Effects of boundary layer thickness on the estimation of equivalent sandgrain roughness in zero-pressure-gradient boundary layers

Abstract

Experiments were conducted in zero-pressure-gradient boundary layers over a rough surface. Profiles of mean velocity and turbulence quantities were acquired using laser Doppler velocimetry at twelve streamwise locations for each of three different freestream velocities. Momentum thickness Reynolds numbers ranged from 1550 to 13,650. The roughness was stochastic with positive skewness, and the ratio of the boundary layer thickness, δ, to the root-mean-square roughness height varied from 55 to 141. The equivalent sandgrain roughness height, k_s, was approximately 4 times the rms roughness height. In the outer region of the boundary layer, the mean velocity and turbulence results were found to be invariant with Reynolds number and δ, when scaled using δ and the friction velocity. The outer region quantities exhibited similarity with equivalent smooth-wall boundary layer quantities, in agreement with results from the literature. The equivalent sandgrain roughness was computed from the mean velocity results using a standard correlation, and was found to vary noticeably and inversely with δ when δ/k_s was less than about 40. The possible existence of a modified correlation to account for the δ/k_s dependence was discussed. Such a correlation might prove useful for extracting k_s values from low δ/k_s data and for predicting the effect of roughness with a known k_s on boundary layers with varying δ/k_s.

Graphical abstract

Appendix

The $\overline{{u^{{\prime 3}} }} /U_{\tau }^{3}$ triple product results in Fig. 13 showed similarity between the rough and smooth wall cases. The other triple products show the same. Figure 

Fig. 16

$\overline{{v{^{\prime}}}^{3}}/{U}_{\tau }^{3}$ triple product profiles in outer coordinates: a smooth-wall, b rough-wall. Symbols and lines as in Fig. 5

16 shows $\overline{{v{^{\prime}}}^{3}}/{U}_{\tau }^{3}$ profiles. The smooth- and rough-wall results again exhibit similarity. The quantity v′³ represents the wall normal transport of v′². Transport of v′² by ejections (v′ > 0) from its peak in Fig. 11 results in the positive v′³ farther from the wall in Fig. 16. Since there is no clear trend in $\overline{{v{^{\prime}}}^{2}}/{U}_{\tau }^{2}$ with Reynolds number in Fig. 11, there is similarly no clear trend in $\overline{{v{^{\prime}}}^{3}}/{U}_{\tau }^{3}$ in Fig. 16. Figure 

Fig. 17

$\overline{{u{^{\prime}}}^{2}v{^{\prime}}}/{U}_{\tau }^{3}$ triple product profiles in outer coordinates: a smooth-wall, b rough-wall. Symbols and lines as in Fig. 5

17 shows $\overline{{u{^{\prime}}}^{2}v{^{\prime}}}/{U}_{\tau }^{3}$ profiles. Similarity is again observed between the rough- and smooth-wall cases in the outer boundary layer, and there does not appear to be significant variation with Reynolds number. The quantity $\overline{{u{^{\prime}}}^{2}v{^{\prime}}}/{U}_{\tau }^{3}$ is most closely related to the wall normal transport of u′², as explained in Volino (2020a). For motions away from the wall (ejections), v′ is positive, resulting in positive u′²v′. Outward motions from the u′² peak at y/δ = 0.1 in Fig. 10 cause positive u′²v′ farther from the wall in Fig. 17, just as these same motions resulted in negative u′³ in Fig. 13. The higher u′² near the wall in the high-freestream-velocity case in Fig. 10 results in higher $\overline{{u{^{\prime}}}^{2}v{^{\prime}}}/{U}_{\tau }^{3}$ for the same case in Fig. 17. The final triple product, $\overline{{u{^{\prime}}v{^{\prime}}}^{2}}/{U}_{\tau }^{3}$, is shown in Fig. 

Fig. 18

$\overline{{u{^{\prime}}v{^{\prime}}}^{2}}/{U}_{\tau }^{3}$ triple product profiles in outer coordinates: a smooth-wall, b rough-wall. Symbols and lines as in Fig. 5

18. In the outer region, there is again similarity between the rough and smooth wall cases. The quantity can be associated with the wall normal transport of v′² and u′v′. In the case of v′², ejections with negative u′ cause negative u′v′², and sweeps with positive u′ cause positive u′v′². In the case of u′v′ (which in the mean is a negative quantity), ejections with positive v′ cause negative u′v′², and sweeps with negative v′ cause positive u′v′². Hence, transport away from the peaks in Figs. 11 and 12 result in the positive u′v′² in the outer part of the boundary layer in Fig. 17. It appears that the larger u′ near the wall for the high-freestream-velocity case in Fig. 10, when associated with the transport of v′², results in higher $\overline{{u{^{\prime}}v{^{\prime}}}^{2}}/{U}_{\tau }^{3}$ for these cases in Fig. 17.

The primary production term for $\overline{u{^{\prime}}v{^{\prime}}}$ is shown in Fig. 18. As with the $\overline{{u{^{\prime}}}^{2}}$ production term in Fig. 14, there is similarity between the rough and smooth wall cases, and no significant dependence on Reynold number (Fig.

Fig. 19

Profiles of primary production term for $\overline{u{^{\prime}}v{^{\prime}}}$ in outer coordinates: a smooth-wall, b rough-wall. Symbols and lines as in Fig. 5

19).