4.1.1. Classification of instantaneous coherent structures

Figure 10 shows the instantaneous coherent structures developed behind the surface-mounted square cylinder in an oblique flow of $45^{\circ }$ at $Re=3000$. Figure 10(a) clearly visualizes the horseshoe, hairpin and KH instability vortices by visualizing the contour of Q-criterion $Q=0.5(\|\overline {\boldsymbol {\varOmega }^*}^2\|-\|\overline {\boldsymbol {S}^*}^2\|)$, where $S^*_{ij}=0.5(\partial u_i^* / \partial x_j^*+ \partial u_j^* / \partial x_i^*$) and $\varOmega ^*_{ij}=0.5(\partial u_i^* / \partial x_j^*- \partial u_j^* / \partial x_i^*$). The horseshoe vortex is established in front of the square cylinder near the bottom surface-cylinder junction region. The horseshoe vortex formation is based on the fact that the shear development of the boundary layer from the bottom surface produces the vertical pressure gradient in front of the square cylinder. Hence, it causes the high momentum downflow stream towards the bottom surface; then the major deflection of this wall occurs in the upstream direction, forming the horseshoe vortex. In this study, the primary and secondary horseshoe vortices, observed due to the strong upstream flow, are developing around the square cylinder's junction region. The relative distance of the primary horseshoe vortex to the frontal surface of the $45^{\circ }$ square cylinder is closer than that of the $0^{\circ }$ square cylinder (Cao et al. Reference Cao, Tamura, Zhou, Bao and Han2022) and circular cylinder (Eckerle & Langston Reference Eckerle and Langston1987). That is because of the high favourable pressure gradient caused by the $45^{\circ }$-inclined front surfaces of the present square cylinder, generating the strong suction to the horseshoe vortex. The primary and secondary vortices remain stable at the distances of $x/w=3.5$ and $x/w=1.9$, respectively, thus indicating that the primary vortex is stronger than the secondary one. The KH vortices of C-shape are detected near the top and rear surfaces of the square cylinder. These vortices are formed due to the instability (breakup) of the shear layers that is highly dependent on the viscosity. Furthermore, it is interesting to note that the breakup of the secondary horseshoe vortex generates a hairpin-shaped vortex at $x/w=2.2$, including the hairpin tail (longitudinal structure) and the hairpin head (lateral structure). The hairpin head rolls up the primary horseshoe vortex as a result of the interaction between two counter-rotating vortex tubes (primary and secondary horseshoe vortices). Otherwise, the hairpin vortex observed in another downstream region is identical to that observed by Tufo et al. (Reference Tufo, Fischer, Papka and Szymanski1999).

Figure 10. Instantaneous coherent structures developed behind the surface-mounted square cylinder at $Re=3000$; (a) isovalues of $Q=10$ represented by grey surfaces; (b) isovalues of $Q=10$ represented by grey surfaces while those of kinetic energy $0.5 \le E^*=0.5 {\boldsymbol {u}^*}^2\Delta x^*\Delta y^*\Delta z^*\le 0.9$ are represented by red surfaces; (c) isovalues of $Q = 10$ are represented by grey surfaces while positive and negative isovalues of the helicity $H^* = \boldsymbol {u}^{*} \boldsymbol {\cdot } \boldsymbol {\omega }^{*} = \pm 10$ are represented by yellow and blue surfaces, respectively.

Figure 10(b) shows the isovalues of $Q=10$ represented by grey surfaces; and those of kinetic energy of $0.5 \le E^*=0.5 {\boldsymbol {u}^*}^2\Delta x^*\Delta y^*\Delta z^*\le 0.9$ are represented by red surfaces. Based on the kinetic energy contour, the formation of the arch-shaped vortex is captured at the distance $x/w=1.5$ downstream. Initially, the C-shape KH instability vortices are asymmetrically formed near two rear surfaces of the square cylinder and develop their size downstream. Then, when the size of two C-shape vortices is sufficiently large, the two vortices combine into one arch-shaped vortex. Hence, the leg and arch parts of the vortex are formed by the instability of the shear layer separated from the rear and top surfaces of the square cylinder, respectively. Time-averaged experimental data (Ahmed, Ramm & Faltin Reference Ahmed, Ramm and Faltin1984; Hucho & Sovran Reference Hucho and Sovran1993; Lienhart & Becker Reference Lienhart and Becker2003; Liu et al. Reference Liu, Zhang, Zhang and Zhou2021) reported C-shaped vortices attached to two side edges of the Ahmed model in oblique flow of $\alpha =30^{\circ }$. However, for $\alpha > 30^{\circ }$, the wake development of C-shaped vortices was not observed. One significant observation in the present result is that, at $\alpha = 45^{\circ }$, the C-shaped KH vortices remain at some distance downstream and then develop into a three-dimensional (3-D) arch-shaped vortex before transforming into complex 3-D structures in the intermediate wake. Otherwise, the current work clearly visualizes the M-shaped instantaneous arch vortex, which is similar to those found by Wang & Zhou (Reference Wang and Zhou2009) and Zhu et al. (Reference Zhu, Wang, Wang and Wang2017) in the case of a circular cylinder. Furthermore, in the context of a square cylinder, the current 3-D arch-shaped vortex has a more distinct shape compared with the 2-D arch-shaped vortex discovered by Kawai et al. (Reference Kawai, Okuda and Ohashi2012).

Figure 10(c) shows isovalues of $Q = 10$ represented by grey surfaces; and the positive and negative isovalues of the helicity $H^* = \boldsymbol {u}^{*} \boldsymbol {\cdot } \boldsymbol {\omega }^{*} = \pm 10$ are represented by yellow and blue surfaces, respectively. The yellow and blue surfaces also signify the clockwise and counterclockwise rotations of the vortex structures, respectively. As shown in the figure, the counter-rotating conical vortex tubes are observed above the top surface of the square cylinder. The clockwise-rotating conical vortex (yellow surface) shows its broken part because of the KH instability. He et al. (Reference He, Ruan, Mehta, Gilliam and Wu2007) and Banks et al. (Reference Banks, Meroney, Sarkar, Zhao and Wu2000) utilized visualization techniques to capture the instantaneous position of conical vortex cores in the selected cross-sections along the streamwise direction while Marwood & Wood (Reference Marwood and Wood1997), Kawai (Reference Kawai1997) and Kawai (Reference Kawai2002) employed the velocity vector plot technique to visualize the time-mean core and reattachment points. The latest numerical model by Ono et al. (Reference Ono, Tamura and Kataoka2008) for a square cylinder in $45^{\circ }$ oblique flow used particle trace, velocity vector and streamwise vorticity techniques to extract the instantaneous core. One important observation in the present result is that the contour of helicity can clearly detect not only the instantaneous core but also the vortex rotation of the conical vortex. Moreover, the helicity contour reconfirms that the primary and secondary horseshoe vortices on each lateral side are counter-rotating in the downstream region (only co-rotating in the near frontal surface); and the hairpin vortex tail and head are also counter-rotating. In the region behind the rear surfaces of the square cylinder, the vortex structures are three-dimensional.

Figure 11 shows the time histories of the pressure coefficients at four probe points located near the upstream edges of the square cylinder at $Re=10\,000$. The pressure values recorded at probe points $a$ and $c$ are nearly the same, similarly at probe points $b$ and $d$. This means that the conical vortex develops its size at the corner of the cylinder's top surface. The switching of the conical vortex occurs at $t^*=115, 135$ and 152, inducing a high jump in the local pressure coefficient ranging from $-$1.2 to $-$0.5. This observation is generally similar to that observed by the numerical method of Ono et al. (Reference Ono, Tamura and Kataoka2008) and experiments of Kawai (Reference Kawai2002) and Nishimura & Kawai (Reference Nishimura and Kawai2010). The only difference is that the asymmetric feature is observed in the present result for the instantaneous pressure coefficients at two symmetric probe points with respect to the wake centreline. The identical asymmetry is also observed in the data obtained by other numerical simulations (Ono et al. Reference Ono, Tamura and Kataoka2008) while the symmetry is observed using an experimental method (Kawai Reference Kawai2002). It can be noticed that the asymmetry might be caused by the limitation of the present computational domain length ($20w$). Nishimura & Kawai (Reference Nishimura and Kawai2010) tried to see the main factor of the wake vortices causing the conical vortex switching by using a splitter plate on the roof surface or in the wake of the square cylinder. They observed that the switching of the conical vortex still occurs with a splitter plate on the roof surface while it does not appear with a splitter plate in the wake, implying a strong effect of wake vortices on the switching. For this interesting point, we will discuss more details of the major factor related to wake vortex structures in the following subsections.

Figure 11. Time histories of pressure coefficients at four probe points (a–d) located near the upstream edges of the square cylinder at $Re=10\,000$.

4.1.2. Dependence of Reynolds number

Figure 12 depicts the instantaneous coherent structures and pressure coefficient contours on the top surface of a square cylinder at $Re=3000$ and 10 000 during a switching process of a conical vortex. While figure 12(a,c) shows the switching process of a conical vortex at $Re =3000$, figure 12(b,d) shows the switching process of a conical vortex at $Re =10\,000$. The coherent structures around the square cylinder are highly changeable because of the Reynolds number effect, although the switching process of a conical vortex occurs independently of the change of Reynolds number. The highly negative pressure suction region is observed to be switched on upstream sides of the top surface of the square cylinder. The appearance of the conical vortex is identical to that observed by numerical and experimental methods (Kawai Reference Kawai1997, Reference Kawai2002; Ono et al. Reference Ono, Tamura and Kataoka2008). However, this appearance was detected based on the time history record of the pressure coefficient along the upstream edges of a square cylinder's top surface while the present observation is made clearer using the Q-criterion contour of the flow field. The switching of the conical vortex is highly correlated with the Kármán vortex shedding behind the square cylinder due to the Reynolds number effect. This Kármán vortex is induced by the arch-shaped vortex deformation, occasionally developing from side to side of the square cylinder. At higher Reynolds number, the pressure suction region spreads along two upstream sides of the cylinder's top surface (shown in figure 12b,d); while it spreads along one upstream side at lower Reynolds number (shown in figure 12a,c). That is because the KH instability forms during the development of a conical vortex at higher Reynolds number ($Re=10\,000$), thus causing the waving reattachment lines. Therefore, it can be noted that the conical vortex pair strongly attaches to the top surface at $Re =10\,000$ while it weakly attaches to the top surface at $Re =3000$. Furthermore, due to the Reynolds number effect, the angle between the reattachment line and the upstream edge of the square cylinder at $Re=3000$ ($28.4^{\circ }$) is larger than that at $Re=10\,000$ ($20.6^{\circ }$). The large pressure suction on the top surface of the square cylinder caused by a conical vortex was previously detected using extraction of the time-mean and root-mean-square pressure coefficient and reattachment lines by Kawai & Nishimura (Reference Kawai and Nishimura1996), Kawai (Reference Kawai1997), Marwood & Wood (Reference Marwood and Wood1997), Kawai (Reference Kawai2002) and Banks (Reference Banks2013). They found that the reattachment line is inclined relative to the square cylinder's edge from $19.5^{\circ }$ to $28.5^{\circ }$ while the vortex core centreline is inclined from $12^{\circ }$ to $14^{\circ }$ to the square cylinder's edge. Therefore, our results fall into the observed range of references. In addition, the distinct observation in the present work is that the variation of the Reynolds number plays a crucial role in determining the inclination of conical vortex reattachment, showing the inverse proportion of the Reynolds number increase to the inclination.

Figure 12. Instantaneous coherent structures and pressure coefficient contour on the top surface of square cylinder at $Re=3000$ and 10 000 during a switching process of a conical vortex; (a,c) the switching process of conical vortex at $Re =3000$ (jfm.2023re3000); (b,d) the switching process of conical vortex at $Re =10\,000$ (jfm.2023re10000). Panels show (a) $Re =3000$, $t^*=195$, (b) $Re =10\,000$, $t^*=125$, (c) $Re =3000$, $t^*=236$, (d) $Re =10\,000$, $t^*=145$.

At $Re=3000$, the arch-shaped vortex is asymmetrical and controlled by the pair of conical vortices. In particular, as shown in figure 12(a), when the conical vortex attaches to the top surface (shown by the reattachment line), the smaller part of the arch-shaped vortex is observed. In an opposite manner, when the conical vortex detaches from the top surface, the larger part of arch-shaped vortex is observed. A similar observation is obtained at higher Reynolds number (shown in figure 12b). Hence, it is fair to note that the formation of the arch-shaped vortex does not depend on the Reynolds number as long as the conical vortex develops on the top surface of the square cylinder. Otherwise, the instability of the horseshoe vortex in the front and wake of the square cylinder occurs at a higher Reynolds number of $Re=10\,000$. The primary and secondary horseshoe vortices are observed at $Re =3000$ while the primary horseshoe vortex is clearly detected at $Re =10\,000$ (the secondary horseshoe vortex is weakly formed).

4.1.3. Undulation of Kelvin–Helmholtz instability

Figure 13 expresses the instantaneous pressure coefficients at two probe points located along the side shear layer developed behind the rear surfaces of the square cylinder at $Re=3000$. Figure 13(a) shows the instantaneous pressure coefficients at two probe points (A and B) located on the $x\unicode{x2013}y$ plane ($z/w=0.5$) while figure 13(b) depicts the contour of the pressure coefficient at four instants of $t^*=195, 236, 277$ and 319. As shown in figure 13(a), the instantaneous pressure coefficients denote the sway process of the shear layer, being related to the Kármán vortex shedding. The sway of the shear layer of low frequency starts from $t^*=175$, where the signal of KH instability of relatively higher frequency appears intermittently with larger amplitude. The visualization of the shear layer sway and the development of KH instability along the shear layers (red and yellow curved lines) are clearly depicted in figure 13(b). During the sway process, the KH instability is signified by small portions of negative pressure coefficient along the shear layers. These KH vortices move downstream with the free-stream velocity and join with the 3-D vortices in the far field. The signals of these low and high frequency vortices are transformed to elucidate the quantitative details of these coherent motions. The identical occurrences of the KH instability in the cylinder's wake are characterized by other investigators of 2-D circular and square cylinders (Prasad & Williamson Reference Prasad and Williamson1997; Lander et al. Reference Lander, Moore, Letchford and Amitay2018; Liu, Hamed & Chamorro Reference Liu, Hamed and Chamorro2018) and surface-mounted square cylinders (Thomas & Williams Reference Thomas and Williams1999). Furthermore, it is interesting to note that, when the shear layer sways itself from up to down (based on top view), the negative pressure suction region caused by the conical vortex also switches itself from one to another upstream side of the square cylinder. This also reconfirms the strong correlation between the switching of the conical vortex and the Kármán vortex shedding, as observed in figure 12. Nishimura & Kawai (Reference Nishimura and Kawai2010) described the interaction between the conical vortex and Kármán vortex in the wake although Bienkiewicz & Sun (Reference Bienkiewicz and Sun1992) and Kawai & Nishimura (Reference Kawai and Nishimura1996) insisted that the conical vortex switching appeared irregularly without the effect of wake vortices. Therefore, the present scrutiny reconfirms the work of Nishimura & Kawai (Reference Nishimura and Kawai2010), revealing the identical frequency of conical vortex and Kármán vortex sheddings.

Figure 13. Instantaneous pressure coefficients at two probe points located along the side shear layer developed behind the rear surfaces of the square cylinder at $Re=3000$; (a) instantaneous pressure coefficients at two probe points (A and B) located on the $x\unicode{x2013}y$ plane ($z/w=0.5$); (b) the contour of pressure coefficient at four instants of $t^*=195, 236, 277$ and 319.

Figure 14 shows the flapping process of KH instability developed on the top surface of a square cylinder where the left-hand side and right-hand side columns indicate the downward and upward flapping, respectively. As shown in figure 14(a,c), two shear layers, developed from the leading and trailing edges of the square cylinder, are observed in flapping-down process. In this stage, the shear layers are strongly suctioned by the low pressure region distributed on the cylinder's surface. In figure 14(e), the separated shear layer remains laminar in the range between the separation point near the leading edge of $x/w=0$ and $x/w=0.25$. The instability region is shown in the range between $x/w=0.25$ and $x/w=1.5$. During this process, the KH vortex starts to shed from the laminar shear layer at $x/w=0.25$ because of the amplification of the disturbance. Then, it evolves and gets larger in size at $x/w=0.45$. The KH vortex is stretched and interacts with other discrete vortices at $x/w=0.8$. The stretching process includes vortex splitting, forming the transition region of the KH instability. In the flapping-down process (shown in figure 14b–d), only one shear layer, developed from the leading edge of the square cylinder, is observed. At this stage, the shear layer undulates upward because of the higher surface pressure distribution compared with the downward process. Due to the upward undulation of the shear layer, the KH vortex starts to shed from the laminar shear layer at $x/w=0$ because of the amplification of the disturbance. Then, it evolves with larger deformation at $x/w=0.65$. The KH vortex is stretched and interacts with other discrete vortices at $x/w=1.2$ by diffusing vorticity, causing vortex breakdown into random eddies. Generally, the KH vortices on the top surface of the surface-mounted square cylinder are identical with those observed by Brun et al. (Reference Brun, Aubrun, Goossens and Ravier2008) for a 2-D square cylinder at $Re=2000$. However, in this study, it is interesting to note that the position, where the KH vortex starts to shed, tends to move upstream along the shear layer during the flapping-up process compared with that during the flapping-down process.

Figure 14. Flapping process of KH instability developed on the top surface of square cylinder at $Re=3000$ where the left-hand side and right-hand side columns indicate downward and upward flapping, respectively; (a,b) the contour of velocity magnitude; (c,d) the contour of vorticity magnitude; (e,f) the contour of Q-criterion.

Figure 15 analyses the spectra ($\phi$) of streamwise velocity fluctuations ($u^{'}=u^{*}-\tilde {u}^{*}$, where $\tilde {u}^{*}$ stands for the moving-averaged streamwise velocity) at nine probe points located along the core of the conical vortex developed on the top surface of the square cylinder at $Re =3000$. As shown in the figure, the frequencies of von Kármán vortex shedding ($f_{VK}$) and KH instability ($f_{KH}$) are clearly observed at six locations from point 1 to point 6, where only the sign of KH instability is identified at points 7, 8 and 9. At the separation point close to the leading edge of the square cylinder (point 1), the signature of the von Kármán vortex is unclear while it is obvious at points 2, 3, 4, 5 and 6, where two peaks of $f_{VK}$ and $0.5f_{VK}$ are detected. From point 3 to point 9, two signatures of KH vortices of $f_{KH}$ and $0.5f_{KH}$ are also observed, revealing the occurrence of vortex pairing. The frequencies $f_{KH}$ and $0.5f_{KH}$ are remarkably noticeable at points 1, 2 and 3, signifying the development of KH vortices along the conical vortex on the top surface of the square cylinder. The frequencies $f_{KH}$ and $0.5f_{KH}$ at points of 4 to 9 are signified by the KH vortices shed from the shear layer of the rear surfaces of the square cylinder. At points 7, 8 and 9, the energy spectrum cascade of the flow approaches the $k^{-5/2}$ slope. The present results of $f_{KH}$ and $0.5f_{KH}$ are identical at all probe points along the shear layer while those spectral frequencies are lower with an increase in distance along the shear layer, as observed by Lander et al. (Reference Lander, Moore, Letchford and Amitay2018). The change of spectral frequencies is because of the reduction of convection velocity of the KH vortices in the shear layer of the 2-D square cylinder; and the reverse flow between the cylinder’s surface and the shear layer also reduces the convection velocity. However, the reverse flow is neglected in the present 3-D square cylinder because of the occurrence of strong conical vortex tubes on the cylinder's top surface, as observed in figure 10(c).

Figure 15. Spectra of streamwise velocity fluctuations ($\phi$) at nine probe points located along the core of the conical vortex developed on the top surface of the square cylinder at $Re =3000$.

Figure 16 expresses the scaling of the ratio of KH and von Kármán frequencies ($f_{KH}/f_{VK}$) as a function of Reynolds number ($Re$). The continuous and dash-dotted lines are the scaling models of square and circular cylinders proposed by Lander et al. (Reference Lander, Moore, Letchford and Amitay2018) and Prasad & Williamson (Reference Prasad and Williamson1997), respectively. The frequency ratio scaling of the square and circular cylinders, represented by the power-law relations, are $f_{KH}/f_{VK}=0.18Re^{0.6}$ and $f_{KH}/f_{VK}=0.0235Re^{0.67}$, respectively. Right-pointing triangle is the LES data of Brun et al. (Reference Brun, Aubrun, Goossens and Ravier2008). Square is the DNS data of Trias et al. (Reference Trias, Gorobets and Oliva2015). Circle is the laser Doppler velocimetry data of Lander et al. (Reference Lander, Moore, Letchford and Amitay2018). Diamond is the present LBE data at $Re =3000$ and 10 000 where the inset depicts the spectra of three components of the velocity fluctuation at probe point 2 $(-0.598w,-0.046w,0.02w)$ at $Re =10\,000$. The present results show a good agreement with the power-law relation of the square cylinder. It is interesting to note that the energy spectrum approaches the $k^{-5/3}$ slope of Kolmogorov's similarity hypothesis at $Re =10\,000$; while the $k^{-5/2}$ slope is obtained at $Re =3000$. The higher inclination of the energy spectra at higher Reynolds number is correlated with the vortex structures formed behind the square cylinder. As can be seen in figure 12, the larger coherent structures (KH instability and arch-shaped vortex) occur in the wake at $Re=3000$ while smaller structures appear at $Re=10\,000$. Hence, it is fair to note that the smaller coherent structures demonstrate the more inclined energy spectrum slope of Kolmogorov's similarity hypothesis, reconfirming the present finding of a Reynolds number effect discussed in the previous subsection. As reported by Prasad & Williamson (Reference Prasad and Williamson1997), Brun et al. (Reference Brun, Aubrun, Goossens and Ravier2008) and Liu et al. (Reference Liu, Hamed and Chamorro2018), the frequency of Kármán vortex shedding slightly increases in the range of $10^3 < Re <5.4*10^4$ for the case of a 2-D circular cylinder while it remains nearly unchanged for the case of a 2-D square cylinder in the range of $1.5*10^4 < Re <7.4*10^4$ (Lander et al. Reference Lander, Moore, Letchford and Amitay2018). Therefore, it primarily depends on the kinematic viscosity of the flow. In the present work, the frequency of Kármán vortex shedding at $Re=10\,000$ (figure 16) is slightly larger than that at $Re=3000$ (figure 15), following the trend obtained by the circular cylinder references. Therefore, it should be noted that the Kármán vortex shedding frequency is not controlled by the spanwise length of the 2-D circular cylinder or the package of vortices (arch-shaped, conical and KH) of a low-AR surface-mounted square cylinder.

Figure 16. Ratio scaling of KH frequency and von Kármán frequency, $f_{KH}/f_{VK}$ as a function of Reynolds number ($Re$). The continuous and dash-dotted lines are the scaling models of square and circular cylinders proposed by Lander et al. (Reference Lander, Moore, Letchford and Amitay2018) and Prasad & Williamson (Reference Prasad and Williamson1997), respectively. Right-pointing triangle ($\triangleright$) is the numerical data of Brun et al. (Reference Brun, Aubrun, Goossens and Ravier2008). Square ($\square$) is the numerical DNS data of Trias et al. (Reference Trias, Gorobets and Oliva2015). Circle ($\circ$) is the experimental data of Lander et al. (Reference Lander, Moore, Letchford and Amitay2018). Diamond ($\blacklozenge$) is the present data, where the inset depicts the spectra of three velocity fluctuation components at probe point 2 $(-0.598w,-0.046w,0.02w)$ at $Re =10\,000$.

4.1.4. Association of coherent structures with flow separation

It was thoroughly discussed in previous sections that the large pressure suction is generated along the upstream edges close to a roof corner of the square cylinder. This local suction is induced by a pair of conical vortices on the roof resulting from flow separation from these edges. Otherwise, the horseshoe, arch-shaped, KH, hairpin and Kármán vortices were also observed. In this section, we will extract the association and correlation with flow separation of these coherent structures. Figure 17 shows the correlation of a conical and KH vortex with a horseshoe vortex at $Re =3000$. Instantaneous contours of vorticity magnitude ($|\boldsymbol {\omega }|$) and velocity vector distribution on $y\unicode{x2013}z$ planes of $x/w=0$ and $x/w=0.9$ are expressed during a switching process of conical vortices at $t^{*}=195$ and $t^{*}=236$. As shown in the figure, the flow pattern changed suddenly from $t^*=195$ to $t^*=236$ and vice versa in the $y\unicode{x2013}z$ plane of $x/w=0$, as the left and right parts of the conical vortex core grew and decayed alternately. The conical vortex part showing oppositely rotating spiral cores attaches closely to the roof surface, corresponding to the maximum suction on the roof directly underneath the conical vortex. This phenomenon is identical to what was observed by Kawai (Reference Kawai1997) and Taniguchi & Taniike (Reference Taniguchi and Taniike1996). The difference in the current study is that we recognize the conical vortex by contour of vorticity magnitude, which is better than the wind-tunnel visualization in the previous research. In $y\unicode{x2013}z$ plane of $x/w=0.9$, the conical vortex core continuously elongates downstream and drives the roof's KH vortex to roll in the clockwise direction, generating the re-entrant part of the arch-shaped vortex expressed in figure 17(a), which was similarly observed by Kawai et al. (Reference Kawai, Okuda and Ohashi2012). However, this study shows that the arch-shaped vortex is formed by the KH vortices from the roof and side surfaces (as also observed in figure 10a). Moreover, during the switching process, the leg parts of the arch-shaped vortex alternately grew and decayed in vorticity magnitude. Simultaneously, the primary horseshoe vortex rolls toward the side surface of the square cylinder ($0.67w$) near the stronger vorticity part. Meanwhile, the opposite primary horseshoe vortex was driven away ($0.8w$) from the side surface of the square cylinder near the weaker vorticity part as a result of the sway process of the side shear layers (figure 13). It was oppositely observed in figure 17(b). Therefore, it demonstrates the strong correlation of KH vortex with horseshoe vortex.

Figure 17. Correlation of conical and KH vortex with horseshoe vortex at $Re =3000$. Instantaneous contours of vorticity magnitude ($|\boldsymbol {\omega }|$) and velocity vector distribution on $y\unicode{x2013}z$ planes of $x/w=0$ and $x/w=0.9$ are expressed during a switching process of conical vortices at $t^{*}=195$ (a) and $t^{*}=236$ (b).

Figure 18 clearly shows not only the association of coherent structures with flow separation, but also the crucial role of the Kármán vortex at $Re=10\,000$ in affecting the oscillating frequency of KH, conical and horseshoe vortex structures. Two probe points A and B are respectively located near the side shear layer and horseshoe vortex to capture the dominant frequencies around these vortices. As seen in figure 18(a), the dominant frequencies at probe point A include the Kármán ($f_{VK}$) and KH ($f_{KH}$) frequencies; meanwhile those at probe point B involve the Kármán and hairpin frequencies ($f_H$ shown in figure 18d). It is interesting to note that the hairpin frequency is lower than the KH frequency; while the Kármán frequency dominates both the side shear layer and the horseshoe vortex. The detection of $f_{KH}$ in the present study is also similar to that observed by Kawai & Nishimura (Reference Kawai and Nishimura1996). Figure 18(b,c) expresses the sway process of the horseshoe vortex at $Re=10\,000$ due to the dominance of the Kármán vortex, which is generated due to the instability of the arch-shaped vortex shown in figure 18(d). The sway of the present horseshoe vortex is in agreement with the observations of Pattenden et al. (Reference Pattenden, Turnock and Zhang2005) and Krajnović (Reference Krajnović2011) for a circular cylinder.

Figure 18. Association of coherent structures at $Re =10\,000$; (a) spectra of streamwise velocity fluctuations ($\phi$) at two probe points A (located near the side separated shear layer) and B (located near horseshoe vortex); (b,c) sway process corresponding to one Kármán vortex shedding; (d) instantaneous coherent structures developed behind the surface-mounted square cylinder at $Re=10\,000$ with isovalues of $Q=10$ coloured by velocity magnitude.

4.2.1. Flow separation

To comprehensively interpret the 3-D complex separation of oblique flow past a surface-mounted square cylinder, critical-point theory is employed to obtain the logical elucidation of this complex flow in a time-mean sense. That is because the instantaneous surface flow pattern is identical to the time-mean surface flow pattern when the shear layer is close to the surface (Cao, Tamura & Kawai Reference Cao, Tamura and Kawai2019). The theory distinguishes the critical points, where the indeterminate streamlines or skin-friction lines are obtained from high-resolution numerical simulation or experiment (Délery Reference Délery2001). Accordingly, the flow topology is introduced by critical points, detachment/reattachment lines, separation/reattachment surfaces and topological rules to describe consistently the complex flow separation. Figure 19 shows classification of the critical point in the $p$–$q$ chart (defined by the eigenvalues $\lambda _1$ and $\lambda _2$ of the velocity derivative tensor, $F_{ij}=\partial \tau _{wi}/\partial x_{j}$, where $\tau _{wi}$ stands for surface shear stress), reproduced following Délery (Reference Délery2001). The purple-filled points represent unstable and stable foci and the centre while the green- and red-filled points stand for isotropic, attachment and separation nodes and saddle points, respectively. In this study, foci, nodes, saddle points, half-saddle points, half-node points, attachment lines and separation lines are respectively denoted by $F$, $N$, $S$, $S^{'}$, $N^{'}$, $(A)$ and $(S)$. These critical points satisfy the topological basic rules (Délery Reference Délery2013) as follows:

(4.1)\begin{equation} \left(\sum N+ \frac{1}{2} \sum N^{'}\right) - \left(\sum S+ \frac{1}{2} \sum S^{'}\right) = 1-n. \end{equation}

For a simply connected domain, $n = 1$; for a doubly connected domain, $n = 2$.

Figure 19. Classification of critical point in the $p$–$q$ chart, reproduced following Délery (Reference Délery2001). The purple-filled points represent unstable and stable foci and the centre while the green- and red-filled points stand for isotropic, attachment and separation nodes and saddle points, respectively.

Figure 20 shows the separated flow topologies projected on the $x\unicode{x2013}y$ planes of $z/w = 0.05, 0.25$ and $0.5$. Figure 20(a i,b i,c i) shows the surface flow patterns of the present simulation results, visualizing the pseudo-streamlines with respect to the velocity vector; figure 20(a ii,b ii,c ii) reconstructs the flow topology based on critical-point concepts from figure 20(a i,b i,c i). The colour shows the contour of the time-mean pressure coefficient ranging from $-1$ to 1.4. Figure 20(a) shows the topology of the bottom-surface flow, represented by foci ($F_1$ and $F_2$), saddle points ($S_1$, $S_2$, $S_3$ and $S_4$), half-saddle points ($S_1^{'}$, $S_2^{'}$ and $S_3^{'}$), nodes ($N_1$ and $N_2$), half-node point ($N_1^{'}$), separation lines ($(S_1)$,$(S_2)$, $(S_3)$, $(S_4)$ and $(S_5)$) and attachment line $(A)$. Generally, the present result of $S_3$, $S_4$, $N_1$ and $S_1$ agrees well with that obtained by Thomas & Williams (Reference Thomas and Williams1999). Furthermore, all critical points of the bottom-surface flow in this study satisfy the topological constraint of a doubly connected domain (4.1), such as ($\sum N+ \tfrac {1}{2} \sum N^{'}) - (\sum S+ \tfrac {1}{2} \sum S^{'})=-1$, where $\sum N=4$, $\sum N^{'}=1$, $\sum S=4$ and $\sum S^{'}=3$. The free-stream flow approaches and diverges from the separation line ($S_1$), where the divergence point is the saddle point $S_1$. The separation line ($S_1$) is the contact line of the secondary horseshoe vortex with the bottom surface. This vortex originates from the saddle point $S_1$, surrounds the square cylinder and extends downstream before entering the free-stream flow. The attachment line ($A$) starts from the node $N_1$ and emanates around the square cylinder, forming the contact line of the primary horseshoe vortex with the bottom surface. Similarly, the primary horseshoe vortex also spreads around the square cylinder, and proceeds downstream before entering the free-stream flow. In the wake region, the shear flows (separated from the half-saddle points $S_2^{'}$ and $S_3^{'}$) and the reversed flows (started from the half-node point $N_1^{'}$) spin towards the symmetrical foci $F_1$ and $F_2$. These two foci $F_1$ and $F_2$ are isolated by separation lines $(S_3)$ and $(S_4)$, passing through the saddle points $S_3$ and $S_4$, respectively. In other bluff body studies, one saddle and two focus points are regularly observed in the time-mean wake while two saddle and two focus points are identified in the present study.

Figure 20. The separated flow topologies projected on the $x\unicode{x2013}y$ planes: (a) $z/w = 0.05$; (b) $z/w = 0.25$; (c) $z/w = 0.5$. Figures (a i,b i,c i) show the surface flow patterns of the present simulation results, visualizing the pseudo-streamlines with respect to the velocity vector; figures (a ii,b ii,c ii) reconstruct the flow topology based on critical-point concepts from the left-hand side figures. The colour shows the contour of the time-mean pressure coefficient ranging from $-1$ to 1.4.

Figure 20(b) expresses the flow topology in the $x\unicode{x2013}y$ plane of $z/w=0.25$, represented by foci ($F_1$, $F_2$, $F_3$ and $F_4$), saddle point $S_1$, half-saddle points ($S_1^{'}$, $S_2^{'}$, $S_3^{'}$, $S_4^{'}$, $S_5^{'}$, $S_6^{'}$, $S_7^{'}$ and $S_8^{'}$), separation lines ($(S_1)$ and $(S_2)$). All critical points also satisfy the topological constraint of a doubly connected domain, where $\sum N=4$, $\sum N^{'}=0$, $\sum S=1$ and $\sum S^{'}=8$. The symmetrical foci $F_1$ and $F_2$ are formed by the reversed flows started from saddle points $S_6^{'}$ and $S_7^{'}$, respectively. The vortex foci $F_3$ and $F_4$ are symmetrical and isolated by the separation line $(S_2)$ of the saddle point $S_1$, which is identical to other bluff body studies. Figure 20(c) expresses the flow topology in the $x\unicode{x2013}y$ plane of $z/w=0.5$, represented by foci ($F_1$ and $F_2$), node $N_1$, half-saddle points ($S_1^{'}$, $S_2^{'}$, $S_3^{'}$, $S_4^{'}$, $S_5^{'}$, $S_6^{'}$, $S_7^{'}$ and $S_8^{'}$). All critical points also satisfy the topological constraint of a doubly connected domain, where $\sum N=3$, $\sum N^{'}=0$, $\sum S=0$ and $\sum S^{'}=8$. The formation of two symmetrical foci $F_1$ and $F_2$ is different from that in the middle plane of $z/w=0.25$. In particular, the reversed flows started from the saddle point $S_6^{'}$ and node $N_1$ roll themselves towards $F_1$; and similarly to $F_2$.

Figure 21 expresses the flow topology in the $x\unicode{x2013}z$ plane of $y/w=0$, represented by foci ($F_1$, $F_2$, $F_3$, $F_4$ and $F_5$), saddle point $S_1$, half-saddle points ($S_1^{'}$, $S_2^{'}$, $S_3^{'}$, $S_4^{'}$, $S_5^{'}$, $S_6^{'}$, $S_7^{'}$, $S_8^{'}$, $S_9^{'}$ and $S_{10}^{'}$). All critical points also satisfy the topological constraint of a doubly connected domain, where $\sum N=5$, $\sum N^{'}=0$, $\sum S=1$ and $\sum S^{'}=10$. In the cylinder–surface junction region of the square cylinder, the foci $F_1$ and $F_2$ represent the primary and secondary horseshoe vortices. On the top surface, the focus $F_3$ is the vortex bubble that arises near the leading edge because of the reversed flow starting from the saddle point $S_6^{'}$. In the wake region, the shear layer developed from the trailing edge of the top surface (half-saddle point $S_{7}^{'}$) proceeds downstream and rolls itself towards the attachment region, originating from the half-saddle point $S_{10}^{'}$. The large recirculation zone of the node centre $F_5$ is manifested between the separated shear layer and small recirculation zone of node centre $F_4$ in the junction region.

Figure 21. The separated flow topology projected on the $x\unicode{x2013}z$ plane of $y/w = 0$; (a) the surface flow pattern of the present simulation results, visualizing the pseudo-streamlines with respect to the velocity vector; (b) the reconstruction of the flow topology based on critical-point concepts. The colour shows the contour of the time-mean pressure coefficient ranging from $-1$ to 1.4.

Figure 22 depicts the flow topologies on the top and leeward surfaces of the square cylinder which are reproduced from the surface streamline contours. As shown in figure 22(a), the symmetrical separation lines of $(S_1)$ and $(S_2)$, originating from the separation point near the upstream corner point of the square cylinder signify the detachment of the conical vortex from the top surface while the attachment line $(A_1)$ is the contact line of the conical vortex with the top surface. Therefore, the conical vortex manifests itself between the separation and attachment lines, forming the symmetrical conical vortices at two regions segregated by the attachment line $(A_1)$. The separation line $(S_2)$ continuously extends on the leeward surface of the square cylinder, as shown in figure 22(b). The $(S_2)$ line is finally prevented by attaching the separation node point $N_2$. Otherwise, the attachment node point $N_1$ is observed to firmly connect with two saddle points $S_1$ and $S_2$. The reversed flows from right to left caused by the adverse pressure gradient are isolated by the separation line $(S_5)$ of the saddle point $S_2$. The saddle point $S_3$ is observed in the cylinder–surface junction region with its two separation lines $(S_6)$ and $(S_7)$.

Figure 22. Reproduction of flow topologies on the square cylinder surfaces from the present results of surface streamline contours: (a) top-surface flow topologies; (b) leeward surface flow topologies.

4.2.2. Time-mean coherent structures

Figure 23 expresses time-mean streamline contours with critical points in the symmetry plane in the junction region near the front cylinder surface and compares the present results with those obtained by Jenssen et al. (Reference Jenssen, Schanderl, Strobl, Unglehrt and Manhart2021). In the reference data, the Reynolds number was investigated at $Re_D=39\,000$ (where $D$ is the diameter of circular cylinder). In the present results, the blue and red contours represent the regions of the negative (called wall-parallel jet) and positive streamwise velocity, respectively. As shown in figure 23(c), these regions are isolated by the black layer, where the streamwise velocity approaches zero. Wall-parallel jet region is formed by the downflow intruding into the bottom surface at half-saddle point $S_3$, then deflecting in the upstream direction, and finally penetrating the bottom surface until the half-saddle point $N_1$. In the reference results shown in figure 23(a,b), a part of the downflow is redirected to the cylinder front surface at node point $V_3$, which is not observed in the present study. The primary and secondary horseshoe vortex centres at foci $V_1$ and $V_2$ are observed in the present study while only a primary horseshoe vortex centre is detected in the reference. The present location of $V_1$ ($x/w+0.205=-0.621$) is closer to the cylinder surface than that of the reference ($x/D=-0.73$, as shown in figure 23b). That is because the favourable pressure gradient at upstream surfaces of $45^{\circ }$ square cylinder is larger than those of a circular cylinder (Baker Reference Baker1980; Pattenden et al. Reference Pattenden, Turnock and Zhang2005; Jenssen et al. Reference Jenssen, Schanderl, Strobl, Unglehrt and Manhart2021). Although the present thickness of the wall-parallel jet is higher than that of the reference because of the occurrence of $V_2$, the present location of $N_1$ ($x/w+0.205=-1.09$) is identical to that of the reference ($x/D=-1.1$, as shown in figure 23b).

Figure 23. Time-mean streamline contours with critical points in the symmetry plane in the junction region near the front cylinder surface; (a,b) particle image velocimetry and LES data of a circular cylinder (Jenssen et al. Reference Jenssen, Schanderl, Strobl, Unglehrt and Manhart2021); (c) present results of a square cylinder where the blue and red contours represent the negative (wall-parallel jet) and positive time-mean streamwise velocities, respectively.

Figure 24 depicts 3-D separation flow visualization in the top-surface region projected on seven $z\unicode{x2013}y$ planes of $x/w$ ranging from $-$0.6 to 0.9. According to the discussion of the development of a symmetrical conical vortex on the top surface of a square cylinder in figure 22(a), seven planes are arranged to capture the development of one time-mean conical vortex on a half-side of the square cylinder's top surface. Each plane visualizes the contour of vorticity magnitude with respect to the pseudo-streamlines. Starting from the separation point, the conical vortex core expands its size with an increase in $x/w$, forming the conical vortex core, as also observed in figure 10(c). From $x/w=-0.6$ to $x/w=0.25$ the conical vortex still attaches the top surface while it moves upwards and rolls in the counterclockwise direction at $x/w=0.6$ and 0.9 because of the sway of the shear layer on the leeward surface. It reconfirms the strong correlation between the sway of the shear layer and the switching of the conical vortex discussed in figure 13. Otherwise, the reattachment line is observed to be more inclined from the upstream edge than the conical vortex core, which is similar to other references (Kawai & Nishimura Reference Kawai and Nishimura1996; Kawai Reference Kawai1997; Marwood & Wood Reference Marwood and Wood1997; Kawai Reference Kawai2002; Banks Reference Banks2013). The 3-D coordinate of the conical vortex core is extracted by figure 25, where the attachment of the conical vortex is clearly shown on the top surface. While the upstream portion creates and time-mean angle of $26.4^{\circ }$ to the upstream edge, the downstream portion of the conical vortex core proceeds aloft when this vortex travels out from the top surface.

Figure 24. Three-dimensional separation flow visualization in the top-surface region projected on seven $z\unicode{x2013}y$ planes of $x/w$ ranging from $-$0.6 to 0.9. Each plane visualizes the contour of vorticity magnitude with respect to the pseudo-streamlines.

Figure 25. The conical vortex core above the top surface of the square cylinder.

The aloft part of the conical vortices continuously travels downstream and establishes the arch-shaped vortex, as observed in figure 26. The arch-shaped vortex and its deformation projected on the $y\unicode{x2013}z$ planes of $x/w$ ranging from 0.75 to 2 is visualized, where left-hand side figures indicate the streamwise time-mean vorticity contours; and right-hand side figure shows the Q-criterion contour of $Q=10$. The horseshoe, side shear layer and conical vortices are clearly captured around the square cylinder while the base streamwise vortex pair is also detected in the moderate wake of the square cylinder. The tip and base vortices are categorized as the streamwise counter-rotating vortices that always occur in the near- and moderate- wake of the 3-D square cylinder (da Silva et al. Reference da Silva, Chakravarty, Sumner and Bergstrom2020; Cao et al. Reference Cao, Tamura, Zhou, Bao and Han2022). In this study, the tip vortex pair is not observed due to the small cylinder AR of 0.5. In the in-plane flows at $x/w=0.75$, the symmetrical arch-shaped vortices are established by two counter-rotating parts, including clockwise (blue contours) and counterclockwise (yellow contours) parts. Each part contains a vortex head and tail. In particular, the upward flows travel vertically from the bottom surface of $z/w=0$ to $z/w=0.45$, then emanate in the opposite transverse directions, thus forming the symmetrical arch-shaped vortices. The development of these vortices is driven by the conical vortices. As a result, the head vortex starts to split from its tail at $x/w=0.9$ because of the aloft part of the conical vortex core. These vortices are partly split at $x/w=1.2$, and they are fully broken into the 3-D vortex structures at $x/w=2$. Simultaneously, the downflow caused by the downwash roll of the shear layer from the top surface (observed in figure 21) occurs to manifest the base vortices. Therefore, it is interesting to note that the formation of the base vortex is highly correlated to the streamwise termination of the arch-shaped vortex.

Figure 26. Visualization of arch-shaped vortex and its deformation projected on $y\unicode{x2013}z$ planes of $x/w$ ranging from 0.75 to 2; (a–d) indicate the streamwise time-mean vorticity contours; (e) shows the Q-criterion contour of $Q=10$.

The arch-shaped vortex core in the wake of the square cylinder is shown in figure 27, where the vortex centres of the core are coordinated from the streamline contours shown in figures 20 and 21. The arch-shaped vortex was rooted by two foci on the $x\unicode{x2013}y$ plane near the bottom surface. Furthermore, the inclination of the arch-shaped vortex core is clearly observed and driven by the aloft part of the conical vortex (as seen in figure 24), which connects to the sway of the separated shear layer from the square cylinder side (as seen in figure 13). The attachment of the downwash flow of the conical vortex pair at the attachment line $(A_1)$ (depicted in figure 22a) exhibits the re-entrance of the arch-shaped vortex, which is identical to that observed by Kawai et al. (Reference Kawai, Okuda and Ohashi2012).

Figure 27. The arch-shaped vortex core in the wake of square cylinder.

4.2.3. Reynolds stresses

To understand the detailed interaction between the coherent structures around the square cylinder and boundary layer on the bottom surface, the transport of turbulent fluctuations is investigated in this section. The Reynolds stresses connecting directly to the turbulence intensity is discussed, including normal ($\overline {u^{'}u^{'}}$, $\overline {v^{'}v^{'}}$ and $\overline {w^{'}w^{'}}$) and shear ($\overline {u^{'}v^{'}}$ and $\overline {w^{'}u^{'}}$) stresses. That is because the Reynolds stress component of $\overline {v^{'}w^{'}}$ is nearly zero due to statistical symmetry. The present examination of the Reynolds stress will shed some light on the characteristics of wake coherent structures in time-mean fluctuation sense, including primary and secondary horseshoe vortices (denoted as PHS and SHS, respectively), side shear layer (SSL), rolled shear layer (RSL), arch-shaped vortex (ASV) and base vortex (BV). Figure 28 depicts the Reynolds normal stress projected on the $x\unicode{x2013}y$ planes of $z/w=0.05, 0.25$ and 0.5 while figure 28(a,d,g) shows the spanwise-averaged Reynolds normal stress of $\overline {u^{'}u^{'}}$, (b,e,h) and (c,f,i) show the transverse- and vertically averaged Reynolds normal stresses of $\overline {v^{'}v^{'}}$ and $\overline {w^{'}w^{'}}$, respectively. For the distribution of $\overline {u^{'}u^{'}}$, at the bottom surface of $z/w=0.05$, the signature of the side shear layer, primary and secondary horseshoe vortices are clearly marked. At $z/w=0.25$, the PHS and SHS disappear while the SSL is intensified together with the existence of the arch-shaped vortex. At $z/w=0.5$, the SSL and ASV are greatly escalated. For the distribution of $\overline {v^{'}v^{'}}$, the attachment region (observed in figure 21) is intensively signified at the $z/w=0.05$ because of the rolled shear layer separated from the cylinder's top surface. The intensity of the shear layer is reduced at $z/w=0.25$ and 0.5, which is opposite to the distribution of $\overline {w^{'}w^{'}}$.

Figure 28. Averaged Reynolds normal stress projected on $x\unicode{x2013}y$ planes of $z/w=0.05, 0.25$ and 0.5; (a,d,g) show the spanwise-averaged Reynolds normal stress of $\overline {u^{'}u^{'}}$; (b,e,h) show the transverse-averaged Reynolds normal stress of $\overline {v^{'}v^{'}}$; (c,f,i) show the vertically averaged Reynolds normal stress of $\overline {w^{'}w^{'}}$.

Figure 29 shows the time-averaged Reynolds shear stress projected on the $x\unicode{x2013}y$ planes of $z/w=0.05, 0.25$ and 0.5. While the left-hand side column shows the Reynolds shear stress of $\overline {u^{'}v^{'}}$, the right-hand side column expresses the Reynolds normal stress of $\overline {w^{'}u^{'}}$. For the distribution of $\overline {u^{'}v^{'}}$, the PHS, SSL and BV are clearly marked at $z/w=0.05$ while the colour contours mean counter rotations. The counter-rotating base vortices reconfirm the identical observation to the previous studies (da Silva et al. Reference da Silva, Chakravarty, Sumner and Bergstrom2020; Cao et al. Reference Cao, Tamura, Zhou, Bao and Han2022). At $z/w=0.25$, only the SSL is intensively magnified while the ASV and SSL are simultaneously detected at $z/w=0.5$. For the distribution of $\overline {w^{'}u^{'}}$, the ASV is clearly observed at $z/w=0.25$ and greatly intensified at $z/w=0.5$ while it is weakly captured at the bottom surface. Figure 30 depicts averaged Reynolds normal and shear stresses projected on the symmetry plane. While the streamwise-, transverse- and vertical-averaged Reynolds normal stresses are respectively shown in figures 30(a), 30(b) and 30(c), the averaged Reynolds shear stress of $\overline {w^{'}u^{'}}$ is expressed in figure 30(d). The rolled shear layers are well captured in the distributions of $\overline {u^{'}u^{'}}$, $\overline {w^{'}w^{'}}$ and $\overline {w^{'}u^{'}}$ while the high intensity of $\overline {v^{'}v^{'}}$ is concentrated on the attachment region. Especially, the highest intensity of $\overline {v^{'}v^{'}}$ is around the half-saddle point $S_{10}^{'}$, as observed in figure 21.

Figure 29. Averaged Reynolds shear stress projected on $x\unicode{x2013}y$ planes $z/w=0.05, 0.25$, and 0.5; (a,c,e) show the Reynolds shear stress of $\overline {u^{'}v^{'}}$; (b,d,f) show the Reynolds normal stress of $\overline {w^{'}u^{'}}$.

Figure 30. Averaged Reynolds normal and shear stresses projected on the symmetry plane of $y/w=0$. (a) Averaged Reynolds normal stress $\overline {u^{'}u^{'}}$. (b) Averaged Reynolds normal stress $\overline {v^{'}v^{'}}$. (c) Averaged Reynolds normal stress $\overline {w^{'}w^{'}}$. (d) Averaged Reynolds shear stress $\overline {w^{'}u^{'}}$.