Turbulence intensity profiles

Influence of skewed three-dimensional sinusoidal surface rough- ness on turbulent boundary layers

7.5 RESULTS AND DISCUSSION

7.5.3 Turbulence intensity profiles

positive correlation betweenk_sk and ∆U⁺. Additionally, the figure reveals that increasing Re_τ leads to an increase in ∆U⁺ for the same rough surface (represented by symbols of different colours). When k_sk increases from negative to zero values in the fully rough regime and with constant ES (S24-06-06N48, S24-06-06 and S24-06-06P48), ∆U⁺ increases by 20%, whereas when k_sk increases from zero to positive values, ∆U⁺ increases by only 10%.

The relationship between k⁺_s and k_sk is shown in Figure 7.7(b), which demonstrates a nearly exponential increase in k_s⁺ as k_sk changes from negative to positive values. For a given surface, an increase inRe_τ leads to an increase ink_s⁺, and this increase becomes more pronounced ask_sk shifts from negative to positive values. An analysis of the fully rough regime with constant ES (S24-06-06N48, S24-06-06 and S24-06-06N48) shows that there was a 70% increase in k_s⁺ when k_sk changed from negative to zero, compared to only a 40% increase when k_sk changed from zero to positive.

The observations from Figures 7.6 and 7.7 demonstrate an increase in C_f, ∆U⁺, and k_s as k_sk increases. Furthermore, the impact of k_sk increasing from negative to zero is greater than whenk_sk increases from zero to positive. These findings align with previous experimental studies. For example,Flacket al.(2020b) found that an increase in drag is larger from negative to zero roughness skewness compared to from zero to positive roughness skewness. Additionally, these results are consistent with numerical studies such as Jelly & Busse (2018), which found that surfaces with more peaks than valleys (positive roughness skewness) produce more drag than negative ones.

The roughness skewness plays a crucial role in impacting turbulence statistics and drag reduction and should not be overlooked in any relationship linkingk_s to real roughness param- eters. In some cases where the roughness skewness is the dominant roughness parameter and changes in other roughness parameters are negligible, k_s can be directly correlated with k_sk. This relationship is demonstrated in Figure 7.8, which depicts that the exponential equation k_s =a×e^b×ksk with a=0.0015 and b=0.8385 accurately predicts k_s for this particular family of roughness with a statistical measure of goodness of fit, R², of 0.97. However, this equation is suitable only for this family of roughness where k_sk is the only variable roughness parame- ter. Nevertheless, it provides a concept of the correlation betweenk_s and k_sk and what type of regression equation best describes this relationship.

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0.5

1 1.5 2 2.5

3 10^-3

Figure 7.8: Equivelant sand grain roughnessksas a function of roughness skewnesskskfor the different 3D sinewave surfaces that haveES=0.2±0.2 at the three differentRe_τ values. The solid black line represents the exponential fitting of the data. The symbols of the rough surface are indicated in Table7.2. N for negative k_sk, Z for zerok_sk andP for positivek_sk. The blue symbols are atU_∞= 10 m/s. The green symbols are atU_∞ = 15 m/s. The red symbols are at

U_∞ = 20 m/s.

Increasing Re_τ for the same rough surface reduces the inner peak value ofu^′+ and shifts the profiles to the right to largery⁺, which is attributed to the transition from the transitionally rough to the fully rough regime. This reduction of the inner peak has been observed byLigrani

& Moffat(1986) when the flow is in a transitionally rough regime until it is fully removed in the fully rough regime.

For better comparison of the u^′ profiles for different rough surfaces, we plot the profiles over all the rough surfaces at matched Re_τ ≈ 3000±300 in Figures7.9(c) and 7.9(d). These figures show that the zero roughness skewness has slightly lower turbulence intensities than positive and negative roughness skewness from the inner region up toy/δ<0.6. However, from Figures7.9(b)and 7.9(d), it can be observed that these small differences can be neglected, and an outer layer similarity can still be valid for this family of roughness.

To investigate the impact of k_sk on turbulence intensities, we employed the diagnostic plot, introduced by Alfredssonet al. (2011), which plotsu^′/U against (U/U_∞). This approach avoids the use of the friction velocityU_τ and the wall-normal positionyand reduces measurement uncertainties. We plotted the diagnostic plot for smooth and rough wall surfaces, as shown in Figures7.10(a)and 7.10(b). The same profiles used in Figures 7.9(a) and 7.9(b)are used here.

The linear relationship u^′/U = 0.286 - 0.255 U/U_∞, which matches the smooth wall data for U/U_∞ >0.6, was used as a reference. All the rough wall profiles were found to be higher than

0 0.5 1 1.5 2 2.5 3

10¹ 10² 10³

0 0.5 1 1.5 2 2.5 3

0 0.5 1 1.5

Figure 7.9: Mean streamwise turbulence intensity profiles over smooth and rough surfaces normalized by inner scaling in(a)and(c)and outer scaling in(b)and(d). The same surfaces used in Figure7.5(a)are plotted in (a)and (b). The same surfaces used in Figure7.5(b)are plotted in(c)and(d). The symbols of our measurements are detailed in Table7.2. Solid black line in(a)and(b): DNS smooth TBL data from Chan et al.Chanet al. (2021) at Reτ ≈2000.

The smooth wall data of Marusic et al.Marusic et al.(2015) is plotted with a black hexagram in (c)and(d).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0 0.2 0.4 0.6 0.8 1

Figure 7.10: (a) and (b): u^′ is normalized by U and plotted against U normalized by U_∞. (c)and (d): u^′is normalized byU√

H and plotted againstU normalized by U_∞. The symbols of our measurements are detailed in Table 7.2. The smooth wall data ofMarusic et al.(2015) is plotted with a black hexagram in(b). Solid black line in(a): DNS smooth TBL data from Chan et al. (2021) at Re_τ ≈ 2000. The solid straight magenta line corresponds to the linear

relationshipu^′/U =0.286 - 0.255U/U_∞ (Alfredssonet al.2012).

the smooth wall profiles. Djenidi et al. (2018) also noticed differences among smooth, 2-D bars and sand-grain roughness in the outer region of TBL when the diagnostic plots were used.

Our results revealed that rough surfaces with zero roughness skewness have the lowest u^′/U, while those with positive roughness skewness have the highest u^′/U. To improve the collapse of smooth and rough wall data in the diagnostic plot, we employed the shape factor, H, which was shown byAbdelazizet al. (2022b) to better collapse the data. In Figures 7.10(c) and 7.10(d), we used the parameter 1/√

H to “weight” the u^′/U profiles, resulting in a much better collapse. Although the physical explanation of why√

Henhances the collapse of different surfaces is yet to be determined and requires further investigation, this approach can be used to present u^′ conveniently for different rough surfaces.