4.1 Mean flow profiles

In figure 9, the mean wall-normal velocity profiles are presented. The vertical blue lines mark the wall-normal locations that are chosen for further analysis in the following sections. These locations are, respectively, in the viscous sublayer, log layer and outer layer spanning a range of $y^{+}$ from 6 to 180. It can be seen that the pressure gradient dramatically changes the log layer and outer layer for these five investigated locations. The typical von Kármán constant $\unicode[STIX]{x1D705}=0.41$ for the classical law of the wall (Von Kármán Reference von Kármán1931),

(4.1) $$\begin{eqnarray}u^{+}=\frac{1}{\unicode[STIX]{x1D705}}\ln (y^{+})+B,\end{eqnarray}$$

works well in the ZPG zones. Yet, for FPG and APG zones, variations of this value have been reported. Nagib & Chauhan (Reference Nagib and Chauhan2008) have emphasized the non-universality of the von Kármán constant $\unicode[STIX]{x1D705}$ through various experiments on canonical flows and showed that FPG leads to higher values of $\unicode[STIX]{x1D705}$ while APG gives lower $\unicode[STIX]{x1D705}$ . They also showed that the constant $B$ can be negative, which is the case for the current flow. This is also consistent with the variations reported by Nickels (Reference Nickels2004) (figure 8 therein), which is repeated here for the present dimensionless pressure gradients $p_{s}^{+}=(\unicode[STIX]{x1D708}/\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}^{3})(\text{d}p/\text{d}s)$ in figure 10(a). Yet, the values for von Kármán constant $\unicode[STIX]{x1D705}$ obtained from this aerofoil case in APG zones are quite different from some earlier reported works of APG boundary layers from flat plates (Lee & Sung Reference Lee and Sung2008; Knopp et al. Reference Knopp, Schanz, Schröder, Dumitra, Cierpka, Hain and Kähler2014). They, however, correspond to quite different flow conditions. Either $\unicode[STIX]{x1D6FD}_{c}$ is much smaller (Lee & Sung Reference Lee and Sung2008) or the local Reynolds number $Re_{\unicode[STIX]{x1D703}}$ is much higher (Knopp et al. Reference Knopp, Schanz, Schröder, Dumitra, Cierpka, Hain and Kähler2014). Yet they all lie on the curve fit proposed by Nagib & Chauhan (Reference Nagib and Chauhan2008) for many FPG, ZPG and APG boundary layers as shown in figure 10(b). Moreover, as mentioned by Lissaman (Reference Lissaman1983) in a review of low-Reynolds-number aerofoils, due to the aerofoil thickness and the angle of attack, for the flow over aerofoils, the turbulent boundary layer after transition can negotiate quite severe APG without separation. This is in essence quite different from flow cases from flat plates for similar pressure gradients. This phenomenon is actually shown in figure 6 as the DNS APG near the TE exhibits a high level while the flow is still attached to the aerofoil surface. Actually, it is one of the design objectives of CD aerofoils to provide higher loading capacity at low Mach numbers (Hobbs & Weingold Reference Hobbs and Weingold1984), which then gives higher pressure gradients. Vinuesa et al. (Reference Vinuesa, Örlü, Vila, Ianiro, Discetti and Schlatter2017) compared a set of high-quality DNS results for similar local $Re_{\unicode[STIX]{x1D703}}$ range APG boundary layers. They, for instance, pinpointed that the data from a DNS (Hosseini et al. Reference Hosseini, Vinuesa, Schlatter, Hanifi and Henningson2016) over a NACA4412 aerofoil at $5^{\circ }$ angle of attack show much severer APG measured by $\unicode[STIX]{x1D6FD}_{c}$ for similar $Re_{\unicode[STIX]{x1D703}}$ range. The strong APG over the aerofoil exhibited a significant effect in the outer flow, and was also noticeable in the incipient logarithmic region and all the way down to the buffer layer. Taking that the CD aerofoil from the current DNS has a blade shape, which has an even higher camber and is operated at a higher angle of attack than the case reported by Hosseini et al. (Reference Hosseini, Vinuesa, Schlatter, Hanifi and Henningson2016), the deviation of the mean velocity profiles from the standard form is therefore expected.

Figure 11.

Reynolds stress components normalized with outer scale along wall-normal direction: (a) FPG, (b) ZPG zones and (c) APG zones.

Figure 12.

Reynolds stress components normalized with inner scale along wall-normal direction: (a) FPG, (b) ZPG zones and (c) APG zones.

The Reynolds stress components $r_{ij}$ are shown in figures 11 and 12. Here the indices $1$ and $2$ , which appear in the legends for $r_{ij}$ , refer to the wall-parallel and wall-normal directions, respectively. In these two figures the wall-normal distance is scaled with the boundary layer thickness $\unicode[STIX]{x1D6FF}$ and the friction velocity $u_{\unicode[STIX]{x1D70F}}$ , respectively, to demonstrate the relative locations of the peaks; and $r_{ij}^{+}$ refers to the Reynolds stress components normalized by $\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}^{2}$ . The boundary layer thickness $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}_{p99}$ is determined by total pressure $p_{t}=(\unicode[STIX]{x1D70C}u^{2}+p_{s})/2$ , where $p_{s}$ is the static pressure at a certain point. The boundary layer reaches its limit when $p_{t}$ reaches $99\,\%$ of its maximum constant value along the wall-normal direction. It is necessary to use $\unicode[STIX]{x1D6FF}_{p99}$ not $\unicode[STIX]{x1D6FF}_{99}$ , the boundary layer thickness determined by $99\,\%$ of the tangential velocity maximum along a certain wall-normal direction, as a reference to calculate the boundary layer thickness because, for loaded aerofoils, the external flow is deflected and will therefore have a non-constant value. Thus, the velocity reference would give a boundary layer thickness much thicker than that obtained by using the total pressure. As shown in figures 11 and 12, the locations of the peak values of $r_{ij}$ , which present the most intense momentum transfer by fluctuating velocities in a turbulent boundary layer, also exhibit a significant change due to the presence of pressure gradients. For the ZPG, the peak values are usually in the viscous sublayer. This is the case for the data shown here in figures 11(b) and 12(b) for the flow at the LE and around mid-chord. Further downstream, with the increase of APG, the flow decelerates and the peak values of $r_{ij}$ extend to the log layer and even into the edge of the outer layer shown as a platform shape in figures 11(c) and 12(c) for $8\leqslant y^{+}\leqslant 103$ , which indicates that the APG acts as an additional factor to help the momentum exchange by fluctuating velocities. This platform shape is consistent with what has been reported by Lee & Sung (Reference Lee and Sung2008) at a similar $Re_{\unicode[STIX]{x1D703}}$ range and by Skaare & Krogstad (Reference Skaare and Krogstad1994) at a much higher $Re_{\unicode[STIX]{x1D703}}$ for an APG flow near separation. As mentioned by Lee & Sung (Reference Lee and Sung2009) from an APG with similar $Re_{\unicode[STIX]{x1D703}}$ from an incompressible DNS approach, hairpin-type vortices are the predominant structures associated with the production of Reynolds stress in the APG flows. The hairpin structures away from the wall in the TE region seen in figure 8(c) support such an argument for the origin of the outer peak of the Reynolds stress in APG. These Reynolds stress profiles further confirm the information presented in figure 4: the intense turbulent mixing for APG zones exists in both the viscous sublayer and the log layer, and even extend to the outer layer.

4.2 Pressure-gradient effects on mean $QR$ evolution as a function of wall-normal distance

For the following analysis, the flow data length scale is firstly rescaled from the initial chord length to the local boundary layer thickness $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}_{p99}$ . Secondly, we introduce a scaling based on the local variance of the velocity-gradient tensor magnitude

(4.2) $$\begin{eqnarray}\text{var}(\unicode[STIX]{x1D63C})=\overline{\text{trace}((\unicode[STIX]{x1D63C}-\overline{\unicode[STIX]{x1D63C}})(\unicode[STIX]{x1D63C}-\overline{\unicode[STIX]{x1D63C}})^{\text{T}})}.\end{eqnarray}$$

This is a measure of the variation of the velocity-gradient tensor magnitude and it can be regarded as a gradient analogue to the TKE. Consequently, the related terms in the $QR$ space are scaled as $\hat{\unicode[STIX]{x1D63C}}=\unicode[STIX]{x1D63C}/\sqrt{\text{var}(\unicode[STIX]{x1D63C})}$ , $\hat{\boldsymbol{Q}}=\boldsymbol{Q}/\text{var}(\unicode[STIX]{x1D63C})$ and $\hat{\boldsymbol{R}}=\boldsymbol{R}/\text{var}(\unicode[STIX]{x1D63C})^{3/2}$ . The p.d.f. values for the $QR$ plots in this paper are multiplied by 10 and normalized by the local maximum p.d.f. value and the colour bars are plotted using a logarithmic scale from 0 to 1.

Lozano-Durán, Holzner & Jiménez (Reference Lozano-Durán, Holzner and Jiménez2015) state that integrating over a statistically inhomogeneous region can considerably bias the Lagrangian statistics. To avoid this bias, the investigated regions in this work are chosen so that the turbulence within the integration area is regarded as quasi-statistically homogeneous. This is achieved by following the procedure reported in Bechlars & Sandberg (Reference Bechlars and Sandberg2017a ). The mean dynamics data presented in this work were integrated in the homogeneous spanwise direction, about an extent of approximately 20 grid points in the streamwise direction and approximately five grid points in the wall-normal direction for each location. This results in $\unicode[STIX]{x0394}x^{+}\approx 100$ and $\unicode[STIX]{x0394}y^{+}\approx 6$ for the location at $y^{+}=24$ . These distances are similar to those in the work of Bechlars & Sandberg (Reference Bechlars and Sandberg2017a ).

As mentioned in previous studies, the teardrop shape is not a universal feature, which is often observed for homogeneous isotropic turbulent (HIT) flows. For wall-bounded flows, the mean evolution of the velocity-gradient tensor invariants has been found to vary strongly with the distance to the wall (Bechlars & Sandberg Reference Bechlars and Sandberg2017a ). This wall effect is observed for all investigated locations in this aerofoil case.

Figure 13.

The mean $QR$ development with distance to the wall over the CD aerofoil. From left to right: FPG ( $18\,\%$ chord), ZPG ( $40\,\%$ chord) and APG ( $92\,\%$ chord) zones. Three wall-normal locations are shown: (a,d,g)  $y^{+}\approx 6$ , (b,e,h)  $y^{+}\approx 24$ , and (c,f,i)  $y^{+}\approx 180$ . The dashed lines divide the $QR$ space into four sectors which cover, respectively, a certain flow topology as mentioned in § 1.

Figure 13 shows the overall $QR$ distribution, in which each row corresponds to regions at $18\,\%$ , $40\,\%$ and $92\,\%$ of the aerofoil chord from FPG to APG zones (left to right) and each column corresponds to a certain region from $y^{+}\approx 6$ to $y^{+}\approx 180$ (top to bottom). The contours cover $95\,\%$ of the events, which means that the black colour represents only $5\,\%$ of the maximum p.d.f. value in each plot. Moreover, to obtain a more quantitative impression, the joint p.d.f. is integrated over the respective sectors in $QR$ space as mentioned in § 1 to calculate the ratio of occurrence from all events in each sector. This is shown in table 1. It can be seen that the Lagrangian dynamics in these three zones have rather different behaviours. In the FPG zone, an oval shape is seen for the location closest to the wall. In the log layer, the teardrop shape is observed. In the outer layer, the contracting vortical mechanism is less present, as can be seen from table 1, which then gives the $QR$ plot a quasi-triangular shape. The shapes seen in the ZPG regions resemble the shapes that were reported in Bechlars & Sandberg (Reference Bechlars and Sandberg2017a ): the outer layer matches the teardrop shape from HIT data in the literature and becomes oval on approaching the wall. The APG has the effect that the $QR$ shape of the outer layer becomes concentrated around the $\unicode[STIX]{x1D6E5}=0$ line with positive $R$ values which is the dividing line of sectors representing the vortical contracting manner and pure straining with flattening character. This means that the energy exchange between the mean kinetic energy and the TKE is stronger than in the case of the ZPG. It is well established that vortex stretching is a major mechanism of the cascading process towards smaller scales. Figure 13(i) suggests that this behaviour is suppressed by the APG as there are fewer events present in the $\unicode[STIX]{x1D6E5}>0$ , $R\leqslant 0$ sector especially for positive $Q$ values, as there are fewer bright areas in the $S$ I sector compared with figure 13(f). This is also quantitatively reflected in table 1 when comparing the percentage in $S$ I between APG and ZPG at $y^{+}\approx 180$ . Closer to the wall at $y^{+}\approx 24$ in the log layer, for positive $R$ values, more flattening strain and less contracting vortical structures are present compared to the ZPG zone with the same wall-normal location. The region closest to the wall produces a $QR$ plot with similar shape to that obtained from the ZPG region. This again indicates that the APG acts as a strong mixing reinforcement in the outer layer and log layer for this turbulent boundary layer. In all, it can be concluded that, from the $QR$ point of view, the viscous sublayer has an invariant character: oval shape independent of the pressure-gradient conditions. Yet the log layer and the outer layer strongly depend on the pressure gradient.

Table 1.

Ratios of the characteristic flow topologies in four sectors as mentioned in § 1 in the $QR$ space from sampling locations over the aerofoil: FPG, $18\,\%$ chord; ZPG, $40\,\%$ chord; and APG, $92\,\%$ chord.

4.3 Component analysis: nonlinearity, viscous and pressure effects

In this part, the contribution from separate terms of the compressible evolution system in (3.7) is studied. For this purpose, three groups of evolution terms are identified as suggested by Bechlars & Sandberg (Reference Bechlars and Sandberg2017a ):

1. (i) $\unicode[STIX]{x1D643}^{\ast }=0$ ;

2. (ii) $\text{trace}(\unicode[STIX]{x1D643})=\unicode[STIX]{x1D72E}=Q=R=0$ ;

3. (iii) $\text{trace}(\unicode[STIX]{x1D643})=\unicode[STIX]{x1D71E}=Q=R=0$ .

These give the simplified invariant dynamics systems from (3.7):

1. (i)

   (4.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}Q}{\text{d}t}=QP-\frac{2}{3}P\,\text{trace}(\unicode[STIX]{x1D643})-3R, & \displaystyle\end{eqnarray}$$

   (4.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}R}{\text{d}t}=-\frac{1}{3}Q\,\text{trace}(\unicode[STIX]{x1D643})+PR; & \displaystyle\end{eqnarray}$$

2. (ii)

   (4.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}Q}{\text{d}t}=-\text{trace}(\unicode[STIX]{x1D63C}\unicode[STIX]{x1D71E}^{\ast }), & \displaystyle\end{eqnarray}$$

   (4.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}R}{\text{d}t}=-\text{trace}(\unicode[STIX]{x1D63C}\unicode[STIX]{x1D63C}\unicode[STIX]{x1D71E}^{\ast }); & \displaystyle\end{eqnarray}$$

3. (iii)

   (4.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}Q}{\text{d}t}=\text{trace}(\unicode[STIX]{x1D63C}\unicode[STIX]{x1D72E}^{\ast }), & \displaystyle\end{eqnarray}$$

   (4.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}R}{\text{d}t}=\text{trace}(\unicode[STIX]{x1D63C}\unicode[STIX]{x1D63C}\unicode[STIX]{x1D72E}^{\ast }). & \displaystyle\end{eqnarray}$$
   Here $\unicode[STIX]{x1D71E}^{\ast }$ and $\unicode[STIX]{x1D72E}^{\ast }$ are the traceless parts of $\unicode[STIX]{x1D71E}$ and $\unicode[STIX]{x1D72E}$ defined in (3.6).

Figure 14.

Ratios of the p.d.f. of the normalized first velocity-gradient invariant $\hat{\boldsymbol{P}}$ divided by the maximum values for the different zones over the aerofoil. Colours follow the legends in figure 9 ( $\cdots \cdots$ ,  $y^{+}\approx 6$ ; – – – –,  $y^{+}\approx 24$ ; ——,  $y^{+}\approx 180$ ).

Group (i) covers all the nonlinear terms, except the traceless part of the source term $\unicode[STIX]{x1D643}$ . For incompressible flows, this gives $\text{trace}(\unicode[STIX]{x1D643})=-2Q$ and the system would reduce to the restricted Euler system. For the current flow with low Mach number $M=0.25$ , the compressibility effects are small. For the evolution equations restricted to the nonlinear terms (as shown in figure 15) this means that there are no significant differences with the restricted Euler system (Vieillefosse Reference Vieillefosse1984). Data plotted in figure 15 are from $92\,\%$ chord in the APG zone at $y^{+}\approx 24$ . The trajectories (black streamlines) caused by nonlinear effects are very similar to those described by Vieillefosse (Reference Vieillefosse1984). Here only data from $92\,\%$ chord in the APG zone at $y^{+}\approx 24$ are shown but the contribution from nonlinear terms from all other investigated locations in the FPG, ZPG and APG zones shows an invariant character. This agreement with the incompressible reference (Vieillefosse Reference Vieillefosse1984) means that the mapping of the $PQR$ space to one of its subsets, the $QR$ space taking $P=0$ for such a low-Mach-number case, is valid. This has also been demonstrated by Bechlars & Sandberg (Reference Bechlars and Sandberg2017a ) for $M=0.5$ turbulent boundary layer data. Moreover, as shown in figure 14, the p.d.f. of the normalized first velocity-gradient invariant $\hat{\boldsymbol{P}}=\boldsymbol{P}/\sqrt{\text{var}(\unicode[STIX]{x1D63C})}$ , or normalized inverse dilatation, of all the investigated regions over the aerofoil exhibits very low absolute $\hat{\boldsymbol{P}}$ values (less than 0.002) for their maximum p.d.f. locations. This confirms that, as mentioned by Suman & Girimaji (Reference Suman and Girimaji2010) in terms of velocity-gradient invariant analysis, at low positive or negative dilatations, the statistical behaviour is similar to that in incompressible flow. In figure 15, the mean trajectories move from negative $R$ values towards positive, forming a round head when $Q$ values increase. The evolution velocity magnitude of the restricted Euler system is small around the origin of the $QR$ space and increases with the distance to the origin. Together with the global $QR$ plots shown in figure 13, it can also be seen that the nonlinear part drives the global shape of the $QR$ plots.

Figure 15.

Nonlinearity effects: the mean $QR$ phase development is shown by trajectories (black) and by the magnitude of the mean phase velocity (contours) restricted to the nonlinear terms of the Navier–Stokes equations. The dashed lines divide the $QR$ space into four sectors which cover, respectively, a certain flow topology as mentioned in § 1.

Figure 16.

Viscous effects: the mean $QR$ phase development is shown by trajectories (black) and by the magnitude of the mean phase velocity (contours) restricted to the viscous terms of the Navier–Stokes equations. The dashed lines divide the $QR$ space into four sectors which cover, respectively, a certain flow topology as mentioned in § 1.

Group (ii) presents the viscous effects by neglecting all terms except the terms that link the viscous diffusion and the velocity-gradient tensor. This part also shows an invariant characteristic across the boundary layer for all investigated regions. Figure 16 is a representative $QR$ plot of this group. The data are extracted at $92\,\%$ chord in the APG zone at $y^{+}\approx 24$ . The trajectories are relatively straight, pointing towards the origin and only show a slight anticlockwise spin. The evolution velocity magnitude is decreasing with decreasing distance to the origin. But overall the restricted viscous diffusion evolution has an invariant character in this turbulent flow across the aerofoil regardless of the pressure-gradient conditions. The nonlinear terms and viscous terms are thus not the major reasons for the change of overall $QR$ distribution in these investigated zones.

Figure 17.

Pressure effects: the mean $QR$ phase development is shown by trajectories (black) and by the magnitude of the mean phase velocity (contours) restricted to the pressure term of the Navier–Stokes equations, respectively. From left to right: FPG ( $18\,\%$ chord), ZPG ( $40\,\%$ chord) and APG ( $92\,\%$ chord) zones. Three wall-normal locations are shown: (a,d,g)  $y^{+}\approx 6$ , (b,e,h)  $y^{+}\approx 24$ , and (c,f,i)  $y^{+}\approx 180$ . The dashed lines divide the $QR$ space into four sectors which cover, respectively, a certain flow topology as mentioned in § 1.

Group (iii) presents the pressure effects by neglecting all terms except the terms that link the pressure with the velocity-gradient tensor. This is the part that is most influenced by the pressure gradient. This is shown in figure 17, in which each row corresponds to regions at $18\,\%$ , $40\,\%$ and $92\,\%$ of the aerofoil chord from FPG to APG zones (left to right) and each column corresponds to a certain region from $y^{+}\approx 6$ to $y^{+}\approx 180$ (top to bottom). For the same wall-normal location in terms of wall unit, the FPG, ZPG and APG zones have rather different behaviours. Furthermore, the contribution of the pressure term varies strongly with the wall-normal location. For FPG zones, the outer-layer results show that the pressure terms totally counteract the nonlinear terms (figure 15) and the mean trajectories show an ‘unrolling’ feature (figure 17 c). In the log region (figure 17 b), a swirl for positive $R$ and $Q$ values is formed in the contracting vortical structures regime sector. Closer to the wall (figure 17 a), two distinct mechanisms, divided by a bow-shaped line, are present. On the left side of this line the pressure also counteracts the nonlinear terms, whereas it supports them on the right side. These features are assumed to be related to the rolling structures of the flow reattachment after transition as shown in figure 8(b). As mentioned by Joshi, Liu & Katz (Reference Joshi, Liu and Katz2014) from PIV measurements of the flow with an acceleration parameter (defined in (2.4)) $K\approx 1.28\times 10^{-6}$ , similar to the investigated FPG region of the current DNS, hairpin packets often form in the FPG boundary layer, but are confined to the near-wall region, and the outer layer frequently consists of extended regions of low turbulence. This is consistent with the current observed trend as the FPG gives the trajectories away from the wall more ‘unrolling’ behaviour. For ZPG zones, the outer-layer results (figure 17 f) show that the pressure terms counteract the nonlinear terms (figure 15) except that a swirl for positive $R$ and $Q$ values is formed in the contracting vortical structures regime. The evolution for this swirl is relatively small across the various locations. On approaching the wall, the swirl becomes ‘unrolling’ and two distinct mechanisms, as for region $y^{+}\approx 6$ in the FPG zones, are present. These results for the ZPG zones qualitatively match the results from the ZPG turbulent boundary layer data of Bechlars & Sandberg (Reference Bechlars and Sandberg2017a ). In the APG zones, both the log layer (figure 17 h) and outer layer (figure 17 i) show strong rolling features in the contracting vortical regime sector as shown by the strong curvature of the mean trajectories. For the location closest to the wall ( $y^{+}\approx 6$ , figure 17 g), the ‘unrolling’ appears and is similar to FPG and ZPG regions. Therefore, at locations closest to the wall, the pressure term contribution in the $QR$ plots is quite invariant. Yet, further from the wall, this contribution varies strongly with the local mean pressure gradient.

Figure 18.

The mean $QR$ development with distance to the wall at the TE and near wake: (a–c) TE ( $98\,\%$ chord); (d–f) near wake ( $5\,\%$ chord from aerofoil TE). The dashed lines divide the $QR$ space into four sectors which cover, respectively, a certain flow topology as mentioned in § 1.

Figure 19.

Anisotropy invariant map for locations at TE (98 % chord, ▴) and near wake ( $x/c=0.05$ to TE, ▪): (a) entire boundary layer and (b) near-wall distribution up to $y^{+}\approx 6$ .

4.4 Effects of the wall: trailing-edge to near-wake flow development

As partly mentioned in the previous sections, the wall has significant effects on the overall $QR$ distribution when compared to HIT flows. This can already be seen from the comparison between the current flow over the aerofoil and HIT flow reported in the literature, as well as from the comparison between the flow near the aerofoil TE and the flow in the near wake. Figure 18 shows the flow at $98\,\%$ aerofoil chord and its adjacent area in the wake $5\,\%$ aerofoil chord downstream of the TE. More precisely, in figure 18, each row corresponds to regions in the boundary layer at TE and in the near wake (left to right) and each column corresponds to either of these two regions from near wall (or near the wake centreline) to locations further from wall (or further from the wake centreline) (top to bottom). The wall-normal locations in the boundary layer at $98\,\%$ aerofoil chord are $y^{+}\approx 6$ , $y^{+}\approx 24$ and $y^{+}\approx 180$ , respectively. From the horizontal comparisons (TE flow with their counterparts following the streamlines in the near wake), the $QR$ distribution in the near wake is clearly seen to be more isotropic: all locations exhibit a teardrop shape whereas the TE flow shows significant differences with varying tangential locations. In order to confirm and to quantify the difference in anisotropy levels, the anisotropy invariant map first introduced by Lumley (Reference Lumley1979) is used. Details about the application of such a tool can be found in Simonsen & Krogstad (Reference Simonsen and Krogstad2005) for instance. Figure 19 shows the distribution of the second and third invariants, $-I_{2}$ and $I_{3}$ , of the Reynolds stress tensor $\unicode[STIX]{x1D63C}_{Re}$ defined as

(4.6) $$\begin{eqnarray}\unicode[STIX]{x1D63C}_{Re}=\left[\begin{array}{@{}ccc@{}}\displaystyle \frac{\overline{\unicode[STIX]{x1D70C}u_{x}u_{x}}}{\overline{\unicode[STIX]{x1D70C}}}-\widetilde{u_{x}}\widetilde{u_{x}} & \displaystyle \frac{\overline{\unicode[STIX]{x1D70C}u_{x}u_{y}}}{\overline{\unicode[STIX]{x1D70C}}}-\widetilde{u_{x}}\widetilde{u_{y}} & \displaystyle \frac{\overline{\unicode[STIX]{x1D70C}u_{x}u_{z}}}{\overline{\unicode[STIX]{x1D70C}}}-\widetilde{u_{x}}\widetilde{u_{z}}\\[12.0pt] \displaystyle \frac{\overline{\unicode[STIX]{x1D70C}u_{y}u_{x}}}{\overline{\unicode[STIX]{x1D70C}}}-\widetilde{u_{y}}\widetilde{u_{x}} & \displaystyle \frac{\overline{\unicode[STIX]{x1D70C}u_{y}u_{y}}}{\overline{\unicode[STIX]{x1D70C}}}-\widetilde{u_{y}}\widetilde{u_{y}} & \displaystyle \frac{\overline{\unicode[STIX]{x1D70C}u_{y}u_{z}}}{\overline{\unicode[STIX]{x1D70C}}}-\widetilde{u_{y}}\widetilde{u_{z}}\\[12.0pt] \displaystyle \frac{\overline{\unicode[STIX]{x1D70C}u_{z}u_{x}}}{\overline{\unicode[STIX]{x1D70C}}}-\widetilde{u_{z}}\widetilde{u_{x}} & \displaystyle \frac{\overline{\unicode[STIX]{x1D70C}u_{z}u_{y}}}{\overline{\unicode[STIX]{x1D70C}}}-\widetilde{u_{z}}\widetilde{u_{y}} & \displaystyle \frac{\overline{\unicode[STIX]{x1D70C}u_{z}u_{z}}}{\overline{\unicode[STIX]{x1D70C}}}-\widetilde{u_{z}}\widetilde{u_{z}}\end{array}\right],\end{eqnarray}$$

where $x$ , $y$ and $z$ are the streamwise, vertical and spanwise directions, respectively. Invariants $I_{2}$ and $I_{3}$ are defined as

(4.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle I_{2}={\textstyle \frac{1}{2}}(\text{trace}(\unicode[STIX]{x1D63C}_{Re})^{2}-\text{trace}(\unicode[STIX]{x1D63C}_{Re}^{2})), & \displaystyle\end{eqnarray}$$

(4.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle I_{3}=\text{det}(\unicode[STIX]{x1D63C}_{Re}). & \displaystyle\end{eqnarray}$$

In figure 19(a), the distribution of the anisotropy level of wall-normal points from the wall to the edge of the boundary layer (up to $\unicode[STIX]{x1D6FF}_{p99}$ ) at the TE (98 % chord) and their counterparts in the near wake ( $x/c=0.05$ to TE) again clearly shows that, in the absence of the wall, the turbulence becomes more isotropic as the values of $I_{3}$ and $-I_{2}$ from the points in the near wake are much closer to the 3D isotropic point where $I_{3}$ and $-I_{2}$ tend to zero. In figure 19(b), it can be seen that the most severe departure from the 3D isotropy for TE points in figure 19(a) is from the locations closer to the wall (from wall to $y^{+}\approx 6$ ) as only the distribution of these near-wall points is plotted in this figure. These near-wall points have a nearly 2D isotropic behaviour. These plots thus further confirm that the trend seen in figures 18(a) and 18(d) comes from the fact that the flow becomes more isotropic in the absence of the aerofoil surface. Furthermore, the comparison between figures 18(a) and 18(d) shows that the wall greatly suppresses the vortical stretching character of the flow as the upper-left part of the $QR$ plot in the near wake is much bigger and rounder than that from the TE.

Another aspect worth mentioning, independent of all the flow conditions, is that, for all the investigated regions over the aerofoil and the near wake, the general mean life cycle of turbulent structures stays unchanged in the $QR$ space. This cycle starts from straining structures and develops to stretching vortical structures and further to contracting vortical structures before the life cycle continues again in the straining structures. This appears to be an invariant character for all the regions in the current flow under consideration.