Mean velocity profile

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

7.5 RESULTS AND DISCUSSION

7.5.2 Mean velocity profile

By referring to Table 7.1, the effect of the roughness skewness parameter on turbulence statistics can be isolated as the surfaces S24-06-06P48, S24-06-06N48 and S24-06-06 share similar roughness parameters except for the skewness value. The positive roughness skewness value of S24-06-06P48 is 0.48, whereas S24-06-06N48 has a negative roughness skewness of the same value of 0.48, and S24-06-06 has a zero roughness skewness value. Figure7.5demonstrates that all rough wall surfaces exhibit higher drag compared to a smooth wall, which is reflected in the downward shift of the mean streamwise velocity profiles over the three-dimensional sinusoidal roughnesses in comparison with a smooth wall.

5 10 15 20 25

10¹ 10² 10³ 10⁴

5 10 15 20 25

Figure 7.5: (a)Mean streamwise velocity profiles over our smooth wall data and 3D sinewave surfaces with constant ES = 0.2. The profiles over smooth and S24-06-06 surfaces at three different freestream velocitiesU_∞ are shown with the same symbols and different colours. Solid black line: DNS smooth TBL data from Chan et al. (2021) at Reτ ≈ 2000. (b) Comparison with velocity profiles over a smooth wall data ofMarusicet al.(2015), (the black hexagram) at matchedRe_τ =3000±300. The symbols of our measurements are detailed in Table7.2. N for negativek_sk,Z for zerok_skandP for positivek_sk. The blue symbols are atU_∞= 10 m/s. The

green symbols are atU_∞= 15 m/s. The red symbols are atU_∞ = 20 m/s.

Figure7.5(a)illustrates theU⁺profiles over the three rough surfaces and compares them with the smooth wall TBL inChan et al. (2021). The profiles overS24-06-06P48 andS24-06- 06N48 are only displayed at the highest Re_τ values. Increasing Reynolds number, by raising the freestream velocity U_∞, reduces the skin friction coefficient and extends the log region in the smooth wall TBL. However, for a given 3D sinusoidal surface, C_f is almost constant with increasing Reynolds number, and the profiles are shifted to the right to a larger y⁺. This indicates that a fully rough regime has been achieved, and the drag is predominantly caused by pressure drag. Figure 7.5(a) shows the profiles over S24-06-06 at three different Reynolds numbers for clarity of the figure, as the other profiles follow the same trend. The profiles of S24-06-06P48, S24-06-06 and S24-06-06N48 are compared, depicted as red symbols in Figure 7.5(a), to investigate the effect of the roughness skewness on the roughness function ∆U⁺. C_f and ∆U⁺ increase when roughness skewness increases from negative to positive values. S24-06- 06P48, with a positive roughness skewness value, has the largest ∆U⁺. ∆U⁺decreases slightly in S24-06-06, which has a zero roughness skewness value, and decreases further inS24-06-06N48, which has a negative roughness skewness value. The increase is more pronounced when moving from a negative to zero roughness skewness value than from zero to a positive one. ∆U⁺increases by 20% from S24-06-06N48 to S24-06-06, whereas the increase is only 10% from S24-06-06 to S24-06-06P48.

Figure7.5(b)displaysU⁺profiles over all the rough surfaces at matched Reynolds number Re_τ =3000±300. The profiles are compared with a smooth wall ofMarusic et al. (2015). Due to wind tunnel constraints, the same Reynolds number could not be achieved in our smooth wall measurements. Figure7.5(b)highlights the significant influence ofES on ∆U⁺reflected in the downward shift of the profiles whenES increases from 0.1 inS24-12-12 to 0.2 inS24-06-06 (filled symbols). The figure also demonstrates the effect of roughness skewness on ∆U⁺when the roughness skewness is the dominant roughness parameter and the changes in other parameters are neglected. Increasing k_sk from negative to positive values increases the downward shift of the velocity profiles. The lowest k_sk value of the surface S24-05-05N63, plotted in large red inverted triangles, has the minimum ∆U⁺ of 6.7. However, the highestk_sk value of the surface S24-05-05P63, plotted in large green triangles, has the maximum ∆U⁺ of 8.8.

The relationship between the shape factor H and C_f as a function of k_sk is plotted in Figures 7.6(a) and 7.6(b), respectively, in order to investigate the effect of k_sk on H and C_f. All of the 3D sinusoidal surfaces listed in Table 7.2 are plotted except for S24-12-12, which has a significantly lower ES compared to the other surfaces. The ES of all other surfaces is approximately 0.18±0.4.

As demonstrated in Figure7.6(a), increasing k_sk from negative to positive values leads to an increase inH. It is noted that for negative and zero roughness skewness, H is higher at the lowest Re_τ value compared to moderate Re_τ for the same surface (the green symbols are higher than the blue symbols). This could indicate that the flow regime is still in a transitional stage for the lowest and middleRe_τ values for negative and zero roughness skewness. However, for positive roughness skewness, the fully rough regime is achieved at lower Re_τ values and H increases with increasingRe_τ for the same surface.

It is also found that in the fully rough regime and with a constantES, the percent change of H from negative roughness skewness to zero roughness skewness is relatively small, around 1.5%. This value doubles from the zero roughness skewness value to the positive roughness skewness value, with an increase of around 3%.

The results in Figure7.6(b)demonstrate that the increase ink_skfrom negative to positive values leads to an increase in the value ofC_f. For a constant roughness surface, an increase in

1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 1.62

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

4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 10^-3

Figure 7.6: (a)The relation between H and k_sk for the different 3D sinewave surfaces that haveES≈0.18±0.4 at the three differentReτ values. (b)The relation betweenCf andkskfor the different 3D sinewave surfaces withES≈0.18±0.4 at the three differentReτ values. The symbols of the rough surface are indicated in Table7.2. N for negativeksk,Z for zeroksk and P for positive k_sk. the left area is for negative roughness skewness, while the right one is for positive roughness skewness values. The blue symbols are atU_∞ = 10 m/s. The green symbols

are atU_∞ = 15 m/s. The red symbols are atU_∞ = 20 m/s.

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

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Figure 7.7: (a) The relation between ∆U⁺andk_sk for the different 3D sinewave surfaces that haveES≈0.18±0.4 at the three differentRe_τ values. (b) The relation betweenk⁺_s andk_skfor the different 3D sinewave surfaces withES≈0.18±0.4 at the three differentRe_τ values. The symbols of the rough surface are indicated in Table7.2. N for negativek_sk, Z for zerok_sk and P for positiveksk. the left area for negative roughness skewness, while the right one is positive roughness skewness values. The blue symbols are atU_∞ = 10 m/s. The green symbols are at

U_∞= 15 m/s. The red symbols are atU_∞ = 20 m/s.

Re_τ slightly decreases C_f, regardless of the value of the roughness skewness. It is noted that in the fully rough regime and when ES is held constant, the percentage increase in C_f as k_sk transitions from negative to zero is more than three times the percentage increase inC_f ask_sk increases from zero to positive values.

The relationship between the roughness function ∆U⁺ and the normalized equivalent sand grain roughness k⁺_s with respect to the roughness skewness k_sk are presented in Figures 7.7(a)and 7.7(b), respectively. The figures exclude the 3D sinusoidal surface S24-12-12, which has a significantly lower ES than the other surfaces. As shown in Figure 7.7(a), there is a

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.