### Analysis of flow velocity profile and drag reduction rate

The spatial development of the boundary layer along stream-wise *x* direction in two cases (uncontrolled smooth flat-plate flow; riblets case) is plotted in Figs. 5 and 6. It shows that the momentum Reynolds number *Re*_{θ} grows approximately linearly along the streamwise direction.The riblets control brings about the increase of *Re*_{θ} (in the riblets region *x* = 22.5 ~ 25 with black dashed box in Figs. 5 and 6), which indicates the thickened boundary layer. The definition of shape factor *H*_{12} is the ratio of the displacement thickness *δ*_{d} to the momentum loss thickness *δ*_{θ} in the boundary layer. The riblets array promotes the shape factor significantly. Before and after riblets array, the shape factor experiences a process of first increasing and then decreasing. When it exceeds the riblets control area, the shape factor and the momentum Reynolds number will quickly return to the same state as that of the uncontrolled case. The larger the shape factor, the less filled the velocity profile, which predicts an enhanced drag reduction, as can be seen from the velocity distribution on the riblets wall.

Figure 7 shows the velocity distribution from the valley to the tip position of the riblets in the boundary layer, where the thickness of the outer layer of the boundary layer δ_{99} is taken as the dimensionless scale parameter and *U* denotes the time-averaged flow velocity. It can be seen that at *y*/δ_{99} > 0.06, the average velocity distributions at different locations in the riblets are the same. The area of influence that produces the velocity variation is mainly in the vicinity of the near-wall riblets and the flow velocity is greater at the valley compared to the tip at the same wall normal position *y*, which is consistent with the experimental observations of Vukoslavcevic, Wallace & Balint [41].

In order to investigate the flow field structure in the inner layer of the boundary layer under riblets control, it is necessary to analyze the mean velocity distribution in the turbulent boundary layer with the wall friction velocity *u*_{τ} and the viscous scale ν/*u*_{τ} as dimensionless parameters. Figure 8 compares the mean velocity *U*^{+}*-y*^{+} variation of the boundary layer for the smooth and riblets-controlled cases, using the local friction velocity *u*_{τ} and the viscous scale ν/*u*_{τ} at the respective locations of the riblets walls to normalize the time-averaged flow velocity *U* and the wall normal height *y*. The dots and solid lines in the figure indicate the velocity distribution curves of the turbulent boundary layer of smooth and trench flat plates, respectively, and the two dashed lines are the linear wall law *U*^{+} = *y*^{+} and logarithmic law curves *U*^{+} = 2.5ln(*y*^{+}) + 5.5, respectively. It can be seen that the log-law curves of the turbulent boundary layer at different positions from the valley to the tip of the riblets have different degrees of upward or downward shifts compared to the smooth flat-plate case, resulting in differences in the velocity magnitude of outer boundary layer *U*^{+}. Among them, the log-law curve is shifted downward near the wave peak position, and the maximum velocity of the outer boundary layer (*U*^{+} ≈ 21) is smaller; The log-law curve at the valley is shifted upward and its maximum velocity in the outer boundary layer reaches a maximum value of *U*^{+} ≈ 46. The upward or downward deviation of the log-law curve and the variation of the magnitude of the outer boundary layer velocity *U*^{+} characterize the difference in the friction velocity *u*_{τ} of the turbulent boundary layer, which leads to the difference in the friction drag on the surface of the flat-plate.

Due to the very small scale of the riblets, it is often difficult for the probe to penetrate into the riblets in engineering experiments, which makes it difficult to determine the wall friction. This involves the selection of the virtual origin of the wall and the determination of the friction velocity. In the numerical simulation of this paper, the tip and valley positions of the riblets are selected as the virtual origin of the riblet wall, and the average velocity profile distribution above the riblets is thus obtained, as shown in Fig. 9. Note that the curves do not include the viscous sublayer data because the velocity distribution of the viscous sublayer varies in the riblets spreading direction for different virtual origin conditions.

In contrast to the velocity profile of the turbulent boundary layer of a smooth flat plate at the same position of the flow direction (*Re*_{θ} = 2320, *Re*_{τ} = 500), the slope of the logarithmic law of the turbulent boundary layer changes very little and rises upward regardless of whether the virtual origin is chosen at the tip or the valley of the riblets. This indicates that the current configuration of the riblets brings about an increase in the thickness of the viscous sublayer at the wall and a decrease in the wall friction velocity, which lead the riblets control to a drag reduction effect. Luchini, Manzo & Pozzi [3], Choi, Moin & Kim [18], Park & Wallace [43] and Wang, Lan & Chen [44] also believe that the low-velocity fluid inside the riblet increases the thickness of the viscous sublayer of the boundary layer, which moves the buffer layer and log-law layer outward, and the fluid momentum and energy exchange near the wall are weakened after the boundary layer is elevated, which reduces the wall shear stress.

Based on the relationship equation of the wall local friction coefficient *c*_{f}, the riblets-controlled drag reduction rate DR of the flat-plate turbulent boundary layer flow is defined as follows:

$$c_{{text{f}}} = frac{{mu_{{text{w}}} left. {{{partial u} mathord{left/ {vphantom {{partial u} {partial n}}} right. kern-nulldelimiterspace} {partial n}}} right|_{{text{w}}} }}{{0.5rho_{infty } u_{infty }^{2} }},$$

(9)

$${text{DR}} = frac{{int_{{A_{{text{f}}} }} {c_{{{text{f0}}}} } – int_{{A_{{text{r}}} }} {c_{{text{f}}} } }}{{int_{{A_{{text{f}}} }} {c_{{{text{f0}}}} } }},A_{{text{f}}} = A_{{text{r}}} cos alpha,$$

(10)

where *c*_{f0} and *c*_{f} denote the friction coefficients of smooth and riblets-controlled flat plates at the same position, respectively, and *A*_{f} and *A*_{r} are the corresponding areas of the two cases.

In Fig. 10, the spatial development of the local friction drag coefficient shows the different drag-reduction effect by the current riblet configuration. Inside the riblet, the skin friction drag decreases in the comparison of the smooth flat-plate case, and it reaches the minimum at the valley position. Gradually upward from the valley, the skin friction drag starts to grow and exceed the basic friction drag, and finally reaches the maximum at the tip position. It shows the drag reduction is not available at any position of the riblet. Especially at the tip, the boundary layer might be enhanced in the loss of groove cavity protection. Similarly, after the riblets area, the local skin friction drag quickly recovered to be consistent with the smooth flat-plate case.

The spanwise distribution of the local friction drag coefficient at the riblet location (corresponding to the smooth flat plate *Re*_{θ} = 2320) is given in Fig. 11. It can be seen that since the riblets are periodically arranged in the spanwise direction, the variation of the local friction coefficient is also periodic. However, it is worth noting that there is a small difference in the magnitude of their fluctuation amplitudes. This is due to the fact that the riblets array does not fill the entire flat plate spanwise space, and the riblets-controlled turbulence boundary layer is affected by the smooth area flow field when approaching the edge position of the controlled region. At the same time, the wall friction reaches the maximum and minimum values at the tip and valley of the riblets, respectively, which indicates that the riblets valley acts as a natural depression barrier zone that facilitates the accumulation of low-speed airflow with a larger viscous sublayer thickness than the tip position.

In comparison with the wall local friction of smooth flat-plate case, the wall local friction coefficients of most positions on the riblets are smaller than those calculated from smooth plate, except for the tip positions of the riblets, which is slightly larger than that of the smooth flat-plate case at the same position. In order to investigate the drag reduction effect of riblets control on the turbulent boundary layer of the entire flat-plate wall surface, the area integral of the wall local friction is performed. Note that the wetted area of riblets plate is 1/cos(*α*) times larger than that of the flat plate, see Eq. (10). Thus, the plane- and time-average drag reduction can be obtained as the integral of wall friction coefficient on the wetted area. This results in a global drag reduction rate DR of 1.276% for the current riblets configuration (*s*^{+} ≈ 30.82, *h*^{+} ≈ 15.41), which is close to the value of 1.2% from Bechert, Bruse, Hage, et al. [4].

### Analysis of velocity fluctuation field

In order to visualize the effect of the riblets on the turbulent structure evolution process, the flow fluctuation velocity *u’* distribution of the spatially fully developed flat-plate turbulence from the smooth region to the riblets region and the downstream flow field is shown in Fig. 12. The dashed box is the riblets-controlled region, and the three normal cross sections correspond to the buffer layer (*y*^{+} = 8.43, 20.35) and the log-law layer (*y*^{+} = 39.22) of the turbulent boundary layer, respectively. Here, the viscous scale of the wall at *x* = 20 in front of the riblets is taken to be dimensionless.

It can be clearly seen that distorted turbulent fluctuation streaks fill the smooth region in front of and behind the riblets region near the wall buffer layer *y*^{+} = 8.43. As these structures flow through the riblets area, they are divided into a straight and elongated structure that flows neatly and orderly downstream along the riblets channel. After passing through the riblets area, the straight streaks structure starts to become distorted again. In addition, as it gradually moves away from the wall *y*^{+} = 20.35, the straight streaks above the riblets gradually start to deform and distort. At *y*^{+} = 39.22, the turbulent fluctuation structures in the smooth region before and after the riblets and in the riblets region are basically similar. This shows that the riblets have a strong rectification effect on the turbulent fluctuation structure in the near-wall region, but this rectification effect will gradually weaken along the normal direction of the wall until it finally disappears. This shows that the effect of the current micro-scale riblets on the turbulent boundary layer flow field at the wall is limited to the near-wall region.

As Fig. 13 quantifies and compares the root mean square (*u*_{rms}, *v*_{rms}, *w*_{rms}) distribution of turbulent fluctuation from the valley to the tip inside the riblets, where the riblets position in the streamwise direction is chosen to correspond to the smooth flat plate at *Re*_{θ} = 2320, and the dashed line is the result of the smooth flat-plate case. It can be seen that, regardless of the riblets or smooth flat-plate case, the flow direction positive stress is the largest among the three directions of Reynolds positive stress, that is, the streamwise turbulent fluctuation intensity is the largest.

The maximum value of *u*_{rms} in the smooth flat-plate case is obtained at *y*/δ_{99} ≈ 0.018 (corresponding to *y*^{+} ≈ 15). Comparing it with the turbulent fluctuation field after applying the riblets control shows that the fluctuation peak is obtained at a position elevated outward the wall. At the same time, the peak magnitude also decreases relative to the smooth flat-plate case, indicating that the turbulence intensity above the riblets decreases after the riblets control is applied. Comparing the turbulent fluctuation intensity at different positions inside the riblets, it can be found that the turbulent fluctuation intensity decreases from the valley to the tip of the riblets in all three directions at the same normal wall position *y*. However, beyond the peak point, the turbulent fluctuation variations at the tip and valley converge quickly. It is worth noting that the maximum value of turbulent streamwise fluctuation *u*_{rms} inside the riblets is not obtained at the tip or valley of the riblets, but somewhere between them. Despite this, the maximum value is still smaller than the peak turbulent fluctuation of the smooth flat-plate case.

Quadrant analysis of Reynolds shear stress according to the sign of *u*’ and *v*’ is good for understanding the evolution of turbulent kinetic energy production caused by riblets in the boundary layer. The first (*u*’ > 0, *v*’ > 0) and third (*u*’ < 0, *v*’ < 0) quadrant events contribute to negative turbulent kinetic energy production, and the second (*u*’ < 0, *v*’ > 0; ejection) and fourth (*u*’ > 0, *v*’ < 0; sweep) quadrant events contribute to positive production. Figure 14 shows that the contribution to the Reynolds shear stress (-overline{u^{prime} v^{prime}} / u_{infty}^{2}) from the second quadrant is the largest in the near-wall region from all four quadrants, which means the dominance of ejection event. At the riblet tip, midpoint and valley, all Reynolds shear stress from the ejection event (second quadrant) is reduced in the comparison of the smooth flat case. From the valley to the tip, the maximum value increases a little successively and their peak locations have a little outward shift. For the Reynolds shear stress of sweep events (fourth quadrant), there is a little decrease inside the riblet in the near-wall boundary layer. At *y*/δ_{99} > 0.1, the value of the smooth flat-plate case becomes a little smaller than that of the riblet flat-plate case, but quickly converges. For the first and third quadrant events, their negative contributions to the turbulent kinetic energy production are small in value relative to the ejection and sweep events. Moreover, the value of the Reynolds shear stress for both smooth and riblet flat cases has no significant change in the whole boundary layer except for a slightly fluctuation at *y*/δ_{99} < 0.06. Figure 15 shows the contour distributions of Reynolds shear stress *u*’*v*’ at two moments. The black dashed line denotes the height of the riblet tip. It is observed that there are a few eddies of greater intensity with large –*u*’*v*’ below the dashed line for the smooth flat-plate case, but almost no eddies inside the riblet. It is because the riblet not only acts as a barrier to the large-scale shear eddies, but also attenuates the ejection and sweep events. Thus, the interaction between the near-wall eddies and the wall under the influence of riblets is weakened.

### Vorticity fluctuation distribution

Figure 16 compares the distribution of streamwise vorticity fluctuation *w*_{x}’ at different moments of the *y*–*z* cross section in the riblets region (*x* = 24.12) for the smooth and riblets-controlled flat-plates cases. It can be seen that the streamwise vorticity fluctuation at the smooth flat plate case is moving close to the wall, while most of the streamwise vortices in the riblets cross-section appear above the riblets tips. This is related to the size of the riblets: when the riblets size cannot accommodate the streamwise vortices, the streamwise fluctuation vortices cannot interact directly with the riblets wall, so the shear stress on the wall decreases. Since the tip position can directly contact with the streamwise fluctuation vortices, the intensity of the streamwise fluctuation vortices at this position is also numerically significantly higher than that of other positions in the riblets.

Figure 17 quantifies the magnitude of the streamwise vorticity fluctuation inside the riblets, where |*w*_{x}’| takes the average of all the streamwise vorticity fluctuation at each observation point 1–5 inside the riblets at the spanwise corresponding positions. When the riblets control is applied, the peak of the streamwise vorticity fluctuation in the turbulent boundary layer decreases and the position of the peak point moves out of the wall when compared to the smooth flat-plate case. When *y*/δ_{99} > 0.1, the vorticity fluctuation distribution inside the riblets is slightly larger than that of the corresponding position of the smooth flat-plate case. At this time, the streamwise vorticity fluctuation along the normal direction of the wall has begun to show a decreasing trend, which indicates that micro-scale riblets lift up the streamwise vortex, weakening the intensity of the streamwise vorticity fluctuation in the near-wall area, but the intensity of the streamwise vorticity fluctuation in the outer layer of the boundary layer away from the wall increases slightly. In addition, it was found that the peak of the streamwise vorticity fluctuation tends to increase gradually from the valley to the tip (positions 1 to 5), with the smallest value of vortex pulsation at the valley position.

The distributions of the wall-normal vorticity fluctuation *w*_{y}’ are given in Fig. 18. Comparing with the results of the smooth flat-plate case, it can be seen that a pair of wall-normal vortices with opposite positive and negative values appears on both sides of almost each riblets tip, which are more neatly arranged in the whole spanwise riblets array space, while the wall-normal vortices of the smooth flat-plate case are also presented in the form of pairs, but the distribution is more scattered.

Figure 19 quantifies and compares the wall-normal vorticity fluctuation distribution for both smooth and riblets flat-plates case. It can be seen that when the riblets control is applied, the wall-normal vorticity fluctuation magnitude increases significantly and the peak point location is slightly higher than that of the smooth flat plate case. In addition, the maximum value of the wall-normal vorticity fluctuation is not obtained at the tip position because the wall-normal pairs of vortices are distributed on both sides of the tip. The value of |*w*_{y}’| decreases from position 2 to tip position 5, but the overall value is still larger than that of the smooth flat-plate case at the corresponding position.

Figure 20 provides a comparative analysis of the distribution of the spanwise vorticity fluctuation for riblets control. It can be found that the spanwise vorticity fluctuation on a smooth flat-plate case is slimmer in shape than that in both streamwise and wall-normal directions, and the intensity of the spanwise vorticity fluctuation gradually decreases along the wall outward the wall-normal direction. When the riblets control is applied, the vorticity fluctuation inside the riblets is no longer distributed monotonically decreasing along the wall direction, but first decreasing, then increasing and finally decreasing (see Fig. 21). In both near-wall and outer boundary layer regions, the spanwise vorticity fluctuation amplitude is generally larger than that of the smooth flat-plate case.

Table 1 specifically gives the comparison of the peak magnitude and maximum wall-normal offset of the vorticity fluctuation in three directions for both smooth and riblets-controlled flat-plates cases. It can be seen that due to the existence of riblets, the peak point of vorticity fluctuation in all three directions is raised outward in different degrees, and the magnitude of streamwise vorticity fluctuation is reduced by 20%, which is consistent with the conclusions of numerical simulation from Choi, Moin & Kim [18]. The difference is that the wall-normal vorticity fluctuation obtained from the simulations in this paper increased by 109% compared to the smooth plate case, but the wall-normal vorticity fluctuation simulated in the literature [18] was slightly reduced compared to the smooth plate case. This may be related to the form of vorticity fluctuation processing, where the vorticity fluctuation *w*‘ is calculated by first taking the absolute value and then averaging the results in the spanwise direction. Since the wall-normal vorticity fluctuation is closely presented in the form of positive and negative pairs of vortices in each riblets area, its vorticity fluctuation value will become cumulatively larger after the absolute value averaging process.

The existence of the microscale riblets structure makes the turbulent coherent structure in the near-wall region change, in which the lift of the vorticity distribution plays a great role. As stated by Dean & Bhushan [16], most of the near-wall vortices are present above the riblets and their influence area is confined to the tip of the riblets. This is consistent with Choi, Moin, & Kim [18] and Martin & Bhushan [13, 14] who suggested that the riblet acts as a lift up of streamwise vortex. The streamline distribution of the flat-plate turbulent boundary layer in the *y–z* section is shown in Fig. 22. It can be clearly observed that the streamwise vortices move close to the wall of the smooth flat-plate, and the intensity of the streamwise vortex in the near-wall area is also larger compared to the riblets-controlled one. Due to the barrier effect of the riblets, large-scale streamwise vortices cannot enter inside the riblets, resulting in a smaller amount of streamwise vorticity inside the riblets.

### Streak structures in the near-wall region

The previous analysis of the riblets flow fluctuation field shows that the rectification of the riblets splits the inner boundary layer flow into a spanwise periodic wall turbulence structure. The streamwise fluctuating turbulence can only flow in an orderly direction along the geometric expansion of the riblets, which limits the spanwise fluctuation of the turbulent flow. At the same time, the flow vortex cannot interact with the near-wall viscous bottom layer due to the lift up effect of the periodic riblets structure, which makes the direct effect of the streamwise vortices on the wall greatly weakened, and thus realizes the effect of wall-bounded turbulence drag reduction. As already pointed out in the analysis of the wall-bounded turbulence velocity profile, the role of the riblets is to lift the velocity distribution in the logarithmic region of the boundary layer upwards as a whole.

In order to visualize the evolution of the turbulent streaks in the near-wall region under riblets control, the three-dimensional contour surface distribution of the transient streamwise velocity of the flat-plate turbulence from the smooth region to the riblets region is shown in Fig. 23. It can be seen that when the flow velocity *u* is low (e.g., *u* = 0.06), the distribution of velocity iso-surfaces in the smooth region is relatively uniform, while the velocity contour surfaces in the riblets region are distributed in straight streaks, similar to the flow of low velocity fluid downstream along the riblets channel. When the flow velocity increases to 0.1, a few ripples appear on the velocity contour surface in the smooth region, indicating fluctuations and instability in the low velocity fluid, while the velocity distribution remains neat streaks on the riblets region and is not affected by upstream velocity fluctuations. When *u* ≥ 0.3, the flow velocity becomes more turbulent on the iso-surface regardless of the smooth area or the riblets area, and the influence of the ribelts on the flow velocity gradually becomes weaker. This indicates that the effect of the riblets is limited to the low velocity fluid near the wall, and makes the low velocity fluid flow along the riblets channel in an orderly manner, thus playing a rectifying role.

This can also be seen in the transient streamwise velocity distribution at *x–z* section shown in Fig. 24. The contour distribution in the near-wall area has many distorted streaks, but after the riblets area the streaks become slender and straight, which forms a sharp contrast with the flow field in the surrounding uncontrolled area. It is also noted that in the downstream region after the riblets control, the thin straight streaks do not immediately return to the turbulent distorted state as before the control region, but extend some distance downstream before finally gradually becoming a distorted and turbulent streak structure. Comparing the magnitude of the transient velocity *u* in the two cases, it can be found that the flow velocity *u* in the riblets area is significantly smaller than that of the smooth flat-plate case at the same location. This indicates that the riblets array provides a good “safe haven” for the low velocity fluid, which is conducive to the slow formation and accumulation of low velocity fluid. The velocity gradient in the middle of the riblets is lower than that in the smooth flat-plate case, implying a lower wall shear stress and therefore a drag reduction effect.

In the spanwise *z*-direction, the two-point correlation function *R*_{u’u’} with the streamwise velocity fluctuation can express the spanwise length scale of low-speed streaks. It is defined as follows,

$$R_{{u^{prime}u^{prime}}} left( {Delta z} right) = frac{{overline{{u^{prime}left( {x,y,z} right)u^{prime}left( {x,y,z + Delta z} right)}} }}{{u_{{{text{rms}}}}^{2} }}$$

(11)

and the streak spacing λ^{+} is twice the spanwise distance, at which the two-point correlation reaches a minimum. Figure 25 presents the distributions of spanwise two-point correlations at *y*^{+} = 8.43, 20.35 and 39.22. The streak spacing obtained from the two-point correlation for both the riblet case and the smooth case is basically at the same level. At *y*^{+} = 8.43, it yields 93.0 for the riblet wall and 95.3 for the smooth wall. It can be seen that the streak spacing by riblets control is slightly reduced, but not much, indicating that the spanwise streak spacing is less affected by riblets. As it moves away from the wall from *y*^{+} = 8.43 to *y*^{+} = 39.22, the streak spacing gradually increases. Moreover, the two-point correlation will gradually tend to 0 at an increasing Δ*z*^{+}, which is due to the decrease of streamwise fluctuation in correlation.

The distributions of the three-dimensional iso-surface for the second-order invariant for the velocity gradient tensor *Q*_{2} = 20, 50, 80, 110 in both riblets-controlled and smooth-plate cases are given in Figs. 26 and 27. The color indicates the magnitude of the instantaneous velocity *u*. The green box indicates the area where the riblets are located. It can be seen that the 3D vortices gradually become less as the *Q*_{2} increases. For those 3D vortices in the near-wall region, their shapes are pulled long and straight as they pass through the riblets region. In an orderly manner, those 3D vortices move along the riblet array in the streamwise *x*-direction. The streamwise velocity of those straight 3D vortices is smaller than that of the surrounding vortices, indicating that the riblets in the near-wall area play a rectifying and slowing effect on the vortex motion. However, for the smooth-plate case, the near-wall area is filled with twisted and disordered vortices with different sizes. The velocity of the near-wall vortices is significantly larger in the case of the smooth-plate case than in the case of the riblets case, which shows a slow near-wall turbulent flow.

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