Wall-drag measurements of smooth- and rough-wall turbulent boundary layers using a floating element

Abstract

The mean wall shear stress, $\overline{\tau }_w$, is a fundamental variable for characterizing turbulent boundary layers. Ideally, $\overline{\tau }_w$ is measured by a direct means and the use of floating elements has long been proposed. However, previous such devices have proven to be problematic due to low signal-to-noise ratios. In this paper, we present new direct measurements of $\overline{\tau }_w$ where high signal-to-noise ratios are achieved using a new design of a large-scale floating element with a surface area of 3 m (streamwise) × 1 m (spanwise). These dimensions ensure a strong measurement signal, while any error associated with an integral measurement of $\overline{\tau }_w$ is negligible in Melbourne’s large-scale turbulent boundary layer facility. Wall-drag induced by both smooth- and rough-wall zero-pressure-gradient flows are considered. Results for the smooth-wall friction coefficient, $C_f \equiv \overline{\tau }_w/q_{\infty }$, follow a Coles–Fernholz relation $C_f = \left[ 1/\kappa \ln \left( Re_{\theta }\right) + C\right] ^{-2}$ to within 3 % ($\kappa = 0.38$ and $C = 3.7$) for a momentum thickness-based Reynolds number, $Re_{\theta } > 15{,}000$. The agreement improves for higher Reynolds numbers to <1 % deviation for $Re_{\theta } > 38{,}000$. This smooth-wall benchmark verification of the experimental apparatus is critical before attempting any rough-wall studies. For a rough-wall configuration with P36 grit sandpaper, measurements were performed for $10{,}500< Re_{\theta } < 88{,}500$, for which the wall-drag indicates the anticipated trend from the transitionally to the fully rough regime .

Appendix

Wall-normal boundary layer profiles of the mean velocity were acquired for both the smooth- and rough-wall configurations. These data allow for an empirical conversion from $Re_{x} \equiv x_F U_{\infty }/\nu$ to $Re_{\theta } \equiv \theta U_{\infty }/\nu$, at the position in the Melbourne wind tunnel where these profiles were taken ($x \approx x_F = 21.0$ m, in the spanwise center). For the smooth-wall case, individual Pitot and static tubes were used to examine the mean velocity. The outside diameter of the Pitot tube was $d_p = 0.98$ mm and the diameter of the total pressure port was ~0.4 mm. The static tube was positioned 13.5 mm above the Pitot tube, and had an outside diameter of $d_s = 1.57$ mm, with the static pressure tab positioned 14.2 mm from the leading edge. For dynamic pressures lower than 100 Pa, a 0–1 Torr pressure gauge transducer was used (Table 1). Otherwise, a 0–10 Torr transducer was employed. A logarithmic wall-normal spacing was selected up to the geometric center of the log-region, beyond which linear spacing was used; typical profiles consisted of 60 points. Post-measurement corrections were applied following Bailey et al. (2013) to yield the wall-normal velocity profiles. Profiles above the rough wall were acquired using hot-wire anemometry (Squire et al. 2016). For each $Re_x$ condition, the momentum thickness was found via numerical integration of the velocity profiles. The obtained $Re_x \rightarrow Re_{\theta }$ conversions are presented in Fig. 15. An empirical relation of Nagib et al. (2007) is superposed to show its agreement (within experimental tolerance) with our current conversion.

Fig. 15

Reynolds number $Re_x$ versus $Re_{\theta }$ above the smooth- and rough-wall configurations at $x \approx x_F = 21.0$ m