Evolution of wall shear stress with Reynolds number in fully developed turbulent channel flow experiments

The Reynolds-number-dependence of wall shear stress (${\tau }_w$) in fully developed, high-aspect-ratio turbulent channel flow is experimentally studied over the range of $230\le {\text{Re}}_{\tau }\le 950$. The principal conclusion of the present work is that beyond ${\text{Re}}_{\tau }\sim 600$, the wall shear stress evolves to an asymptotic state in which the statistical moments, probability density function, and power spectra of ${\tau }_w$ become independent of Reynolds number. In this state, the turbulent intensity, skewness and kurtosis of the wall shear stress take on values that are slightly larger than those previously measured, because prior estimates of these quantities did not resolve the full range of wall shear stress fluctuations, which extended beyond 10 standard deviations above the mean. For ${\text{Re}}_{\tau }\gtrsim 600$, the spectra of ${\tau }_w$ were found to collapse very well when using a wall-unit-based normalization, providing further evidence of an asymptotic state of the wall shear stress at large Reynolds numbers.

Figure 1

The dependence of the (a) average and (b) RMS wall shear stress on friction Reynolds number (${\text{Re}}_{\tau }\equiv u_{\tau }h/\nu$, where $h$ is the channel half-width).

Figure 2

The dependence of the intensity of the wall shear stress fluctuations on friction Reynolds number (${\text{Re}}_{\tau }$). (a) Present work (linear-linear plot). (b) Present work compared with the work of others (linear-log plot). 95% confidence intervals estimated by Bayesian bootstrap analysis [58] using 200 replications are depicted in (a) by way of error bars.

Figure 3

The dependence of the skewness ($\square$) and kurtosis ($\circ$) of the wall shear stress fluctuations on friction Reynolds number (${\text{Re}}_{\tau }$). 95% confidence intervals estimated by Bayesian bootstrap analysis [58] using 200 replications are depicted by way of error bars.

Figure 4

Normalized probability density functions (PDFs) of the wall shear stress fluctuations for ${\text{Re}}_{\tau }=232$, 449, 677, and 895. (a) Linear-linear axes. (b) Log-linear axes. Recall ${\tau }_w^{\prime }={\tau }_w-\langle {\tau }_w\rangle$.

Figure 5

Normalized PDFs of the wall shear stress fluctuations multiplied by (a) ${({\tau }_w/{\tau }_{w_{\text{RMS}}})}^2$, and (b) ${({\tau }_w/{\tau }_{w_{\text{RMS}}})}^4$. The area under these curves are equal to the normalized second and fourth moments of the wall shear stress. The area under the open circles represents the contribution to the moment that would be unresolved if fluctuations greater than 5 standard deviations were not accounted for. ${\text{Re}}_{\tau }=895$.

Figure 6

Power spectral densities of the wall shear stress fluctuations for ${\text{Re}}_{\tau }=232$, 449, 677, and 895. (a) Dimensional spectra. (b) Spectra nondimensionalized using wall units.