (b) Traditional outer-scaled TKE. DNS data are from Lee and Moser [13], and Kawamura and co-workers [14, 15] (triangles).

Figure 4

New inner scaling. (a) TKE dissipation on log-linear axes. (b) TKE on log-linear axes. (c) TKE dissipation on log-log axes. (d) TKE on log-log axes.

Figure 5

(a) New outer-scaled TKE dissipation. (b) New outer-scaled TKE.

Figure 6

(a) Reynolds-number dependence of the $k_{\max }^{+},{\langle uu\rangle }_{\max }^{+},{\epsilon }_{k,\mathrm{w}}^{+}$, and $U_{\mathrm{ctr}}^{+}$. The $k_{\max }^{+}$ and ${\epsilon }_{k,\mathrm{w}}^{+}$ data are from the DNS of Lee and Moser [13] and Kawamura and co-workers [14, 15]. $U_{\max }^{+}$ data are from Hutchins and Marusic [17] and Metzger and Klewicki [18]. $U_{\mathrm{ctr}}^{+}$ are from the DNS [13, 14, 15] and the Princeton superpipe data (circles) [19, 20]. (b) Ratios of $U_{\mathrm{ctr}}^{+}/{\epsilon }_{k,\mathrm{w}}^{+}$ and $k_{\max }^{+}/{\epsilon }_{k,\mathrm{w}}^{+}$. Data are from the DNS of Lee and Moser [13].