3.1 Integral quantities

In order to assess the effect of the APG on TBLs we first need to define a parameter to measure its magnitude. This measurement is made in terms of the Clauser pressure-gradient parameter $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FF}^{\ast }/\unicode[STIX]{x1D70F}_{w}\,\text{d}P_{e}/\text{d}x_{t}$, where $\unicode[STIX]{x1D6FF}^{\ast }$ is the displacement thickness and $P_{e}$ is the pressure at the boundary-layer edge (Clauser Reference Clauser1954, Reference Clauser1956). Figure 2(a) shows the evolution of the Clauser pressure-gradient parameter along the chord on both sides of the wing sections (recall that the NACA0012 at $0^{\circ }$ angle of attack is a symmetric case and the results shown here are based on that symmetry). First, note that the computation of $\unicode[STIX]{x1D6FD}$ is started at, approximately, $x/c=0.2$. The reason is that, as mentioned above, the calculation of $\unicode[STIX]{x1D6FF}_{99}$ is based on the diagnostic scaling (Vinuesa et al. Reference Vinuesa, Bobke, Örlü and Schlatter2016), which requires the turbulent intensity. Thus, it can only be used after the boundary layer becomes fully turbulent, i.e. after approximately $x/c=0.2$ from the leading edge. Therefore, all those quantities that depend on the boundary-layer thickness or boundary-layer edge velocity will be computed at those locations. With respect to the results shown in figure 2(a), it can be observed that the pressure side of the NACA4412 shows, for most of the surface, a negative value of $\unicode[STIX]{x1D6FD}$ which corresponds to a favourable pressure gradient (FPG). Second, regarding the NACA0012 and the suction side of the NACA4412, it is clear that the NACA4412 exhibits a much stronger APG with respect to the NACA0012, due to the airfoil camber and the angle of attack.

Figure 2. Clauser pressure-gradient parameter $\unicode[STIX]{x1D6FD}$ as a function of (a) the distance from the leading edge and (b) the friction Reynolds number. The black circle on (b) indicates the case in which the NACA0012 and the NACA4412 exhibit matching local values of $\unicode[STIX]{x1D6FD}$ and $Re_{\unicode[STIX]{x1D70F}}$.

Apart from the assessment of APG effects on TBLs, the other aim of this paper is to analyse the effect of the flow history, i.e. $\unicode[STIX]{x1D6FD}(x)$. In order to account for this effect, we define an integrated APG magnitude, $\overline{\unicode[STIX]{x1D6FD}}(Re_{\unicode[STIX]{x1D703}})=(Re_{\unicode[STIX]{x1D703}}-Re_{\unicode[STIX]{x1D703},0})^{-1}\int _{Re_{\unicode[STIX]{x1D703},0}}^{Re_{\unicode[STIX]{x1D703}}}\unicode[STIX]{x1D6FD}(Re_{\unicode[STIX]{x1D703}})\,\text{d}Re_{\unicode[STIX]{x1D703}}$ (where $Re_{\unicode[STIX]{x1D703},0}$ defines the point where the integration is started), as a function of the momentum-thickness Reynolds number. This method was first proposed by Vinuesa et al. (Reference Vinuesa, Örlü, Sanmiguel Vila, Ianiro, Discetti and Schlatter2017b), who were able to obtain the skin-friction curve of APG TBLs by using this parameter and ZPG TBL data. In the evaluation of flow-history effects it is desired that local APG and Reynolds-number effects are avoided. Figure 2(b) shows that there is only one case in which we have the same local APG magnitude ($\unicode[STIX]{x1D6FD}\simeq 3.5$) and same friction Reynolds number ($Re_{\unicode[STIX]{x1D70F}}\simeq 362$) for both wing sections. This will be considered as the matching $\unicode[STIX]{x1D6FD}{-}Re_{\unicode[STIX]{x1D70F}}$ case which will allow the analysis of the flow history effect on the TBL. Note that this specific case requires the results of the NACA0012 at $x_{ss}/c=0.95$ for which $\text{ETT}^{\ast }=30$, therefore convergence of the results at this location is ensured.

Figure 3. Normalised pressure coefficient $C_{p}$ for the two wing sections under study as a function of the distance from the leading edge.

Figure 4. Streamwise evolution of (a) the friction Reynolds number and (b) the momentum-thickness Reynolds number.

Figure 3 shows the pressure coefficient, $C_{p}$, along the chord of the NACA4412, both on the pressure and suction sides, and the NACA0012 (note that the $C_{p}$ is the same on both sides). The pressure coefficient is presented in normalised form, where the pressure coefficient at the stagnation point $C_{p,st}$ is unity, therefore $C_{p,st}=(P_{st}-P_{\infty })/(1/2\unicode[STIX]{x1D70C}U_{\infty }^{2})=1$. This is achieved by setting the ambient pressure $P_{\infty }$ to fulfil this equation. The $C_{p}$ distributions show a very clear trend with respect to the level of pressure gradient. Near the leading edge, the lowest $C_{p}$ corresponds to the case with higher curvature (i.e. the NACA4412 wing section on the suction side) in which the flow is most accelerated. Nevertheless, as the flow approaches the trailing edge, the effect of the APG can be clearly distinguished. The suction side of the NACA4412 (i.e. the region with the strongest APG) exhibits a higher rate of increase in $C_{p}$ due to the flow being decelerated by the APG. On the other hand, on the pressure side of the NACA4412, which is subjected to a favourable pressure gradient, the $C_{p}$ trend decreases along the chord as the flow is accelerated due to the FPG. Lastly, the NACA0012, which corresponds to a mild APG, shows a decreasing $C_{p}$ along the chord but at a slower rate than on the suction side of the NACA4412.

Figure 4(a) shows the streamwise evolution of the friction Reynolds number, which is also computed after $x/c=0.2$ as discussed above. It can be observed that the friction Reynolds number in the FPG TBL shows continuous growth along the surface of the wing section whereas both APG TBLs show an initial increase in friction Reynolds number up to a point where it starts to decrease. In the case of the NACA0012 (i.e. mild APG), there is a subtle decrease of $Re_{\unicode[STIX]{x1D70F}}$ at, approximately, $x_{ss}/c=0.9$ after reaching a maximum value of $Re_{\unicode[STIX]{x1D70F}}=372$ while the TBL of the NACA4412 (i.e. strong APG) shows a steep reduction of $Re_{\unicode[STIX]{x1D70F}}$ which begins at, approximately, $x_{ss}/c=0.8$ where $Re_{\unicode[STIX]{x1D70F}}=369$. This decreasing behaviour is the result of the APG becoming very strong and reducing significantly the skin friction. On the other hand, the momentum-thickness Reynolds number $Re_{\unicode[STIX]{x1D703}}$ is presented in figure 4(b). In this case, all the boundary layers show an increasing $Re_{\unicode[STIX]{x1D703}}$ behaviour, the growth rate being larger for increasing APG. This is directly related to the increase in the mean streamwise velocity deficit of the boundary layer due to the presence of the APG.

Other integral properties of interest in the assessment of the APG effect on TBLs are the skin-friction coefficient $C_{f}=2(u_{\unicode[STIX]{x1D70F}}/U_{e})^{2}$ and the shape factor $H=\unicode[STIX]{x1D6FF}^{\ast }/\unicode[STIX]{x1D703}$. In figure 5(a) we show the skin-friction coefficient of the wing sections as a function of $Re_{\unicode[STIX]{x1D703}}$, together with the ZPG correlation of this value obtained by Nagib, Chauhan & Monkewitz (Reference Nagib, Chauhan and Monkewitz2007). The figure shows that the APG TBLs exhibit an almost constant decrease of the skin-friction coefficient while the skin-friction reduction in the ZPG is not so strong as $Re_{\unicode[STIX]{x1D703}}$ increases. It can be observed that the trend of $C_{f}$ for both wing sections is similar and it can also be noted that there is no mean separation of the flow along the wing sections since the skin-friction coefficient does not show negative values for any of the cases analysed. Figure 5(b) shows the evolution of the shape factor as a function of $Re_{\unicode[STIX]{x1D703}}$, together with the ZPG correlation by Monkewitz, Chauhan & Nagib (Reference Monkewitz, Chauhan and Nagib2007). Note that the ZPG trend exhibits the expected decreasing behaviour with $Re_{\unicode[STIX]{x1D703}}$, whereas both APG TBLs show an increasing trend related to the thickening of the boundary layer due to the APG.

Figure 5. Evolution of (a) the skin-friction coefficient and (b) the shape factor as a function of the momentum-thickness Reynolds number. The results of the ZPG are computed by means of correlations obtained by Nagib et al. (Reference Nagib, Chauhan and Monkewitz2007) for $C_{f}$ and by Monkewitz et al. (Reference Monkewitz, Chauhan and Nagib2007) for $H$.

Table 1 shows the values of the integral parameters discussed above for the two wing cases at several streamwise positions, together with selected ZPG profiles from the database by Eitel-Amor et al. (Reference Eitel-Amor, Örlü and Schlatter2014) at approximately matching $Re_{\unicode[STIX]{x1D70F}}$, which will be analysed in detail below.

Table 1. List of analysed cases with the corresponding integral properties. ZPG data at similar $Re_{\unicode[STIX]{x1D70F}}$ from the database by Eitel-Amor et al. (Reference Eitel-Amor, Örlü and Schlatter2014). The case with matched $\unicode[STIX]{x1D6FD}$ and $Re_{\unicode[STIX]{x1D70F}}$ values is indicated in figure 2.

3.2 Turbulence statistics

In this section we focus on the analysis of the most relevant turbulence statistics computed from the simulations. As discussed in § 2, the turbulence statistics were obtained by spanwise and temporal averaging. The terms required to compute all the statistical terms are first expressed in the spectral-element mesh, and then spectrally interpolated on profiles normal to the wing surface. The statistics are expressed in terms of the directions tangential ($t$) and normal ($n$) to the wing surface. The comparison with the ZPG TBL follows the proposal by Monty et al. (Reference Monty, Harun and Marusic2011) of analysing cases with matching $Re_{\unicode[STIX]{x1D70F}}$.

3.2.1 Tangential mean-velocity profiles

Figure 6 shows the tangential mean-velocity profiles of the cases described in table 1. We first focus on the cases at matched $x_{ss}/c=0.4$ and 0.75, in which both the local $\unicode[STIX]{x1D6FD}$ and the integrated APG magnitude differ. The first finding is that the wake region is more prominent for higher APG, as observed among others by Spalart & Watmuff (Reference Spalart and Watmuff1993), Monty et al. (Reference Monty, Harun and Marusic2011): in the cases at $x_{ss}/c=0.4$ the effect is mild, especially in the NACA0012; however, at $x_{ss}/c=0.75$, this effect is significant in both wings, especially in the NACA4412 due to the stronger APG. Another observation that can be drawn from these plots is the effect of the APG on the buffer region of the turbulent boundary layer. Despite the fact that this effect cannot be observed at $x_{ss}/c=0.4$, the stronger APGs at $x_{ss}/c=0.75$ exhibit a significant shift of the inner-scaled mean velocity downwards within the buffer layer at increasing APG, an effect first reported by Spalart & Watmuff (Reference Spalart and Watmuff1993), which may be overemphasised in low-Reynolds-number boundary layers. Besides the shift in the inner-scaled mean velocity by the APG, the profiles also exhibit a larger slope in the overlap region as shown in the insets in figure 6(a,b) with increasing APG, as observed by Spalart & Watmuff (Reference Spalart and Watmuff1993) and Vinuesa et al. (Reference Vinuesa, Hosseini, Hanifi, Henningson and Schlatter2017a) among others, and analysed by Nagib & Chauhan (Reference Nagib and Chauhan2008) through the characterisation of the dependence of the von Kármán coefficient $\unicode[STIX]{x1D705}$ on flow geometry and pressure gradient. These effects are related to effect of the APG on the large-scale structures (Maciel et al. Reference Maciel, Gungor and Simens2017; Sanmiguel Vila et al. Reference Sanmiguel Vila, Örlü, Vinuesa, Schlatter, Ianiro and Discetti2017), which affect the momentum transfer across the whole boundary layer (Vinuesa et al. Reference Vinuesa, Hosseini, Hanifi, Henningson and Schlatter2017a).

Figure 6. Inner-scaled tangential mean-velocity profiles at (a) $x_{ss}/c=0.4$ and (b) $x_{ss}/c=0.75$. (c,d) Show the case with matched $\unicode[STIX]{x1D6FD}$ and $Re_{\unicode[STIX]{x1D70F}}$ values, in inner and outer scaling respectively. Refer to table 1 for the parameters of the cases.

Figure 7. Inner-scaled non-zero Reynolds stresses corresponding to (a) $x_{ss}/c=0.4$ and (b) $x_{ss}/c=0.75$. (c,d) Show the case with matched $\unicode[STIX]{x1D6FD}$ and $Re_{\unicode[STIX]{x1D70F}}$ values, in inner and outer scaling respectively. The colours indicate: tangential (—— (blue)), wall-normal (—— (red)), spanwise (—— (green)) velocity fluctuations and Reynolds-shear stress (—— (black)). Refer to table 1 for the parameters of the cases.

On the other hand, figure 6(c,d) shows the mean velocity profile at matching $\unicode[STIX]{x1D6FD}-Re_{\unicode[STIX]{x1D70F}}$ conditions with different scalings. The inner-scaled mean-velocity profiles show a different trend when compared with the other two cases regarding the behaviour in the buffer layer, i.e. the velocity in the buffer layer is slightly lower in the ZPG than in the NACA0012 at this location. This in principle would also contradict the results by Spalart & Watmuff (Reference Spalart and Watmuff1993), although it can be argued that at this particular location the viscous units may not be adequate to scale the profile in the NACA0012 case due to the fact that, as shown in figure 2(b), here, the friction Reynolds-number curve starts to show a decreasing trend. This suggests that, under these conditions, inner scaling may be inappropriate, as also reported by Maciel et al. (Reference Maciel, Wei, Gungor and Simens2018). Therefore, for the matching $\unicode[STIX]{x1D6FD}{-}Re_{\unicode[STIX]{x1D70F}}$ case we will use the outer scaling based on $U_{e}$. It can then be observed in figure 6(d) that the outer-scaled mean velocity of the ZPG on the buffer and overlap regions is much higher than the APG cases, where the flow is significantly decelerated. Furthermore, when comparing both wing sections (both with same local APG and $Re_{\unicode[STIX]{x1D70F}}$), the mean-velocity profile of the NACA4412 is shifted downwards in the buffer layer, suggesting that the flow history (i.e. the integrated APG) has a noticeable effect in the mean-velocity profiles.

3.2.2 Reynolds stresses

Further understanding of APG and flow-history effects on turbulent boundary layers can be acquired through the analysis of the Reynolds-stress tensor. Figure 7 shows the non-zero Reynolds stresses for each of the cases included in table 1. Following the same procedure as for the mean-velocity profiles, we first focus on the analysis of the profiles at $x_{ss}/c=0.4$ and $x_{ss}/c=0.75$, shown in (a) and (b) respectively. Already at $x_{ss}/c=0.4$ the effect of the APG is noticeable, and at $x_{ss}/c=0.75$ the effect becomes greatly intensified. The inner-scaled near-wall peak of the tangential velocity fluctuations increases with the APG (as previously documented for instance by Monty et al. (Reference Monty, Harun and Marusic2011)). In addition, the APG has an even more significant effect on the outer region of the boundary layer, exhibiting an outer peak for very strong APG TBLs (Skåre & Krogstad Reference Skåre and Krogstad1994) in the tangential component of the velocity fluctuations, but also larger intensities in the other terms of the Reynolds-stress tensor. This is linked to the energisation of the outer region of the boundary layer in the presence of an APG (Harun et al. Reference Harun, Monty, Mathis and Marusic2013).

In the matching $\unicode[STIX]{x1D6FD}{-}Re_{\unicode[STIX]{x1D70F}}$ case we encounter a situation similar to that of the mean flow regarding the scaling in inner units. In particular, we observe that the NACA0012, which has been subjected to a milder accumulated APG than the NACA4412, exhibits a stronger inner-scaled near-wall peak. Due to the complexity of flow-history effects, it is not clear whether the accumulated effect of $\unicode[STIX]{x1D6FD}$ affects the inner and outer regions equally, and given the fact that the $Re_{\unicode[STIX]{x1D70F}}$ shows a decreasing trend in the NACA0012 case at this location, inner scaling might not be the most appropriate choice. In order to obtain additional insight into the inner and outer regions in these two boundary layers, we scale the components of the Reynolds-stress tensor in outer units. This approach was also adopted by Harun et al. (Reference Harun, Monty, Mathis and Marusic2013) in their APG experiment, to show that the observed larger values of the streamwise velocity fluctuations were not due to the lower friction velocity, which is a consequence of the APG, but due to more intense large-scale turbulent fluctuations. Harun et al. (Reference Harun, Monty, Mathis and Marusic2013) reached the conclusion that, in outer scaling, the outer peak of the streamwise velocity fluctuations increases with APG magnitude, whereas the near-wall peak exhibits the opposite trend. This is associated with the increased wall-normal convection of near-wall turbulence, which is also produced by the APG, and as will be discussed below affects the power-spectral density in the outer region. Interestingly, when using the edge velocity $U_{e}$ to scale the Reynolds stresses in figure 7(d) we can observe a behaviour similar to the one reported by Harun et al. (Reference Harun, Monty, Mathis and Marusic2013), where the NACA4412 profile, subjected to the strongest accumulated APG, exhibits the largest outer peak and the smallest peak in the near-wall region. Note that in this study we employ the edge velocity $U_{e}$ for outer scaling given the fact that in PG TBLs there is not a clear definition of the outer velocity scale (unlike in ZPGs, where the free-stream velocity is constant), since the streamwise pressure gradient leads to a mean velocity gradient for $y_{n}>\unicode[STIX]{x1D6FF}_{99}$. Furthermore, Vinuesa et al. (Reference Vinuesa, Bobke, Örlü and Schlatter2016) argued that $U_{e}$, obtained using the method reported in their study, constitutes a robust outer scale over a wide range of Reynolds numbers and pressure gradients.

In order to further assess the effect of APG and flow history on TBLs, the database by Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017) will be added to the analysis. In particular, we will consider the cases with constant values of $\unicode[STIX]{x1D6FD}(x)=1$ and 2 over significant parts of the simulation domain. For the purpose of the present work, these flat-plate cases exhibit an approximately 3 % variation in $\unicode[STIX]{x1D6FD}$ in the areas of interest, i.e. from points II to I and from IV to III in the $\unicode[STIX]{x1D6FD}=1$ and 2 cases, respectively. Note that due to the transition from laminar to turbulent flow produced by the tripping, the integrated APG magnitude $\overline{\unicode[STIX]{x1D6FD}}$ is computed by integrating from the location at which the flow can be considered to be turbulent, i.e. at approximately $Re_{\unicode[STIX]{x1D70F}}\simeq 150$. Here, we consider a lower limit in $Re$ than that of Vinuesa et al. (Reference Vinuesa, Örlü, Sanmiguel Vila, Ianiro, Discetti and Schlatter2017b) in order to have a sufficient integration length for some of the matching cases discussed below. Although the resulting values of $\overline{\unicode[STIX]{x1D6FD}}$ vary with the choice of the origin for integration, the trends documented below do not depend on this particular choice. Figure 8 shows the four conditions in which there are matching $\unicode[STIX]{x1D6FD}{-}Re_{\unicode[STIX]{x1D70F}}$ cases between the flat plates and the wing sections. Additional information from these four cases is provided in table 2, with the corresponding $\overline{\unicode[STIX]{x1D6FD}}$ values.

Figure 9 shows the inner-scaled tangential velocity fluctuations for the cases summarised in table 2. This specific Reynolds-stress tensor component has been selected because it allows us to analyse the effect of the flow history both on the near-wall peak and the outer region. The first observation that can be made is the almost negligible effect of the integrated APG magnitude $\overline{\unicode[STIX]{x1D6FD}}$ on the near-wall peak, where only small differences among cases can be observed. This result suggests that the inner peak scaled in wall units is not closely related to the accumulated flow history, but its deviation with respect to the ZPG curve indicates that it is more strongly dependent on the local APG magnitude $\unicode[STIX]{x1D6FD}$. This would indicate that the near-wall region adapts more quickly to the imposed pressure gradient than the outer region (Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017). The other observation that can be made from these results is the strong effect of the flow history (i.e. the integrated $\overline{\unicode[STIX]{x1D6FD}}$) on the outer region of the boundary layer, showing consistency with the fact that the inner-scaled tangential velocity fluctuations in the outer layer increase in magnitude with larger integrated APG magnitudes.

Figure 8. Clauser pressure-gradient parameter as a function of the friction Reynolds number on the two wing sections and two of the flat-plate APG cases by Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017). The Roman numerals indicate the various matching cases under study.

Table 2. Data from the wing sections and APG flat plates (Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017) at matching local values of $\unicode[STIX]{x1D6FD}$ and $Re_{\unicode[STIX]{x1D70F}}$ indicated in figure 8 with Roman numerals from I to IV.

Figure 9. Inner-scaled tangential velocity fluctuations for cases (a) I, (b) II, (c) III and (d) IV (as defined in figure 8). Refer to table 2 for the description of the cases. The constant-$\unicode[STIX]{x1D6FD}$ and ZPG cases are obtained from Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017) and from Schlatter et al. (Reference Schlatter, Li, Brethouwer, Johansson and Henningson2010), respectively.

Figure 10. (a,c,e) Turbulent kinetic energy budgets and (b,d,f) production of $\overline{u_{t}^{2}}$ and $\overline{u_{n}^{2}}$. Profiles at (a,b) $x_{ss}/c=0.4$, (c,d) $x_{ss}/c=0.75$ (in inner scaling) and (e,f) case at matched $\unicode[STIX]{x1D6FD}$ and $Re_{\unicode[STIX]{x1D70F}}$ values (in outer scaling). The colours in the left panels correspond to: production (—— (blue)), dissipation (—— (red)), turbulent transport (—— (green)), viscous diffusion (—— (brown)), velocity–pressure-gradient correlation (—— (black)) and convection (—— (magenta)).

3.2.3 Turbulent kinetic energy budgets

Following the assessment of the mean flow and the Reynolds stresses, we now analyse the effect of the APG and flow history on the TKE budget terms across the TBL. In figure 10(a,c,e) we show the TKE budgets for the cases listed in table 1. Among the main results, the APG has a significant effect on the turbulent production across the boundary layer but most remarkably in the outer region, where strong APGs exhibit an outer peak. This result was already observed in the experiment by Skåre & Krogstad (Reference Skåre and Krogstad1994), who linked the emergence of the outer peak in turbulent production with the increase in turbulent shear stresses in the outer region due to the APG. It is worth mentioning that, due to the strong increase in TKE production in the outer layer, the dissipation of TKE is also greatly amplified in this region. Thus, the dissipation becomes very relevant in the outer region of the APG TBLs as well. Note that the dissipation does not only increase in the outer region but also it suffers a noticeable intensification across the whole boundary layer, more prominently at the wall. This large dissipation at the wall is then balanced by a strong viscous diffusion produced by the APG. The remaining terms of the TKE budget also are affected by the APG but the differences are smaller. On the other hand, the outer-scaled TKE budget of the matching $\unicode[STIX]{x1D6FD}{-}Re_{\unicode[STIX]{x1D70F}}$ case, shown in figure 10(e), exhibits a behaviour consistent with the one observed in the analysis of the Reynolds-stress tensor: the turbulent production and the dissipation are lower near the wall when $\overline{\unicode[STIX]{x1D6FD}}$ is larger, whereas it is the opposite far from the wall.

Within the TKE budget, the turbulent production term is of particular relevance for this study as it is directly related to the energisation of the TBL that will be thoroughly assessed hereafter by means of spectral analysis. The TKE production $P^{k}$ in a statistically two-dimensional flow (where the mean spanwise velocity is zero) can be expressed as $P^{k}=1/2(P_{\overline{u_{t}^{2}}}+P_{\overline{u_{n}^{2}}})$, where,

(3.1)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}P_{\overline{u_{t}^{2}}}=-2\left(\overline{u_{t}^{2}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{t}}{\unicode[STIX]{x2202}x_{t}}}+\overline{u_{t}u_{n}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{t}}{\unicode[STIX]{x2202}x_{n}}}\right),\\ P_{\overline{u_{n}^{2}}}=-2\left(\overline{u_{n}^{2}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{n}}{\unicode[STIX]{x2202}x_{n}}}+\overline{u_{t}u_{n}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{n}}{\unicode[STIX]{x2202}x_{t}}}\right).\end{array}\right\}\end{eqnarray}$$

The production of $\overline{u_{t}^{2}}$ and $\overline{u_{n}^{2}}$ are shown in figure 10(b,d,f) for the cases shown in table 1 in order to determine their contribution to the overall TKE production. It can be observed that, at low APGs, $P_{\overline{u_{t}^{2}}}$ is the only term contributing to $P^{k}$; on the other hand, stronger APGs show a small negative contribution from $P_{\overline{u_{n}^{2}}}$ (note that this term remains close to zero in the ZPG cases), together with a significant increase in $P_{\overline{u_{t}^{2}}}$. The comparison with the ZPG shows that the negative contribution of $P_{\overline{u_{n}^{2}}}$ to the total turbulent production is only characteristic of APG TBLs because of the strong wall-normal velocity present in this type of flow, whereas in ZPG TBLs the wall-normal velocity in this region is significantly lower (Vinuesa et al. Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018). Therefore, the APG not only greatly increases the TKE production through the second term in $P_{\overline{u_{t}^{2}}}$ as discussed by Skåre & Krogstad (Reference Skåre and Krogstad1994), but also the vertical motion induced by the APG extracts TKE through the first term in $P_{\overline{u_{n}^{2}}}$ (which is the only term that can have a negative contribution to the production in APGs).

4.1 One-dimensional spanwise power-spectral densities

We start by considering the inner-scaled one-dimensional spanwise pre-multiplied power-spectral density of the tangential velocity fluctuations $k_{z}\unicode[STIX]{x1D719}_{u_{t}u_{t}}^{+}$, as well as the same spectra for the wall-normal $k_{z}\unicode[STIX]{x1D719}_{u_{n}u_{n}}^{+}$ and spanwise fluctuations $k_{z}\unicode[STIX]{x1D719}_{ww}^{+}$. We also computed the power-spectral density of the Reynolds-shear stress $k_{z}\unicode[STIX]{x1D719}_{u_{t}u_{n}}^{+}$, and expressed all of them as a function of the inner-scaled wall-normal distance $y_{n}^{+}$ and the inner-scale spanwise wavelength $\unicode[STIX]{x1D706}_{z}^{+}$. The calculation of the one-dimensional power-spectral density is performed by means of the fast Fourier transform (FFT) because of the periodic boundary conditions imposed in the spanwise direction. Figure 11 shows the inner-scaled pre-multiplied spectra of the selected Reynolds-stress tensor terms. Note that the main objective of this section is to assess the different mechanisms of energisation due to APG and due to the increment in Reynolds number. To this end, we also consider ZPG TBL spectra at $Re_{\unicode[STIX]{x1D70F}}=305$ and 2480, where the lower $Re$ is the closest, from the ZPG database, to the $Re_{\unicode[STIX]{x1D70F}}$ in the wings under study.

Figure 11. Inner-scaled pre-multiplied spanwise power-spectral density of the (a) tangential, (b) wall-normal, (c) spanwise velocity fluctuations and (d) Reynolds-shear stress of the NACA0012 and NACA4412 wing sections at $x_{ss}/c=0.4$ (– – –) and $x_{ss}/c=0.75$ (——). We also show ZPG TBL spectra at $Re_{\unicode[STIX]{x1D70F}}=305$ (– – –) and $Re_{\unicode[STIX]{x1D70F}}=2480$ (——) from the database by Eitel-Amor et al. (Reference Eitel-Amor, Örlü and Schlatter2014). The dotted lines represent the small-scale bounds of the lowest ZPG contour at $Re_{\unicode[STIX]{x1D70F}}=2480$. The contours correspond to $k_{z}\unicode[STIX]{x1D719}_{u_{t}u_{t}}^{+}=(1.0,2.1,3.4)$; $k_{z}\unicode[STIX]{x1D719}_{u_{n}u_{n}}^{+}=(0.2,0.5,0.8)$; $k_{z}\unicode[STIX]{x1D719}_{ww}^{+}=(0.2,0.6,1.2)$ and $k_{z}\unicode[STIX]{x1D719}_{u_{t}u_{n}}^{+}=(0.2,0.5,0.8)$.

Firstly, we focus on the evolution of the power-spectral density of the ZPG with Reynolds number. It can be observed that the energy of the smallest scales near the wall is unaffected by the variation in $Re_{\unicode[STIX]{x1D70F}}$, similarly to the largest scales in the near-wall region. On the other hand, the noticeable differences in the power-spectral density with APG suggest that the energising mechanisms of the APG are different to those of large Reynolds numbers (Vinuesa et al. Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018). One of the main observations that can be drawn from the power-spectral density in figure 11 is the energisation of the smallest scales in the near-wall region (i.e. for $y_{n}^{+}\approx 15$) with adverse pressure gradient, an effect which is not present in high-Reynolds-number flows. In order to quantify this effect, in figure 12(a) we show the contribution to the various Reynolds stresses, at $y_{n}^{+}=15$, of the (spanwise) scales below $\unicode[STIX]{x1D706}_{z}^{+}=44$. This is the wavelength of the lowest ZPG contour at $Re_{\unicode[STIX]{x1D70F}}=2480$. As can be observed in this figure, the integrated contribution to the various Reynolds stresses from the smaller scales increases with the APG magnitude, a fact that supports the mentioned energisation mechanism. Furthermore, in figure 12(b) we show the relative contribution of these scales to the Reynolds stress at the same location, and these trends also show the progressively larger contribution of the smaller scales for increasing $\unicode[STIX]{x1D6FD}$. For instance, the integrated contribution of the inner-scaled tangential velocity fluctuations with $\unicode[STIX]{x1D706}_{z}^{+}<44$ corresponds to approximately 3 % of the near-wall peak in the ZPG, whereas this doubles to approximately 6 % in the NACA4412. The energisation of the small scales close to the wall is not an artefact of the viscous scaling, as can be observed in figure 13 where the power-spectral density is scaled in outer units (note that this conclusion also holds if the spanwise wavelength is scaled in outer units), and it is therefore a genuine effect of the APG. Furthermore, the large scales in the near-wall region are also energised by the APG. Despite the fact that in the outer region both mechanisms (i.e. APG and Reynolds number) show similar effects on the large scales of the TBL, the analysis is focused on the small scales in this region where the most noticeable differences between the cases are found. Furthermore, figure 11 shows a significant contribution of small-scale energy (with $\unicode[STIX]{x1D706}_{z}^{+}\simeq 100$) in the outer region, i.e. for $y_{n}^{+}$ larger than approximately 100, of all the Reynolds-stress tensor terms, in contrast to what is observed in the ZPG cases. Whereas the ZPG shows a decrease in the power-spectral density of these scales when increasing the Reynolds number, the boundary layers subjected to APG exhibit an increasing trend of the power-spectral density of the small scales in the outer region, a behaviour that is most prominent in the NACA4412 case (i.e. stronger APG distribution along the surface of the wing section). This result suggests that the APG promotes the wall-normal transport of the smaller scales from the near-wall region towards the outer region of the turbulent boundary layer, as discussed by Vinuesa et al. (Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018), which generates the observed increment in the power-spectral density of small scales far from the wall.

Figure 12. (a) Integral of the inner-scaled pre-multiplied spanwise power-spectral density of the various Reynolds stresses (see figure 11) with respect to $\unicode[STIX]{x1D706}_{z}^{+}$ for values of $\unicode[STIX]{x1D706}_{z}^{+}<44$. (b) Ratio of the integral for values up to $\unicode[STIX]{x1D706}_{z}^{+}<44$ and the integral along the full range of $\unicode[STIX]{x1D706}_{z}^{+}$. Analysis for $y_{n}^{+}=15$.

Focusing on the general characteristics of the power-spectral densities, further observations can be made. The first one is the emergence of the spectral outer peak (i.e. a second contour of largest power-spectral density $k_{z}\unicode[STIX]{x1D719}_{u_{t}u_{t}}^{+}=3.4$ at $y^{+}\approx 100$ and $\unicode[STIX]{x1D706}_{z}^{+}\approx 300$) in the tangential fluctuations (figure 11a) for strong APG TBLs such as the NACA4412 at $x_{ss}/c=0.75$. However, this is not an effect only of the APG as it appears also in ZPG TBLs at very high Reynolds numbers, as shown by Hutchins & Marusic (Reference Hutchins and Marusic2007a) and Eitel-Amor et al. (Reference Eitel-Amor, Örlü and Schlatter2014). According to Hutchins & Marusic (Reference Hutchins and Marusic2007b), the outer spectral peak is always present but, due to the lack of separation of scales at low Reynolds number, both the inner and outer peaks coincide and they cannot be noticed separately. Nevertheless, note that in the case of ZPG one can observe both spectral peaks for $Re_{\unicode[STIX]{x1D70F}}\geqslant 1700$ (Hutchins & Marusic Reference Hutchins and Marusic2007b), whereas, with a strong APG, a clear distinction between both spectral peaks is achieved for much lower $Re_{\unicode[STIX]{x1D70F}}$, as observed in the NACA4412; thus, the separation of scales is clearly enhanced by the APG. Subsequently, it is interesting to note what the effect of the APG is on the spectral inner peak of the tangential fluctuations. Whereas the inner-scaled near-wall peak in the statistics shown in figure 7 grows with increasing APG, the spectral near-wall peak is reduced, when scaled in viscous units, as the APG is increased. At the same time, the energy is spread over a wider range of scales in $\unicode[STIX]{x1D706}_{z}^{+}$ at a certain wall-normal location. Moreover, the power-spectral densities of the wall-normal fluctuations and Reynolds-shear stress show that the energetic scales are located farther from the wall than those in the other velocity components, a fact that complies with the wall-normal component of the scales being limited in the near-wall region due to the impermeability of the wall (Townsend Reference Townsend1976).

Figure 13. Outer-scaled pre-multiplied one-dimensional power-spectral density of the (a) tangential, (b) wall-normal, (c) spanwise velocity fluctuations and (d) Reynolds-shear stress of the NACA0012 and NACA4412 wing sections at matching $\unicode[STIX]{x1D6FD}$ and $Re_{\unicode[STIX]{x1D70F}}$ values, and ZPG at $Re_{\unicode[STIX]{x1D70F}}=305$ (Eitel-Amor et al. Reference Eitel-Amor, Örlü and Schlatter2014). The dotted lines represent the small-scale bounds of the lowest ZPG contour at $Re_{\unicode[STIX]{x1D70F}}=305$. The contours correspond to $k_{z}\unicode[STIX]{x1D719}_{u_{t}u_{t}}/U_{e}^{2}=(1,2.5,4)\times 10^{-3}$; $k_{z}\unicode[STIX]{x1D719}_{u_{n}u_{n}}/U_{e}^{2}=(0.3,0.8,1.3)\times 10^{-3}$; $k_{z}\unicode[STIX]{x1D719}_{ww}/U_{e}^{2}=(0.5,1.2,2)\times 10^{-3}$ and $k_{z}\unicode[STIX]{x1D719}_{u_{t}u_{n}}/U_{e}^{2}=(0.3,0.8,1.3)\times 10^{-3}$.

So far we have analysed the effect of APG magnitude on the energetic scales in TBLs; however, we are also interested in the assessment of the flow-history effect on these scales. Therefore, we follow the same approach as with the turbulence statistics and compute the power-spectral density of both wing sections for the matching $\unicode[STIX]{x1D6FD}-Re_{\unicode[STIX]{x1D70F}}$ case. Recall that this specific case requires the use of outer scaling since $u_{\unicode[STIX]{x1D70F}}$ is not an appropriate scaling parameter at $x_{ss}/c=0.95$ in the NACA0012 case. Figure 13 shows the outer-scaled pre-multiplied one-dimensional power-spectral density of the non-zero Reynolds stresses for the two wings, together with the power-spectral density of the ZPG at $Re_{\unicode[STIX]{x1D70F}}=305$ (Eitel-Amor et al. Reference Eitel-Amor, Örlü and Schlatter2014), which is a value of $Re_{\unicode[STIX]{x1D70F}}$ similar to those of the NACA profiles. The first observation that can be drawn from figure 13 is that the small scales in the outer region of the NACA4412 case are more energetic than in the NACA0012 case, suggesting that the flow history has an effect on the vertical displacement of the smallest scales in the near-wall region towards the outer layer. This is further quantified in figure 14, in which we illustrate both the absolute and relative contributions to the outer-scaled Reynolds stresses in the outer region from the smallest scales. In particular, we consider the wall-normal location $y_{n}^{+}=100$, and then define a cutoff wavelength $\unicode[STIX]{x1D706}_{z,c}$ which follows the least energetic ZPG contour, as marked in figure 13. Also in this case the smallest scales contribute more to the Reynolds stresses with the APG than with the ZPG. Interestingly, and since the two wings have the same local $\unicode[STIX]{x1D6FD}$, the increasing trends in figure 14 are observed for the average APG magnitude $\overline{\unicode[STIX]{x1D6FD}}$. In the tangential velocity fluctuations the value of $\unicode[STIX]{x1D706}_{z,c}^{+}=100$ (see figure 13a) at $y_{n}^{+}=100$. For this case, the scales smaller than $\unicode[STIX]{x1D706}_{z,c}$ are responsible for approximately 8 % of the outer-scaled fluctuations in the ZPG, whereas this fraction is above 11 % in the NACA4412. Also in the outer region, one can observe that the large scales are more energetic when subjected to a greater $\overline{\unicode[STIX]{x1D6FD}}$, therefore the integrated APG has an energising effect on all of the scales in the outer layer. These results are consistent with those from figure 11 and the turbulence statistics. On the other hand, figure 7(d) showed a decreasing trend of the inner peak in outer scaling with increasing $\overline{\unicode[STIX]{x1D6FD}}$, a characteristic that can be observed in figure 13(b). This figure shows that the energetic content of the spectral inner peak in outer scaling is significantly reduced for large values of the integrated APG.

Figure 14. (a) Integral of the outer-scaled pre-multiplied spanwise power-spectral density of the Reynolds stresses (see figure 13) with respect to $\unicode[STIX]{x1D706}_{z}^{+}$ for values of $\unicode[STIX]{x1D706}_{z}^{+}<\unicode[STIX]{x1D706}_{z,c}^{+}$. (b) Ratio of the integral for values up to $\unicode[STIX]{x1D706}_{z}^{+}<\unicode[STIX]{x1D706}_{z,c}^{+}$ and the integral along the full range of $\unicode[STIX]{x1D706}_{z}^{+}$. Analysis for $y_{n}^{+}=100$ and the values of $\unicode[STIX]{x1D706}_{z,c}^{+}$ given in figure 13.

4.2 Linear coherence spectra

One of the main conclusions obtained in the one-dimensional spectral analysis of the TBLs is that the energisation mechanisms of the APG and high-$Re$ ZPGs are different due to the presence of energetic small scales in the outer layer when the TBL is subjected to an APG, in addition to other differences found between the cases. The assessment of the energisation mechanisms can be extended by computing the correlation between the scales across the boundary layer as proposed by Baars, Hutchins & Marusic (Reference Baars, Hutchins and Marusic2017). This requires the study of the linear coherence of the tangential velocity fluctuations at a reference wall-normal location in the near-wall region $y_{R}$ and the rest of the locations normal to the wing surface through the computation of the linear coherence spectrum (LCS),

(4.1)$$\begin{eqnarray}\unicode[STIX]{x1D6FE}^{2}(y_{n},y_{R};\unicode[STIX]{x1D706}_{z})\equiv {\displaystyle \frac{|\langle \tilde{u} _{t}(y_{n};\unicode[STIX]{x1D706}_{z})\tilde{u} _{t}^{\ast }(y_{R};\unicode[STIX]{x1D706}_{z})\rangle |^{2}}{\langle |\tilde{u} _{t}(y_{n};\unicode[STIX]{x1D706}_{z})|^{2}\rangle \langle |\tilde{u} _{t}(y_{R};\unicode[STIX]{x1D706}_{z})|^{2}\rangle }}.\end{eqnarray}$$

Figure 15. Inner-scaled pre-multiplied one-dimensional power-spectral density $k_{z}\unicode[STIX]{x1D719}_{u_{t}u_{t}}^{+}$ (——) and linear coherence spectra $\unicode[STIX]{x1D6FE}^{2}$ of the tangential velocity fluctuations with respect to the near-wall region (– – –). (a) ZPG TBL at different Reynolds numbers and (b) wing sections at $x_{ss}/c=0.75$ from the leading edge. The contours correspond to $k_{z}\unicode[STIX]{x1D719}_{u_{t}u_{t}}^{+}=(1.0,2.1,3.4)$ and $\unicode[STIX]{x1D6FE}^{2}=(0.2,0.4,0.6,0.8)$. The reference location for $\unicode[STIX]{x1D6FE}^{2}$ was $y_{R}^{+}=2$.

Here, $\tilde{u} _{t}(y_{n};\unicode[STIX]{x1D706}_{z})$ is the Fourier transform of the tangential velocity at the wall-normal location $y_{n}$, in either the spanwise direction or in time; $\tilde{u} _{t}^{\ast }(y_{n};\unicode[STIX]{x1D706}_{z})$ denotes the complex conjugate of the tangential velocity at the wall-normal location $y_{n}$, $\langle \cdot \rangle$ designates ensemble averaging and $|\cdot |$ corresponds to the modulus. Figure 15 shows the power-spectral densities presented in figure 11 together with the LCS obtained using as reference the location $y_{R}^{+}=2$, for the APG TBLs at $x_{ss}/c=0.75$ and for ZPG TBLs at different Reynolds numbers. This figure indicates that there are significant differences in the LCS from APGs and ZPGs at higher $Re$. Focusing first on figure 15(a), it can be observed that the outer region of the turbulent boundary layer at low $Re$ shows very low coherence with the near-wall region whereas at higher $Re$ the effect is completely different; in the outer region the spectral outer peak exhibits a significant correlation with the near-wall region. This is consistent with the experimental results presented by Baars et al. (Reference Baars, Hutchins and Marusic2017), who related this effect with the attached-eddy hypothesis described by Townsend (Reference Townsend1976). Nevertheless, in figure 15(b), despite the fact that the APG generates a strong spectral outer peak even at a lower $Re$, the correlation between the outer region and the near-wall region is very low, completely the opposite to what is observed in the ZPG at high $Re$. This result indicates that in TBLs subjected to strong APGs (at low $Re$) there is a greater independence between the outer and inner regions, possibly implying that the largest turbulent structures may not be attached to the wall. Thus, our results suggest that when the $Re$ is low and the APG is strong, the attached-eddy hypothesis may not be applicable and therefore they confirm that the energisation mechanisms of APGs and increasing $Re$ are different. This is in agreement with the work by Kitsios et al. (Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017), who reported that very strong APGs lead to a flow behaviour closer to that of a free-shear layer rather than a ZPG TBL, and with Vinuesa et al. (Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018) who observed different energising mechanisms in the outer region due to APG and $Re$. It is also important to note that the observed effects of APGs on TBLs are relatively less intense at progressively higher $Re$ (Vinuesa et al. Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018).

Figure 16. Inner-scaled two-dimensional pre-multiplied power-spectral density of the tangential velocity fluctuations $k_{z}k_{t}\unicode[STIX]{x1D719}_{u_{t}u_{t}}^{+}$ at $y_{n}^{+}\simeq 15$. (a) ZPG TBL data at different $Re_{\unicode[STIX]{x1D70F}}$ (Eitel-Amor et al. Reference Eitel-Amor, Örlü and Schlatter2014), (b) comparison of the ZPG TBL at different $Re_{\unicode[STIX]{x1D70F}}$ with spectra of the wing sections at $x_{ss}/c=0.4$ and 0.75. The contours correspond to $k_{z}k_{t}\unicode[STIX]{x1D719}_{u_{t}u_{t}}^{+}=(0.15,1)$. The dashed straight line in (a) is given by $\unicode[STIX]{x1D706}_{z}^{+}=1.5\unicode[STIX]{x1D706}_{t}^{+}$.

4.3 Two-dimensional power-spectral densities

In this section we use the spanwise and temporal information from the time series to compute the two-dimensional power-spectral density and further extend the analysis of APG and flow-history effects on TBLs. The two-dimensional power-spectral density is computed as a function of the inner-scaled spanwise wavelength $\unicode[STIX]{x1D706}_{z}^{+}$ and the inner-scaled temporal period $\unicode[STIX]{x1D706}_{t}^{+}$. As explained in § 4.1, the power-spectral density in terms of the spanwise wavelength is computed by means of the FFT, whereas the power-spectral density in terms of the temporal period is computed through Welch’s overlapping window method due to the non-periodicity of the temporal signal.

We begin with the analysis of the two-dimensional pre-multiplied spectra in the near-wall region at a wall-normal distance $y_{n}^{+}\simeq 15$. Figure 16(a) shows the effect that increasing $Re$ produces on the two-dimensional spectra in ZPG TBLs, whereas figure 16(b) compares this effect with the impact of APGs. In figure 16(a) it can be observed that $Re$ has a significant effect on the largest scales at that location and a less noticeable effect on the smallest ones, with only small variations in the spectral density contours up to $Re_{\unicode[STIX]{x1D70F}}=1170$. This can be directly compared with the work by Hoyas & Jiménez (Reference Hoyas and Jiménez2006) and Sillero et al. (Reference Sillero, Jiménez, Moser and Malaya2011), who discussed two-dimensional spectra at the same location in a turbulent channel and compared the two-dimensional spectra of the ZPG TBL and the turbulent channel in the near-wall region, respectively. In the former, Hoyas & Jiménez (Reference Hoyas and Jiménez2006) discovered that the two-dimensional spectrum of the tangential velocity fluctuations in channel flow scales very well in viscous units, except for the spectral ridge that appears in the right-top corner of the spectral maps, which follows the line $\unicode[STIX]{x1D706}_{z}\sim \unicode[STIX]{x1D706}_{x}$ with increasing $Re$. In the ZPG TBL, the behaviour of the spectral ridge coincides with the one in the channel, as shown also by Sillero et al. (Reference Sillero, Jiménez, Moser and Malaya2011), although the low-$Re$ effect at the smallest scales was not discussed in any of these works. Focusing now on the APG effects, figure 16(b) shows that the various spectral maps do not collapse in viscous units, as in the case of the ZPG TBL or the channel. It can be observed that, despite the fact that in the lower edge of the spectra the contours coincide in all the cases, the rest of the two-dimensional spectral contour lines exhibit different behaviours for the various $\unicode[STIX]{x1D6FD}$ and $Re$ values. In the case of the APG, the energisation of the scales takes place across all the scales and does not generate a specific ridge as in the high-$Re$ ZPG TBL, at least for the Reynolds numbers under investigation here. In addition, it can be observed that the strong APG has a large effect on the smallest scales. Regarding the differences among flows, the spectral ridge that appears at high $Re$ in the channel and the TBL corresponds to the near-wall footprint of large scales in the logarithmic region (Hoyas & Jiménez Reference Hoyas and Jiménez2006), whereas in the wing sections this ridge does not appear because there are not such large structures in the boundary layer (this is clearly seen in the spanwise spectra shown in figures 11 and 15). Therefore, one of the main observations that can be made from this analysis is that the APG strongly energises all the scales but does not generate a relevant number of larger structures. This is in contrast to the increment in $Re$, which generates noticeably larger structures with a modulation and superposition impact on the near-wall region, but interestingly, the higher $Re$ does not directly energise the turbulent structures in the near-wall region. This result is consistent with the finding obtained from the LCS since the ZPG does show a significant footprint of large scales near the wall (i.e. the spectral ridge), whereas the APG does not exhibit such a high correlation with larger scales in the near-wall region.

Figure 17 shows the two-dimensional spectra at $y_{n}^{+}\simeq 150$, a location already selected by del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) and Chandran et al. (Reference Chandran, Baidya, Monty and Marusic2017) in their studies of spectra in channel flow and ZPG TBL, respectively. Note that, despite focusing mainly on one-dimensional spectra, Jiménez et al. (Reference Jiménez, Hoyas, Simens and Mizuno2010) compared the two-dimensional spectra of a ZPG TBL and a turbulent channel also at the buffer layer, showing very similar contours in the two flows, differing in intensity but matching in shape. This similarity of the ZPG TBL with the channel allows us to compare our ZPG TBL results with those presented by del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004). Figure 17(a) shows the effect of increasing $Re$ on the two-dimensional spectra in ZPGs, where the $Re_{\unicode[STIX]{x1D70F}}=2480$ case coincides with the lowest $Re$ studied experimentally by Chandran et al. (Reference Chandran, Baidya, Monty and Marusic2017). Our results are consistent with those presented by del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) and Chandran et al. (Reference Chandran, Baidya, Monty and Marusic2017), since the various bounds of the spectra coincide: there is a linear relation $\unicode[STIX]{x1D706}_{z}\sim \unicode[STIX]{x1D706}_{t}$ in the small-scale region marked by the dashed line whereas the larger scales, in addition to the lower edge of the spectra, show a square-root relation $\unicode[STIX]{x1D706}_{z}^{+}\sim (\unicode[STIX]{x1D706}_{t}^{+})^{1/2}$ represented by the dotted lines. Furthermore, these spectra in the overlap layer show very similar results to those in the near-wall region, with the main difference being that, in this case, the ridge lies along $\unicode[STIX]{x1D706}_{z}^{+}\sim (\unicode[STIX]{x1D706}_{t}^{+})^{1/2}$ and is related to the largest scales, as discussed by del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004), whereas the rest of the scales show similar inner-scaled energy at different Reynolds numbers, as was observed close to the wall. On the other hand, once again the APG TBLs exhibits a different behaviour with respect to the ZPG TBLs as shown in figure 17(b). It can be observed that at these $Re$ there is not a clear ridge at which the energy of the scales is increased, but rather the APG energises all the scales, this increment being more significant in the smallest scales. A possible explanation for the small-scale energy increase in the overlap layer could be due to the vertical transport of the smallest scales near the wall towards the outer region. However, some similarities can be noticed between the APG and the low-$Re$ ZPG. In both cases the spectral contours are bounded by $\unicode[STIX]{x1D706}_{z}^{+}\sim (\unicode[STIX]{x1D706}_{t}^{+})$ at the top and $\unicode[STIX]{x1D706}_{z}^{+}\sim (\unicode[STIX]{x1D706}_{t}^{+})^{1/2}$ at the bottom, although these bounds are shifted (upwards and downwards in the upper and lower edges, respectively) when subjected to an APG, an effect that does not take place in high-$Re$ ZPG TBL. Therefore, it is clear that the energisation mechanism of the APG is completely different than the mechanism of increasing $Re$. Also note that the APG TBL shows a two-dimensional spectrum similar to the equivalent ZPG TBL with same $Re_{\unicode[STIX]{x1D70F}}$ in terms of the relation between $\unicode[STIX]{x1D706}_{z}^{+}$ and $\unicode[STIX]{x1D706}_{t}^{+}$, but the APG broadens the range of energetic scales.

Figure 17. Inner-scaled two-dimensional pre-multiplied power-spectral density of the tangential velocity fluctuations $k_{z}k_{t}\unicode[STIX]{x1D719}_{u_{t}u_{t}}^{+}$ at $y_{n}^{+}\simeq 150$. (a) ZPG TBL at various $Re_{\unicode[STIX]{x1D70F}}$ (Eitel-Amor et al. Reference Eitel-Amor, Örlü and Schlatter2014) and (b) comparison of the ZPG at different $Re_{\unicode[STIX]{x1D70F}}$ with spectra of the wing sections at $x_{ss}/c=0.4$ and 0.75. The contours correspond to $k_{z}k_{t}\unicode[STIX]{x1D719}_{u_{t}u_{t}}^{+}=0.15$. The dashed straight lines represent $\unicode[STIX]{x1D706}_{z}^{+}\sim \unicode[STIX]{x1D706}_{t}^{+}$ and the dotted lines correspond to $\unicode[STIX]{x1D706}_{z}^{+}\sim (\unicode[STIX]{x1D706}_{t}^{+})^{1/2}$.