3.1. Manipulation of the LSS

The wall-parallel PIV data allowed the investigation of how the surface roughness manipulates the turbulent structures that are typical of the boundary layer. In particular, the focus of this investigation was the analysis of the LSS.

The first step of the investigation was concerned with the detection of the LSS. It is worth mentioning that for the high-speed streaks, a similar procedure can be applied.

The detection of the LSS was based on the definition of a threshold on the local value of the streamwise velocity, as suggested by Iuso et al. (Reference Iuso, Di Cicca, Onorato, Spazzini and Malvano2003) and Schoppa & Hussain (Reference Schoppa and Hussain2002). Regions of streamwise fluctuating velocity $u(x_i,z_i) < -u_{thr}$ are first identified in the image. For the present investigation, $u_{thr} = 0.5 u_\tau$. Within each identified region, the ($x_c, z_c$) locations of local minima of $u$ in $z$ are identified as streak centres. The $z$ positions where $u$ exhibits a zero-crossing are recognized as the streak borders. Furthermore, the spanwise distance between the centroids of the two closest LSS detected within a snapshot is defined as the streaks’ spacing $\lambda _z$. The procedure is then repeated for all the streaks educed from each snapshot.

Different approaches were also tested, such as the one described by Lin et al. (Reference Lin, Laval, Foucaut and Stanislas2008) based on the ratio between the mean and the standard deviation of the streamwise velocity, which led to similar results.

As evidenced in previous investigations (Choi Reference Choi2002; Iuso et al. Reference Iuso, Di Cicca, Onorato, Spazzini and Malvano2003; Quadrio & Ricco Reference Quadrio and Ricco2004; Ricco Reference Ricco2004; Yao, Chen & Hussain Reference Yao, Chen and Hussain2019), active flow control techniques such as the spanwise oscillation of the wall can lead to significant variations in the statistical distribution of the near wall streaks. Variations are also expected in the present investigations, although more limited due to the fact that the control technique is passive, and as such, it yields much lower values of drag reduction.

Following the eduction approach described above, the streaks’ spacing $\lambda _z$ and their width $w_S$ are analysed initially. A statistical description of the two parameters, streaks’ spacing ($\lambda _z^+$) and width ($w_S^+$), is instrumental to determine the effect of the wall manipulation on the structure of the boundary layer. It is worth mentioning explicitly that the quantities are normalized in inner units using the values of the friction velocity calculated for each case from a Clauser fit applied to the logarithmic region of the mean velocity profile measured by Cafiero & Iuso (Reference Cafiero and Iuso2022). In figure 3, the probability density functions (p.d.f.s) of $\lambda _z^+$ and $w_S^+$ obtained from the Smooth, RLong, RS1 and RS2 cases at the two investigated values of $Re_\theta$ are reported. At the lowest value of $Re_\theta$, corresponding to $s^+\approx 7$, the effect of the manipulation is only weakly evidenced on the distributions of the LSS spacing ($\lambda _z^+$), as reported in figure 3(a). In particular, the effect of the ribbed geometry, regardless of the specific manipulation, is such that larger values of the streaks’ spacing are less probable. The peak of the p.d.f., which is found at $\lambda _z^+\approx 130$ for the Smooth case, shifts towards smaller values at $\lambda _z^+\approx 110$. The limited variations of the p.d.f. are in good agreement with the direct measurements of the friction drag reported in table 3. Indeed, the RS cases yielded small values of the drag reduction, with similar values attained also with the RLong case.

Figure 3. Probability density function of the streaks’ (a,c) spacing and (b,d) width. Data are collected at (a,b) $Re_\theta =2200$ and $y^+=35$, (c,d) $Re_\theta =2900$ and $y^+=50$.

Similar considerations can also be drawn for the streaks’ width (figure 3b), at $Re_\theta =2200$. The main difference in the p.d.f.s can be found in the fact that ribbed geometries yield greater probabilities of streaks width in the range of values between 25 and 45. The p.d.f.s of both spacing and width show a similar behaviour, with a horizontal shift of the curves for all the manipulated cases towards lower values of both $\lambda _z^+$ and $w_S^+$.

At $Re_\theta =2900$, corresponding to $s^+\approx 11$, the sinusoidal riblets were found to be more effective in terms of drag reduction than the longitudinal ones, as documented in table 3. The effectiveness in terms of drag reduction is also reflected in a more prominent manipulation of the LSS.

The p.d.f.s of the streaks’ spacing are reported in figure 3(c). Both the Smooth and RLong cases show a peak at $\lambda _z^+\approx 110$, which is in good agreement with the literature on the topic. The sinusoidal cases, instead, show a rather different behaviour. The p.d.f. is skewed towards smaller values of $\lambda _z^+$, peaking at $\lambda _z^+\approx 110$. Small differences are instead evidenced between RS1 and RS2 cases. Moreover, the streaks’ spacing values greater than $\lambda _z^+\approx 150$ are less probable in the sinusoidal cases.

The p.d.f.s of the streaks’ width obtained at $Re_\theta =2900$, corresponding to $s^+\approx 11$, are reported in figure 3(d). While the Smooth and RLong cases feature distributions of the width that are not significantly different from those reported in figure 3(b), this is not so for the sinusoidal cases. In particular, the p.d.f. exhibits a plateau in the range of $w_S^+$ between 25 and 45, with a corresponding lower probability of finding streaks with values of $w_S^+$ greater than 50.

On the basis of the results presented above, a first summary of the effect of the manipulation can be drawn. In the sinusoidal cases, the LSS are closer together and thinner. This effect increases at values of $s^+$ where the RS cases are more effective in reducing drag. This effect will be linked to the drag reduction mechanism later in the paper.

The conditional averages of the mean streamwise velocity and wall-normal vorticity profiles, calculated across the streaks, can provide further evidence of the topology of the streaky structures.

Figure 4 shows the conditionally averaged streamwise velocity fluctuations ($\hat {u}^+$) and wall-normal vorticity ($\hat {\omega }_y^+$) profiles. Starting from the streak's centre ($z_c$) at all the streamwise locations corresponding to a streak, the streamwise velocity and wall-normal vorticity values in the range $-70< z-z_c<70$ are stored and then averaged over all the educed streaks.

Figure 4. Conditionally averaged (a,c) streamwise velocity fluctuations and (b,d) wall-normal vorticity profiles, calculated across the LSS. Data are collected at (a,b) $Re_\theta =2200$ and $y^+=35$, (c,d) $Re_\theta =2900$ and $y^+=50$.

The profiles are plotted in a reference frame that is centred on the streak, $(z-z_c)^+$. Here and in the following, the hat symbol will be used to indicate that the average is performed on points belonging to the identified LSS.

At $Re_\theta =2200$, both the streamwise velocity and the wall-normal vorticity profiles do not show significant differences between the investigated cases. Figure 4(a) shows that all cases exhibit a velocity deficit associated with the presence of the LSS, at least concerning RS2. The RS cases are characterized by a narrower profile, attaining a plateau at smaller values of $|(z-z_c)^+|$. The wall-normal vorticity profiles reported in figure 4(b) highlight these differences further. The sinusoidal cases are characterized by a steeper slope of the $\hat {\omega }_y^+$ profile at $(z-z_c)^+=0$. Furthermore, even though only limited differences are detectable, the spanwise distance between the peaks is reduced in the RS cases, consistently with results reported in figure 3(b). Indeed, this distance can be associated with the typical streaks’ width.

At $Re_\theta =2900$, which corresponds to $s^+\approx 11$, much clearer differences can be detected between the investigated cases. The sinusoidal cases evidence a stronger velocity deficit in the centre of the streak (figure 4c). Correspondingly, the above-mentioned narrower profile of the streak is further evidenced at values of $s^+$ yielding higher values of drag reduction (see table 3).

The conditional averaged wall-normal vorticity profiles (figure 4d) show these differences further. In particular, the sinusoidal cases exhibit a much steeper gradient at the centre of the streak, associated with a narrower velocity profile. Differently from the case at a lower Reynolds number, the sinusoidal case also evidences larger values of the peak with respect to the Smooth and RLong cases. Even though the differences between the RS1 and RS2 cases are small, the results seem to suggest an effect of the amplitude of the sinusoidal pattern; nevertheless, a broader parametric space, with more investigated amplitudes $a$, is needed to support this observation. The results of figures 4(b) and 4(d) show that the sinusoidal riblets enhance the wall-normal vorticity at the streaks’ edges, with corresponding large values of in-plane shear.

Further insights into the effect of the riblet geometry can be evidenced by analysing the p.d.f. of the modulus of the in-plane projection of the velocity gradient, $|\boldsymbol {\nabla } \hat {u}|_{x,z}$, calculated in correspondence of the LSS edges. Figures 5(a) and 5(b) further support the conclusions that have been drawn when discussing the conditional velocity and vorticity profiles.

Figure 5. Probability density functions of the modulus of the velocity gradient ($|\boldsymbol {\nabla } \hat {u}|_{x,z}$) projected on the $x$–$z$ plane calculated at the streaks’ edges, for (a) $Re_\theta =2200$ and $y^+=35$, (b) $Re_\theta =2900$ and $y^+=50$.

The riblet cases at $Re_\theta =2200$ (figure 5a) show a shift towards greater values of the peak of the modulus of the velocity gradient. In particular, the RLong, RS1 and RS2 cases feature an increased probability of $|\boldsymbol {\nabla } \hat {u}|_{x,z}>0.12$. This suggests that the riblet cases are generally characterized by more intense shear regions; furthermore, the amplitude of the sinusoidal geometry seems to introduce even stronger values of the velocity gradients.

At larger values of the Reynolds number, $Re_\theta =2900$, the RLong case features a behaviour that closely resembles the Smooth case. Conversely, the p.d.f.s of $|\boldsymbol {\nabla } \hat {u}|_{x,z}$ in the sinusoidal cases are characterized by a shift of the peak towards greater values of $|\boldsymbol {\nabla } \hat {u}|_{x,z}$, in agreement with the larger values of wall-normal vorticity evidenced in figure 4.

The p.d.f. of the conditional wall-normal vorticity calculated at the edges of the LSS is reported figure 6. At $Re_\theta =2200$, the effect of the riblet geometry is such that smaller values of vorticity become less probable, with a corresponding increment of the probability associated with the tails of the p.d.f., namely for $|\hat {\omega }_y^+|>0.15$. Correspondingly, the peak of the p.d.f. shows a progressive shift towards larger values of $|\hat {\omega }_y^+|$ when the manipulated cases are considered.

Figure 6. Probability density functions of the wall-normal vorticity $\hat {\omega }_y$ calculated at the streaks’ edges. Data are collected at (a) $Re_\theta =2200$ and $y^+=35$, (b) $Re_\theta =2900$ and $y^+=50$.

At $Re_\theta =2900$, the RS cases show a much clearer distinction with respect to both the Smooth and RLong cases. In particular, the RS cases lead to a shift in the maxima of the p.d.f. towards greater values of $|\hat {\omega }_y^+|$. In addition to this, a higher probability of occurrence of values $|\hat {\omega }_y^+|>0.08$ is detected, with a trend that seems to suggest the effect of the amplitude of the sinusoidal riblet. On the other hand, the RLong case features a p.d.f. of the wall-normal vorticity that closely resembles that of the Smooth case.

Schoppa & Hussain (Reference Schoppa and Hussain2002), using transient growth analysis of the streaks, have shown that large values of $\omega _y$ are generally associated with streaks that are more prone to instability, and as such give rise to streamwise vortices. In the present case, the greater values of wall-normal vorticity can be responsible not only for the streaks’ instability but also for their fragmentation and/or weakening.

Table 4 summarizes the number of educed streaks for each of the investigated cases. In particular, the results are presented in the form of percentage variation with respect to the Smooth case ($\Delta N_{S} = 100\times (N_{S} - N_{S}^{S})/N_{S}^{S}$), where $N_{S}$ indicates the number of streaks in the ribbed cases, and $N_{S}^{S}$ the number of educed streaks in the Smooth case. The results seem to suggest that there is a fundamentally different behaviour when comparing the RLong with the RS cases. The latter indeed show a consistent increase in the number of the streaks with respect to the Smooth case. At larger values of the Reynolds number, which correspond to higher effectiveness of the riblets in terms of drag reduction, the number of educed streaks increases.

Table 4. Percentage variation of the number of educed LSS. The smooth case is taken as a reference.

The results presented in figures 3 and 4, and table 4, allow us to further elaborate on the mechanism by which sinusoidal riblets manipulate the near-wall turbulence. The sinuous geometry leads to streaks that are characterized by larger values of the wall-normal vorticity. This makes the LSS more prone to instability, thus causing their fragmentation. The fragmentation is reflected in two different aspects: (i) more streaks are educed in the RS cases; (ii) the mean spacing between streaks, as well as their width, is reduced in the RS cases.

Cafiero & Iuso (Reference Cafiero and Iuso2022) suggested a possible mechanism leading to the drag reduction evidenced by the sinusoidal riblets, which was associated with the imprint of the geometry manipulation onto the flow field. The spatial periodic inversion triggered by the sinusoidal geometry yields secondary flows and a cross-stream velocity component with an inverted direction at each bend. This mechanism leads to enhanced wall-normal vorticity, along with possible effects on the streamwise and spanwise vorticity that need further analysis in future investigations. The results presented in this paper support the previous hypothesis, linking the fragmentation of the LSS to the spatial variation of vorticity.

A statistical analysis based on two-point correlations of the velocity fluctuations in the wall-parallel ($x$–$z$) plane will serve the purpose of elucidating the effects of the wall manipulations on the structure of the LSS.

The two-point correlation between any two quantities $A$ and $B$ is defined as

(3.1)\begin{equation} R_{AB}(\Delta x, \Delta z)= \frac{\overline{A(x,z)\,B(x+\Delta x,z+\Delta z)}}{\sigma_A \sigma_B}, \end{equation}

where $\sigma _A$ and $\sigma _B$ are the standard deviations of $A$ and $B$, respectively, and $\Delta x$ and $\Delta z$ are the in-plane separations between the two components. The overbar notation indicates the ensemble average operation over the acquired snapshots. It is worth mentioning that the current analysis is carried out considering that the location at which the correlation is computed is beyond the roughness layer, and as such, homogeneity can be assumed in $x$ and $z$. Following Cheng et al. (Reference Cheng, Wong, Hussain, Schröder and Zhou2021) and Modesti et al. (Reference Modesti, Endrikat, Hutchins and Chung2021), we estimate the value of the roughness layer as $\delta _r^+ = 0.62 s^+$, which in the most limiting case, $Re_\theta =2900$, corresponds to $\delta _r^+=6.57$.

Figure 7(a) shows the two-point correlation $R_{uu}$ calculated at $Re_\theta =2900$ and $y^+=50$, corresponding to the case where the sinusoidal riblets yield the largest values of drag reduction. The contour shows the representative Smooth case, whilst figure 7(b) shows the comparison between the four investigated cases for two generic values of the contour line ($R_{uu}=0.5$ solid line, and $R_{uu}=0.2$ dashed line) for the four investigated cases.

Figure 7. (a) Contour plot of the normalized two-point correlations $R_{uu}$ calculated for the Smooth case. (b) Comparison between the four investigated cases for two generic contour lines: $R_{uu}=0.5$ (solid line) $R_{uu}=0.2$ (dashed line). Data are collected at $Re_\theta =2900$ and $y^+=50$.

The Smooth case exhibits iso-lines of the correlation that are elongated in the streamwise direction, which are characteristic of the existence of the low- and high-speed streaks. The RLong case shows a behaviour that is comparable with the Smooth case, with a more elongated correlation lobe, and same width; this result is in good agreement with the data shown in figure 3, which reported values of the streaks’ spacing and width that are similar between the Smooth and RLong cases. It is important to point out that the streamwise extent of the iso-lines of correlation is limited by the size of the illuminated region (see § 2). Therefore, it is not surprising if the extent of the correlation map in the streamwise location is sharply interrupted, at least for the Smooth case, in correspondence with $\Delta x^+\approx 1000$.

The sinusoidal cases show an altered structure of the two-point correlation maps: both the streamwise and spanwise extents of the correlation lobe are indeed reduced. The amplitude of the riblet seems to play a role in how the correlation map is modified: larger values of the amplitude correspond to reduced extents of the correlation maps. This is confirmed quantitatively in table 5, where the values of the streamwise integral length scale ($L_x^+$) are calculated by integrating the $R_{uu}$ correlation map (figure 7) at $\Delta z^+=0$ and reported. In this case, the RLong case yields values of $L_x^+$ greater than those attained by the Smooth wall (about 17.1 $\%$). On the other hand, both sinusoidal cases feature a reduction in the streamwise extent of the correlation, of 6.9 $\%$ and 25.4 $\%$ for RS1 and RS2, respectively. A possible explanation for the opposite behaviour between the RLong and RS cases might be associated with a different underlying physics that leads to the drag reduction. The same trend is indeed retrieved also in the values of the spanwise correlation length $L_z^+$, calculated by integrating the $R_{uu}$ correlation map (figure 7) at $\Delta x^+=0$. The effect of the surface manipulation on $L_z^+$ is quite striking, with the sinusoidal cases featuring a significant reduction. This behaviour is consistent with the manipulation of the near-wall structures by means of the surface geometry, which in turn leads to attenuated and less energetic structures. A possible interpretation for this result can be found in the mechanism leading to the fragmentation of individual structures when manipulated. The effect of the ribbed surface is indeed such that LSS are weakened and, as already pointed out when discussing the p.d.f. of the streaks’ spacing ($\lambda _z^+$), brought closer together. This is associated with the reduced values of $L_z^+$. This mechanism is represented schematically in figure 8.

Figure 8. Schematic representation of the fragmentation occurring in the case of sinusoidal manipulation.

Table 5. Values of the longitudinal ($L_x^+$) and lateral ($L_z^+$) integral length scales calculated for the four investigated cases at $Re_\theta =2900$ and $y^+=50$.

The effect of the sinusoidal geometry on the structure of the turbulent boundary layer is further confirmed by the analysis of the $R_{uw}$ correlation maps (figure 9). The contour plot shows two lobes of correlation of opposite sign, located on either side of $\Delta z^+=0$. The positive (negative) correlation lobe is located at $\Delta z^+>0$ ($<0$). For the representative condition of LSS ($u<0$), the positive lobe of correlation corresponds to a negative spanwise velocity ($w<0$), i.e. an entrainment towards the centre of the streak. An opposite behaviour can be envisaged instead for the negative correlation lobe. The distance between the two peaks indicates a measure of the streaks’ width. The results are indeed consistent with the description: the spanwise extent of the correlation lobe reduces for the manipulated cases, with RS2 attaining the smallest size.

Figure 9. (a) Contour plot of the normalized two-point correlations $R_{uw}$ calculated for the Smooth case. (b) Comparison between the four investigated cases for one generic contour line, $R_{uw}=\pm 0.04$ (solid/dashed line, respectively). Data are collected at $Re_\theta =2900$ and $y^+=50$.

3.2. The VISA conditional averages

The confirmation of the fragmentation mechanism can be sought in the conditional averages calculated across the edges of the low- and high-speed streaks. The edges of the streaks are characterized by strong velocity gradients in the streamwise and spanwise directions, and they are generally associated with values of the local variance of the velocity that are greater than the variance calculated over the entire dataset. When the latter condition is verified, an event is identified.

The approach that has been followed is based on VISA (Johansson, Alfredsson & Kim Reference Johansson, Alfredsson and Kim1991), which represents the spatial counterpart of the variable interval temporal average (VITA) (Blackwelder & Kaplan Reference Blackwelder and Kaplan1976). Each instantaneous image is divided into a structured grid. The size of each cell of the grid is then defined, and for the present investigation, a streamwise extent $\Delta \tilde {x}^+=200$ and spanwise extent $\Delta \tilde {z}^+=100$ have been considered, corresponding to numbers of grid points $n_x$ and $n_z$ in the streamwise and spanwise directions, respectively. At each index $(i,j)$ of the grid, the value of the local variance of the streamwise velocity is computed as

(3.2)\begin{equation} \overline{{u}^2}^+_{loc} = \frac{1}{n_x n_z}\sum_{i=1}^{n_x} \sum_{j=1}^{n_z} (u^+(i,j))^2-\left(\frac{1}{n_x n_z}\sum_{i=1}^{n_x} \sum_{j=1}^{n_z} u^+(i,j)\right)^2, \end{equation}

where $\overline {{u}^2}^+_{loc}$ indicates a local average, calculated over the cell of extent $\Delta \tilde {x}^+$, $\Delta \tilde {z}^+$. For each instantaneous image, a corresponding map of the local variance is obtained. Locally, the points that are characterized by values of the local variance $\overline {{u}^2}^+_{loc}$ greater than $k$ times the global variance $\overline {u^2}^+$ (i.e. calculated over all the snapshots) are identified as VISA events:

(3.3)\begin{equation} \overline{{u}^2}^+_{loc}>k \, \overline{u^2}^+.\end{equation}

For the present investigation, $k$ has been set to 1.

Di Cicca et al. (Reference Di Cicca, Iuso, Spazzini and Onorato2002a) argued that the VISA events that satisfy the condition of (3.3) are generally found at the edges of the streaks since those regions are characterized by large values of intermittency. Figure 10 shows a representative instantaneous realization of the flow field with overlaid locations of the VISA events. The locations of the events indeed correspond to the regions of intense shear at the edges of the streaks.

Figure 10. The VISA events detected on a representative instantaneous realization of the flow field overlaid on the colour map of the streamwise velocity fluctuation.

Among all the identified events, further conditioning is then applied. In particular, the VISA events that are characterized by $\partial u/\partial x<0$ and $\partial u/ \partial z>0$ are selected. The selection of these events is purely arbitrary; indeed, by combining the streamwise and cross-stream derivatives of the streamwise velocity component, it is possible to reconstruct all the flanks of a sinuous streak.

Each of the identified events is then averaged considering a window centred on the event location, of extent $\Delta \tilde {x}^+$ in the streamwise direction and $\Delta \tilde {z}^+$ in the cross-stream direction. Figure 11 shows the averaged VISA event of the cross-stream velocity component obtained for the Smooth, RLong, RS1 and RS2 cases with overlaid streamlines. The source of the streamlines is the same for all cases.

Figure 11. VISA conditionally averaged spanwise velocity ($\tilde {w}^+$) for events characterized by $\partial u/\partial x<0$ and $\partial u/\partial z>0$, with overlaid streamlines, for cases (a) Smooth, (b) RLong, (c) RS1, and (d) RS2. Data are collected at $Re_\theta =2900$ and $y^+=50$.

Here and in the following, a tilde will be used to indicate VISA quantities. In all cases, the contour shows a region of intense shear, with values of $\tilde {w}^+$ that change in sign when passing from $\tilde {x}^+<0$ to $\tilde {x}^+>0$, or similarly from $\tilde {z}^+<0$ to $\tilde {z}^+>0$. This leads to the consideration that the centre of the event is a region of the flow where the streamlines roll up, hence leading to large values of the wall-normal vorticity. Further to this, it can be argued that the manipulated cases show streamlines that are denser than the Smooth case, which can be addressed directly to the larger values of the wall-normal vorticity.

The VISA conditioned profiles of the streamwise ($\tilde {u}^+$) and cross-stream ($\tilde {w}^+$) velocity components extracted at $\tilde {z}^+=0$ are reported in figures 12(a,b). As could also be inferred from the contour plots in figure 4, the effect of the manipulation is such that the ribbed geometry leads to a less intense gradient of the streamwise velocity component compared with the Smooth case. On the other hand, there is no significant difference among the manipulated cases. The cross-stream VISA profiles measured in the streamwise direction show instead an opposite trend with respect to those of figure 12(a). All the ribbed cases display an enhancement of the peak-to-peak value of $\tilde {w}^+$, with the longitudinal case featuring the largest values. This result seems to be at odds with results reported in § 3.1, when describing the effect of the sinusoidal riblets on the wall-normal vorticity. However, the leading term associated with the wall-normal vorticity is $(\partial \tilde {u}/\partial z)^+$, with $(\partial \tilde {w}/\partial x)^+$ being instead significantly smaller.

Figure 12. The VISA conditionally averaged streamwise ($\tilde {u}^+$) and spanwise ($\tilde {w}^+$) velocity profiles extracted at (a,b) $z^+=0$ and (c,d) $x^+=0$ for VISA events characterized by $\partial u/ \partial x<0$ and $\partial u/\partial z>0$, with overlaid streamlines. Data are collected at $Re_\theta =2900$ and $y^+=50$.

To confirm that the results are indeed consistent, we show the joint p.d.f. of the velocity gradients $(\partial \tilde {u}/\partial z)^+$, $(\partial \tilde {u}/\partial x)^+$ calculated for the identified events in figure 13(a). In particular, the spanwise gradient is linked to the wall-normal vorticity, while the streamwise one relates to the most energetic VISA events. The manipulated cases are characterized by greater velocity gradients $(\partial \tilde {u}/\partial z)^+$ in the spanwise direction, with a corresponding attenuation of the streamwise velocity gradient $(\partial \tilde {u}/\partial x)^+$. This is identified more clearly in figure 13(b). Indeed, consistently with the larger values of wall-normal vorticity evidenced at the streak edges by the RS cases, the p.d.f. shows that despite the similarity between the manipulated and Smooth cases, the RS ones yield a higher probability of large values of $(\partial \tilde {u}/\partial z)^+$.

Figure 13. (a) Joint p.d.f. of the gradient of the VISA conditioned streamwise velocity calculated along the streamwise ($(\partial \tilde {u}/\partial x)^+$) and cross-stream ($(\partial \tilde {u}/\partial z)^+$) directions at the centre of each VISA event for the (i) Smooth, (ii) RLong, (iii) RS1 and (iv) RS2 cases. (b) Marginal p.d.f. of $(\partial \tilde {u}/\partial z)^+$. Data are collected at $Re_\theta =2900$ and $y^+=50$.

3.3. The VISA conditioned turbulent kinetic energy budget

The identification of the VISA events is further exploited to elucidate the contribution of the LSS to the production term of the turbulent kinetic energy equation. The planar PIV available for the current experiments provides us access to only some of the terms involved in the turbulent kinetic energy budget, which can be formulated as follows:

(3.4)\begin{equation} \underbrace{\frac{\bar{U}_k}{2}\,\frac{\partial \overline{q^2}}{\partial x_k}}_{\mathcal{A}} = \underbrace{-\overline{u_i^{\prime} u_j^{\prime}}\,\frac{\partial \bar{U}_i}{\partial x_j}}_{\mathcal{P}}- \underbrace{\frac{\partial}{\partial x_k} \left( \frac{\overline{u^{\prime}_k q^2}}{2}+\frac{\overline{u^{\prime}_k p}}{\rho} \right)}_{\mathcal{T}} + \underbrace{\frac{\nu}{2}\,\frac{\partial^2 \overline{q^2}}{\partial x_m^2}}_{\mathcal{D}_v}-{\varepsilon},\end{equation}

with $q^2={u^{\prime }_i}^2$. Here, $\mathcal {A}$, $\mathcal {P}$ and $\mathcal {T}$ indicate the advection, production and transport of turbulent kinetic energy, while $\mathcal {D}_v$ and $\varepsilon$ indicate the viscous diffusion and dissipation due to the fluctuating rate of strain. For the present case, given the value of the Reynolds number, $\mathcal {D}_v$ is deemed to be negligible with respect to $\varepsilon$. The turbulent transport $\mathcal {T}$ is constituted of the triple velocity correlation $\overline {u^{\prime }_k q^2}$ and the pressure velocity correlation $\overline {u^{\prime }_k p}$.

The purpose of this subsection is to highlight the effect of the surface manipulation on the production term of the turbulent kinetic energy equation. The analysis is carried out on the detected VISA events, in order to focus on how the production and transport are affected by the surface manipulation, limited to the streaks’ edges. Given the limited spatial resolution, along with the absence of some of the terms measured with the current PIV set-up, it was not possible to provide information about the dissipation term ($\varepsilon =2 \nu (\overline {s_{ij}s_{ij}})$). On the other hand, considering the wall-normal distance at which the measurements are taken, namely $y^+=50$, the transport term attains values that are close to zero (Pope Reference Pope2000) and as such can be not focal for the present investigation.

Figure 14 shows the production term ($\tilde {\mathcal {P}}$) calculated for the Smooth, RLong, RS1 and RS2 cases from the VISA conditioned flow field, overlaid with the iso-lines of $\tilde {u}^+=\pm 0.5$. The effect of the surface manipulation is immediately visible in the distribution of the production term. In particular, the introduction of the ribbed surface leads to the attenuation of the production term, regarding both positive and negative terms. All the cases are characterized by two intense lobes, mostly contained within the streak. The identified iso-lines are indeed representative of the shear region between a low-speed streak and a high-speed streak: given the specific conditioning that is considered for the VISA events, which takes into account those events that are characterized by $\partial u/\partial z>0$ and $\partial u/\partial x<0$, values $\tilde {z}^+>0$ correspond to locations that are outside the LSS. Values $\tilde {z}^+=0$, i.e. the centre of the VISA events, are indeed associated with the streaks’ edge.

Figure 14. Colour map of the VISA conditionally averaged production ($\tilde {\mathcal {P}}$), overlaid with the isolines $\tilde {u}^+=\pm 0.5$, for the (a) Smooth, (b) RLong, (c) RS1 and (d) RS2 cases. Data are collected at $Re_\theta =2900$ and $y^+=50$.

The analysis of the individual terms contributing to the turbulence production, reported here only for the Smooth case for the sake of brevity, shows that the leading term in the turbulence production is $\tilde {\mathcal {P}}\approx \tilde {\mathcal {P}}_{xx}=- \overline {uu}\,\partial \bar {u}/\partial x$, while the remaining three terms (considering the available PIV measurements) are at least one order of magnitude smaller. It must be noted explicitly that the investigated plane neglects the production term that is associated with the strongest component of shear ($\partial \bar {u}/\partial y$) and shear stress ($\overline {uv}$).

The attenuation of the turbulence production in the wall-parallel plane can be ascribed to the already highlighted fragmentation of the near-wall LSS, in the cases of the sinusoidal riblets. In particular, as evidenced in figure 14(a), the Smooth case yields intense regions of positive/negative production at the streak boundaries, associated with the more intense streamwise and cross-stream shear rates in the conditional VISA events of figure 12.

All the riblet cases (figures 14b–d) are characterized by an attenuation of the turbulence production (both positive and negative). It is interesting to notice how despite there being only small differences between the longitudinal and sinusoidal cases, RLong (figure 14b) features a more intense protrusion of the turbulence production within the core of the LSS, i.e. towards $z^+<0$ with respect to RS1 (figure 14c) and RS2 (figure 14d). Indeed, the latter cases are characterized by values of the turbulence production mostly confined at the edges of the streaks (i.e. the centre of the VISA event). This is also in line with the general observation of narrower streaks as highlighted in all the sinusoidal cases.

Figure 15. Colour maps of the components of the turbulence production (a) $\tilde {\mathcal {P}}_{xx}$ and (b) $\tilde {\mathcal {P}}_{xz}$, and corresponding components of the velocity gradient (c) $\partial \bar {\tilde {u}}/\partial x$ and (d) $\partial \bar {\tilde {u}}/\partial z$ calculated over the VISA events, overlaid with the isolines $\tilde {u}^+=\pm 0.5$.

For one representative condition only, namely the Smooth case, figure 15 shows the comparison between the production terms associated with the normal $\overline {\tilde {u}\tilde {u}}$ and shear $\overline {\tilde {u}\tilde {w}}$ stresses. In particular, figures 15(a,b) are representative of the ${\tilde {\mathcal {P}}}_{xx}$ and $\tilde {{\mathcal {P}}}_{xz}$ terms, respectively, and figures 15(c,d) give the associated values of the mean velocity gradients $\partial \bar {\tilde {u}}/\partial x$ and $\partial \bar {\tilde {u}}/\partial z$, respectively. The first immediate conclusion is that the term ${\tilde {\mathcal {P}}}_{xx}$ is at least one order of magnitude greater than ${\tilde {\mathcal {P}}}_{xz}$. This is due to the fact that despite the term $\partial \bar {\tilde {u}}/\partial z$ being greater than $\partial \bar {\tilde {u}}/\partial x$, the stress term $\overline {\tilde {u}\tilde {w}}$ attains values close to zero, for all the investigated cases. Conversely, the normal stress term $\overline {\tilde {u}\tilde {u}}$ counterbalances the smaller values of the velocity gradient $\partial \bar {\tilde {u}}/\partial x$, thus yielding greater production values. The comparison of the total production reported in figures 14 and 15 also shows that the regions of negative/positive production are determined mostly by the sign of $\partial \bar {\tilde {u}}/\partial x$. Considering the specific case of VISA events, the regions of negative production are ascribed, for example, to regions of the flow located at the boundaries between LSS and the surrounding flow. Indeed, when a fluid particle moves from the inner to the outer part of LSS, we have the local value $\partial \bar {\tilde {u}}/\partial x>0$, with corresponding stretching and negative production (Onorato & Iuso Reference Onorato and Iuso2001). An opposite consideration can be made for the regions of the flow characterized by $\partial \bar {\tilde {u}}/\partial x<0$, which yield $\tilde {\mathcal {P}}>0$. This mechanism is also represented schematically in figure 16. It is noteworthy that $\mathcal {P}_{xx}$ is the leading term within the data accessible to the current PIV set-up. Figure 17 reports the turbulence production measured in the wall-normal ($x$–$y$) plane from previous experimental campaigns, at the same value of the Reynolds number, $Re_\theta =2900$. In this case, no conditional averages are applied to the data, thus being indicative of the full components of the Reynolds stresses and shear rates. The leading term is the one related to the wall-normal shear rate, $P_{xy}=-\overline {uv}\,\partial {\bar {u}}/ \partial y$.

Figure 16. Schematic representation of the relation between the velocity gradient and the values of the turbulence production.

Figure 17. Turbulence production normalized in inner units calculated in the wall-normal plane at $Re_\theta =2900$. The data are extracted from the dataset of Cafiero & Iuso (Reference Cafiero and Iuso2022).

The data show that all the surface manipulations lead to an attenuation of the turbulence production, which is particularly significant for values $12< y^+<70$. On the other hand, the differences evidenced between the ribbed surfaces are quite limited, with the RS1 case showing slightly smaller values than RLong and RS2, which could be considered to be within the uncertainty of the measurements.