3.2.1 Streamwise velocity-fluctuating structures

In this section, we examine the modification of large-scale structures in the log layer as the APG strength increases to identify the relationship between the APG and the large-scale features and its connection to the statistics discussed above. First, in order to gain some idea about spatial features of large-scale structures in APG flows, we visualize instantaneous streamwise velocity-fluctuating ( $u^{\prime }$ ) structures at $y/\unicode[STIX]{x1D6FF}=0.15$ (a nominal upper extent of the log layer). Figure 7(a,b) shows the instantaneous flow fields for ZPG and mild APG flows for the entire domain ( $x/\unicode[STIX]{x1D703}_{in}<1500$ ). In addition, figure 7(c,d) exhibits instantaneous snapshots of moderate and strong APG flows ( $m=-0.25$ and $-0.32$ ). It should be noted that the computational domain for the two strong APG cases with $m=-0.25$ and $-0.32$ is larger than those of the ZPG and mild APG cases due to the large growth rate of the boundary layer thickness. As shown, characteristics of outer-region large-scale turbulent structures are highly sensitive to the strength of the APG, and the development and dynamics of the structures wholly differ.

Figure 7.

Instantaneous view of $u^{\prime }$ -structures normalized by $U_{0}$ at $y/\unicode[STIX]{x1D6FF}=0.15$ for ZPG and APG TBL flows. The boundary layer thickness in each panel is obtained at the centre of the computational domain. (a) Zero pressure gradient flow with $m=0.0$ , (b) APG flow with $m=-0.15$ , (c) APG flow with $m=-0.25$ and (d) APG flow with $m=-0.32$ . The rectangular boxes in (a–d) are included for a detailed analysis of the flows later. The computational domain sizes for $m=-0.25$ and $-$ 0.32 are larger than those for $m=0.0$ and $-$ 0.15.

Because the appearance of the large-scale structures in the raw flow fields in figure 7 is very complex, with multiple scales and varying strength levels, the extraction of characteristic lines for large-scale structures is more helpful to identify the spatial organization of the structures. Here, the characteristic lines are detected based on the streak detection algorithm in Lee et al. (Reference Lee, Lee, Choi and Sung2014). The algorithm is briefly explained below.

Step 1: The instantaneous flow field is filtered using a two-dimensional Gaussian filter on the cross-stream plane to attenuate the small-scale features of $u^{\prime }$ . The standard deviation of the Gaussian filter along the wall-normal direction is determined based on the half-width of the two-point correlation coefficient of the streamwise velocity fluctuations with a contour of 0.5 in the spanwise direction to separate the negative $u^{\prime }$ -structure from adjacent negative $u^{\prime }$ -structures.

Step 2: Spectral low-pass filtering is applied to the Gaussian-filtered flow field with a cutoff wavelength of $\unicode[STIX]{x1D706}_{z}/\unicode[STIX]{x1D6FF}=0.5$ to extract the large-scale signal $\hat{u}$ .

Step 3: The characteristic lines ( $CL$ s) for the large-scale structures on the $xz$ plane are obtained from the local extrema of the Gaussian- and low-pass-filtered field $\hat{u}$ :

(3.1) $$\begin{eqnarray}\displaystyle CL=\left\{\begin{array}{@{}ll@{}}1,\quad & \text{where }\unicode[STIX]{x2202}\hat{u} /\unicode[STIX]{x2202}z=0\text{ and }\hat{u} >u_{th},\\ -1,\quad & \text{where }\unicode[STIX]{x2202}\hat{u} /\unicode[STIX]{x2202}z=0\text{ and }\hat{u} <-u_{th},\\ 0,\quad & \text{otherwise}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

The threshold ( $u_{th,raw}$ ) used to identify the negative $u^{\prime }$ -structure in the raw flow field is carefully chosen to be 10 % of the free-stream velocity ( $U_{0}$ ). In addition, because the velocity fluctuations are weakened by the two filters, the threshold value for the detection of $CL(u_{th})$ is corrected based on the ratio of the correlation functions, e.g. $u_{th}(\boldsymbol{x})=f(\boldsymbol{x})u_{th,raw}$ , where $f(\boldsymbol{x})=\langle u(\boldsymbol{x})\hat{u} (\boldsymbol{x}^{\prime })\rangle /\langle u(\boldsymbol{x})u(\boldsymbol{x}^{\prime })\rangle$ . For more information, the readers can refer to the earlier paper of Lee et al. (Reference Lee, Lee, Choi and Sung2014).

Figure 8.

Instantaneous flow fields of the $\unicode[STIX]{x1D6FF}$ -scaled $u^{\prime }$ -structures in the equilibrium regions at $y/\unicode[STIX]{x1D6FF}=0.15$ (on the $xz$ plane). The boundary layer thickness in each panel is obtained at the centre of the equilibrium region. White lines indicate $CL=-1$ for the streamwise dimension of the negative $u^{\prime }$ -structures. (a) Zero pressure gradient flow, (b) APG flow, $m=-0.15$ , (c) APG flow, $m=-0.25$ , and (d) APG flow, $m=-0.32$ . The flow fields are extracted from the rectangular boxes in figure 7. The rectangular boxes with red dashed lines in (a,b) indicate very long structures with lengths greater than ${\sim}5\unicode[STIX]{x1D6FF}$ .

Figure 8 shows the instantaneous flow fields of the $\unicode[STIX]{x1D6FF}$ -scaled $u^{\prime }$ -structures in the equilibrium regions (rectangular boxes in figure 7) at $y/\unicode[STIX]{x1D6FF}=0.15$ . Here, $CL=-1$ is depicted by the white solid lines in each panel. Although the rectangular boxes for all cases are shown for the equilibrium regions in figure 7, the flows under the ZPG and APGs when $m=-0.15$ and $-0.25$ reach states of equilibrium further upstream, as shown in figure 2(c). Because the coordinate origin in the streamwise direction is defined at a location where the pressure gradient is applied in the present study, and the boundary layer thickness for each flow is obtained in the middle of the equilibrium region, the $x$ axis label in figure 8 differs with an increase of $m$ . Figure 8(a) displays that the ZPG flow includes a number of long negative $u^{\prime }$ -streaks (black colour contour) in the log layer with multiple scales around them with varying strengths. In particular, the rectangular boxes with red dashed lines highlight that very long meandering structures with lengths greater than ${\sim}5\unicode[STIX]{x1D6FF}$ travel with streamwise coherence, and the negative $u^{\prime }$ -structures are often inclined towards the spanwise direction at angles of approximately $\pm 0{-}7^{\circ }$ , consistent with earlier observations in canonical wall-bounded turbulent flows (Hutchins & Marusic Reference Hutchins and Marusic2007a ; Lee & Sung Reference Lee and Sung2011; Baltzer, Adrian & Wu Reference Baltzer, Adrian and Wu2013). As the mild APG ( $m=-0.15$ ) is applied to the TBL flow (figure 8 b), regimes of very long meandering negative $u^{\prime }$ -structures with lengths greater than ${\sim}5\unicode[STIX]{x1D6FF}$ are also found (rectangular boxes) with spanwise angles. However, as the APG strength is increased further in figure 8(c,d), the streamwise scales of structures based on the $CL$ s appear to be reduced in the equilibrium region, although large-scale structures are observed with large spanwise inclination angles in raw flow fields (figure 7).

The distinctive view of the large-scale $u^{\prime }$ -structures in figure 8 compared with the observation in the raw flow field (figure 7) for the two strong APG TBL flows is due to small-scale features in the large-scale patterns. For the ZPG TBL flow shown in figure 7(a), a long negative $u^{\prime }$ -structure includes multiple scales of motion with varying strengths along the streamwise direction. Several of the strong segments of the small negative $u^{\prime }$ -structures (the dark blue contour) appear to be streamwise-aligned coherently to create a long $u^{\prime }$ -structure; these small-scale motions consist of slightly streamwise-elongated negative-velocity fluctuation peaks that decay in magnitude rapidly with the distance from the peak location. A similar spatial feature of small-scale motions is observed for the mild APG flow in figure 7(b), and concatenation of the small motions induces wavering or spanwise-inclined very long motions. However, as the pressure gradient increases further, the number of strong small-scale negative fluctuations decreases significantly within the large-scale pattern, and the existing small-scale motions are scattered and disconnected but often organized along large-scale lines. Thus, the spatial organization of the $CL$ suggests that the small-scale structures aligned along the large-scale pattern for the strong APG flows are not coherently aligned to form large-scale structures, contrary to ZPG and mild APG flows. However, because the $CL$ above is identified with a relatively high threshold value, the choice of a lower threshold value captures the large-scale pattern of the $u^{\prime }$ -structure, similar to those in figure 7(c–d), although the identified large-scale structure is very weak in strength.

Figure 9 displays an isosurface of the swirling strength ( $\unicode[STIX]{x1D706}_{ci}$ ) from the present simulation. The data are extracted from the equilibrium region for each flow in figures 7 and 8, and the contour levels are depicted using the streamwise velocity fluctuations ( $u^{\prime }/U_{0}$ ) to identify the connection between the vortical structures and the $u^{\prime }$ -structures. The swirling strength $\unicode[STIX]{x1D706}_{ci}$ is calculated from the three-dimensional velocity gradient tensor (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999). For the ZPG flow, the predominance of hairpin vortices throughout the domain is clear, with the hairpins coherently aligned in the streamwise direction to create a train of hairpins (Adrian Reference Adrian2007). Because the hairpin heads and legs induce retarded streamwise momentum beneath them, and given that the fluid in the outside of the hairpins convects downstream more rapidly due to the forward induction of the hairpins, large-scale positive and negative $u^{\prime }$ -structures are created. The mild APG flow in figure 9(b) also reveals a multitude of vortices and arches coherently aligned in the streamwise direction; these vortices are closely associated with the negative $u^{\prime }$ -structures. However, as the APG strength is increased further, it is difficult to describe the structures as an ordered hairpin forest. Most of them are significantly distorted and are oriented randomly, although individual hairpins with complete shapes are observed. Therefore, for strong APG flows, the streamwise coherence of the vortices appears to be lost, at least compared with ZPG and mild APG flows, and less streamwise coherence of the hairpins for packet organization is created. The observation in figure 9 is consistent with the spatial organization of the $CL$ in figure 8. Finally, it is worth mentioning that although the present observation provides clear evidence for the predominance of hairpin vortices in the ZPG and mild APG TBL flows, similar to previous studies in ZPG TBL flows up to $Re_{\unicode[STIX]{x1D703}}=4700$ (Adrian Reference Adrian2007; Wu & Moin Reference Wu and Moin2010; Dennis & Nickels Reference Dennis and Nickels2011a ; Jodai & Elsinga Reference Jodai and Elsinga2016), there still remain issues for the presence of a ‘forest of hairpins’ in high-Reynolds-number ZPG flow (Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010; Schlatter & Őrlü Reference Schlatter and Őrlü2012).

Figure 9.

Isosurfaces for the swirling strength ( $\unicode[STIX]{x1D706}_{ci}$ ) in ZPG and APG flows: (a) ZPG flow, (b) APG flow, $m=-0.15$ , (c) APG flow, $m=-0.25$ , and (d) APG flow, $m=-0.32$ . The contour level is $\unicode[STIX]{x1D706}_{ci}\unicode[STIX]{x1D703}_{in}/U_{0}=0.2$ , as coloured by the streamwise velocity fluctuations ( $u^{\prime }/U_{0}$ ).

To provide statistical evidence for variation of the streamwise scales as the pressure gradient increases, two-point spatial correlations of the streamwise velocity fluctuations in the log layer are calculated for the ZPG and APG flows. The two-point correlation coefficient is defined as follows: $R_{uu}(r_{x},r_{y},r_{z})=\langle u^{\prime }(x_{r},y_{r},z_{r})u^{\prime }(x_{r}+r_{x},y_{r}+r_{y},z_{r}+r_{z})\rangle /\unicode[STIX]{x1D70E}_{u^{\prime }}^{2}$ , where spatial shifts from the references $x_{r}$ , $y_{r}$ and $z_{r}$ are represented as $r_{x}$ , $r_{y}$ and $r_{z}$ , with $\unicode[STIX]{x1D70E}_{u^{\prime }}$ as the root mean square of $u^{\prime }$ . For a horizontal plane, $r_{y}=0$ , and the correlation coefficient is independent of the reference position of $z_{r}$ due to flow homogeneity in the spanwise direction. In addition, although the correlation contours are shown at a fixed $x_{r}$ in the present study, the streamwise variation of the contours is negligible within the equilibrium region. In order to highlight large-scale features, the correlations are outer-scaled and are depicted while focusing on low contour levels (Hutchins & Marusic Reference Hutchins and Marusic2007a ; Baltzer et al. Reference Baltzer, Adrian and Wu2013). For the ZPG TBL data in figure 10(a), the general appearance of the correlation with a streamwise-elongated positive-correlation region flanked by a negative-correlation region in the spanwise direction and an X-pattern with a zero-crossing point at approximately $x/\unicode[STIX]{x1D6FF}=3.0{-}3.5$ is very similar to that observed in a previous experimental study of ZPG TBLs (Hutchins & Marusic Reference Hutchins and Marusic2007a ). It is noted that a spanwise symmetric condition, implying reflectional symmetry of $R_{uu}$ with respect to the $r_{z}/\unicode[STIX]{x1D6FF}=0$ line, is not applied in the calculation to show the dominant imprint of the patterns for large-scale $u^{\prime }$ -structures with spanwise-inclined characteristics.

Figure 10.

Contours of two-point spatial correlations of the streamwise velocity fluctuations at $y_{r}/\unicode[STIX]{x1D6FF}=0.15$ . The reference streamwise location is $x_{r}/\unicode[STIX]{x1D703}_{in}=1125$ for each flow. (a) Zero pressure gradient flow, (b) APG flow, $m=-0.15$ , (c) APG flow, $m=-0.25$ , and (d) APG flow, $m=-0.32$ . The solid and dashed lines correspondingly indicate $R_{uu}=0.05$ and $-$ 0.05.

Compared with the ZPG data, the APG data when $m=-0.15$ , as shown in figure 10(b), indicate that the streamwise scale of the large-scale structures (solid line) based on the streamwise extent of a region for which $R_{uu}\geqslant 0.05$ is increased with a shallow spanwise-angled X-shaped pattern (shown in yellow). However, further increase of the APG strength, shown in figure 10(c,d), causes continuous shortening of the streamwise length scales for the large-scale structures in the log layer. The X-shaped correlation patterns with large inclination angles towards the spanwise direction (shown in yellow) when $m=-0.25$ and $-$ 0.32 are created by the highly spanwise-inclined structures, as observed in the instantaneous flow fields in figure 7, because the patterns of the low-level contours for the correlations can be understood as an imprint of the large- and very-large-scale structures in the flows. It should be noted that because the two-point correlations above provide averaged structures with a wide range of scales passing through the streamwise event location, small-scale structures may affect the characteristics of the large-scale structures near the event location. However, additional calculations for two-point spatial correlations of spectral-filtered large-scale streamwise velocity fluctuations with a cutoff wavelength of $\unicode[STIX]{x1D706}_{z}/\unicode[STIX]{x1D6FF}=0.5$ demonstrate that there is a negligible influence of the small-scale motions on our results (not shown here). As the magnitude of the APG is increased, the spanwise length scale is continuously increased in figure 10.

Figure 11.

Variations of the (a) streamwise and (b) spanwise length scales based on two-point spatial correlations $R_{uu}$ as $m$ varies. The reference location in the streamwise direction is in the middle of the equilibrium region for each flow, and the length scales are extracted based on the region of $R_{uu}\geqslant 0.05$ , as shown in figure 9 (black solid line).

In addition to the length scales in the log layer, variations of the streamwise and spanwise length scales for several wall-normal locations from the wall are estimated in figure 11 based on $R_{uu}$ . For the ZPG flow, the streamwise length scale normalized by the boundary layer thickness increases to $y/\unicode[STIX]{x1D6FF}=0.15$ and then gradually decreases far from the wall (Lee & Sung Reference Lee and Sung2011). For the mild APG flow, although the decreasing trend is similar to that of the ZPG flow, the streamwise coherence when $y/\unicode[STIX]{x1D6FF}<0.4$ is larger than that for the ZPG flow. Further increase in the pressure gradient demonstrates that the streamwise length scales of the structures are continuously reduced when $y/\unicode[STIX]{x1D6FF}<0.4$ . However, far from the wall, the streamwise length scales for the large-scale structures are similar irrespective of the APG strength. For the spanwise length scales shown in figure 11(b), the continuous increase is obvious for $y/\unicode[STIX]{x1D6FF}<0.3$ as the pressure gradient increases, consistent with the observation in the instantaneous view. However, this situation is reversed far from the wall. The wall-normal variation of the length scales in figure 11 provides evidence that the streamwise coherence of the large-scale motions in the outer layer ( $y/\unicode[STIX]{x1D6FF}<0.4$ ) is not directly correlated with the energy levels of the large-scale structures in the TBLs under the APGs (figure 4). As conjectured in the introduction, the present results suggest that for a mild APG flow with an approximate range of $\unicode[STIX]{x1D6FD}<2.0$ , the large-scale organization pattern of the velocity fluctuations in the log layer is more elongated in the streamwise direction than that of a ZPG flow, but for moderate and strong APG flows with an approximate range of $\unicode[STIX]{x1D6FD}>2.0$ , the spatial coherence of the large-scale motions is shortened in the streamwise direction.

Because the log region large-scale structures for the strong APG flows are highly meandering in the spanwise direction or are spanwise-inclined due to the streamwise alignment of highly spanwise-offset streaks (Hutchins & Marusic Reference Hutchins and Marusic2007a ; Baltzer et al. Reference Baltzer, Adrian and Wu2013), the spatial organization may mask the true streamwise length scales of $u^{\prime }$ -structures when measured along the $r_{z}=0$ line of correlations (Hutchins & Marusic Reference Hutchins and Marusic2007a ). In order to provide further evidence of the reduced streamwise coherence of strong APG flows, the ensemble-averaged population of the streamwise length scales for the lines of $CL=-1$ is examined. Figure 11 displays the averaged population for the streamwise length of the negative $u^{\prime }$ -structures ( $L_{CL}$ ) in the log layer. The population is calculated from the ratio of the number of $CL$ s in each length ( $N_{CL}$ ) to the total number of $CL$ . Although not shown, the population distribution, defined as the number of $CL$ s per unit area, shows a trend similar to that in the present results. It is clear that the population is highest for small-scale structures in ZPG and APG flows. For the mild APG flow with $m=-0.15$ , the number of large- and very-large-scale structures is slightly increased compared with the ZPG flow. The maximum streamwise length found for the mild APG flow is approximately ${\sim}10\unicode[STIX]{x1D6FF}$ , whereas the streamwise extent of the longest structure for the ZPG flow is approximately ${\sim}8\unicode[STIX]{x1D6FF}$ . As the pressure gradient is further increased, the population of the large-scale structures clearly decreases with reduced streamwise extents of the longest structures. For the strong APG flows with $m=-0.25$ and $-$ 0.32, the number of small-scale structures ( ${<}0.5\unicode[STIX]{x1D6FF}$ ) accounts for more than 60 % of all negative $u^{\prime }$ -structures. This result indicates that the reduced streamwise coherence for the strong APG flows is mostly due to the increased population of the small-scale structures and the reduced population of the large-scale structures. It is important to note that although the population distribution for $u^{\prime }$ -structures is influenced by the threshold value chosen, a careful inspection of the population while varying the threshold value shows a consistent result for ZPG and APG flows due to the relatively weak streamwise coherence for strong APG flows (figure 8). Dennis & Nickels (Reference Dennis and Nickels2011b ) noted that changing the level within a reasonable margin does not alter the distribution greatly, although their algorithm differs from that in the present study. In addition, when the population distribution of the negative $u^{\prime }$ -structures is estimated using raw data with a threshold value (Ganapathisubramani, Longmire & Marusic Reference Ganapathisubramani, Longmire and Marusic2003; Lee & Sung Reference Lee and Sung2011; Baltzer et al. Reference Baltzer, Adrian and Wu2013), it is confirmed that the population of large-scale structures decreases for strong APG TBL flows.

Figure 12.

Population distribution for the streamwise length of the lines of $CL=-1$ in the log layer as $m$ is varied. The streamwise length $L_{CL}$ is determined by the most upstream and downstream positions of each $CL$ , as shown in the inset.

Figure 13.

Identical to figure 10, but for skin-friction velocity-fluctuating structures. The solid and dashed lines indicate contour levels of 0.05 and $-$ 0.02 respectively.

3.2.2 Skin-friction velocity-fluctuating structures

Another indication of the reduction of the streamwise coherence for the log region large-scale structures can be found based on the skin-friction features, as large-scale structures convecting in the log layer of the TBL have a clear footprint in the near-wall region (Abe, Kawamura & Choi Reference Abe, Kawamura and Choi2004; Hutchins & Marusic Reference Hutchins and Marusic2007a ; Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2009; Hutchins et al. Reference Hutchins, Monty, Ganapathisubramani, Ng and Marusic2011). Figure 13 shows the two-point correlation contours for the skin-friction velocity fluctuations as the APG strength increases. As expected, the streamwise length scale for the mild APG TBL flow is longer than that for the ZPG TBL flow, consistent with the observation in figure 10. As the pressure gradient becomes stronger, the length scale is continuously shortened with larger spanwise length scales. Interestingly, the correlation in the strong APG TBL flows in figure 13(d) shows that the contours at $r_{x}/\unicode[STIX]{x1D6FF}=0$ are rounded, and a slight change of the contour level for large-scale structures (e.g. $R_{uu}=0.01$ ) induces relatively large variation in the spanwise length scale, while the contours for the large-scale components in the ZPG and relatively weak APG flows in figure 13(a–c) are streamwise-elongated with little influence on the spanwise extent according to the contour level. It is noted that for the correlation of the streamwise velocity fluctuations in figure 10(d), there is no rounded contour pattern near the event location, and the choice of a contour level lower than $R_{uu}=0.05$ does not result in large variation in the spanwise scale of the structures. The emergence of the circular contour pattern in the correlation of the skin-friction velocity fluctuations is most likely due to the high spanwise growth rate of near-wall large-scale structures in the APG flow. Figure 11 shows that the largest increase of the spanwise length scale associated with the large-scale motions under the strongest APG flow is observed near the wall compared with the ZPG flow, while the near-wall streamwise length scale is shortened the most. The decomposition of the two-point correlation contour in figure 13(d) into large- and small-scale components shows that the rounded contour is present in the large-scale component (not shown here), supporting our conjecture.

It is reported that large-scale $u^{\prime }$ -structures in the log layer are closely associated with the wall-shear stress fluctuations in a turbulent channel flow (Abe et al. Reference Abe, Kawamura and Choi2004). In addition, in a numerical study of a turbulent channel flow, Hwang (Reference Hwang2013) showed that large-scale structures with a spanwise length scale wider than $\unicode[STIX]{x1D706}_{z}^{+}=100$ play an important role in the generation of turbulent skin friction based on filtered simulations. For an APG flow, it is shown that the skin-friction coefficient decreases with an increase in the APG effect, as depicted in figure 2 (Lee & Sung Reference Lee and Sung2009; Monty et al. Reference Monty, Harun and Marusic2011), and this has often been explained by the fact that the near-wall streaks are significantly weakened for the APG flow due to the wider spanwise spacing of streamwise vortices (Na & Moin Reference Na and Moin1998; Lee & Sung Reference Lee and Sung2009). In addition, in a DNS study of a turbulent channel flow with a lower curved wall, Marquillie, Ehrenstein & Laval (Reference Marquillie, Ehrenstein and Laval2011) reported that the shape and intensity of the streaks in APG wall turbulence contribute to local onset of the instability, thus modifying the self-sustained near-wall cycle mechanism (Jiménez & Pinelli Reference Jiménez and Pinelli1999). In the present study, we find that the relatively reduced streamwise coherence of the large-scale $u^{\prime }$ -structures in the log layer induces the shortening of the large-scale skin-friction structures responsible for the generation of the turbulent skin-friction drag in figure 13. Thus, the less coherent large-scale organization of the structures in the log layer may be another cause of the decrease in the skin-friction drag for the strong APG flows. However, the reduction of the skin-friction coefficient for the mild APG flow ( $m=-0.15$ ), while the streamwise coherence of the outer-region large-scale structures is slightly increased compared with the ZPG flow, suggests that the near-wall streamwise vortices are still important with regard to the skin-friction drag.

Figure 14.

Premultiplied spanwise energy spectra of the streamwise velocity fluctuations $k_{z}\unicode[STIX]{x1D6F7}_{uu}/u_{\unicode[STIX]{x1D70F}}^{2}$ at $x_{r}/\unicode[STIX]{x1D703}_{in}=1125$ : (a) ZPG flow, (b) APG flow, $m=-0.15$ , (c) APG flow, $m=-0.25$ , and (d) APG flow, $m=-0.32$ . The contour level is varied from 0.2 to 2.6 in intervals of 0.2. The black and white dashed lines are shown to simplify the recognition of the inner and outer peak locations in the maps, and the dash–dot lines are to indicate the cutoff spanwise wavelength to separate large- and small-scale motions.

3.2.3 Spanwise energy spectra

In an attempt not only to provide an overall picture of the energy distributions in APG TBL flows but also to identify the dominant contribution of the wavelengths to the increase of the Reynolds stresses, premultiplied spanwise energy spectrum maps of the streamwise velocity fluctuations in the equilibrium regions are calculated. Given the two-point correlation function of each spanwise Fourier mode, $S_{uu}(x_{r},y_{r},k_{z})=\langle \hat{u} ^{\ast }(x_{r},y_{r},k_{z})\hat{u} (x_{r},y_{r},k_{z})\rangle$ , the spectrum in figure 14 is defined as

(3.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{uu}(x_{r},y_{r},k_{z})=S_{uu}(x_{r},y_{r},k_{z})+S_{uu}(x_{r},y_{r},-k_{z})=2Re\{S_{uu}(x_{r},y_{r},k_{z})\}, & & \displaystyle\end{eqnarray}$$

where $\hat{u}$ is the spanwise Fourier component of $u^{\prime }$ corresponding to the wavenumber $k_{z}$ ( $=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{z}$ ), and the superscript $^{\ast }$ indicates the conjugate pair of $\hat{u}$ . In the figure, the cutoff spanwise wavelength of $\unicode[STIX]{x1D706}_{z}/\unicode[STIX]{x1D6FF}=0.5$ to separate large- and small-scale features is depicted by horizontal dashed dot lines. For the ZPG TBL flow in figure 14(a), an energy peak, referred to as the ‘inner peak’, is observed at $y^{+}=12$ with a mean spanwise wavelength of the order of 100 in wall units, i.e. a typical signature of the near-wall cycle of self-sustained quasi-streamwise vortices (Jiménez & Pinelli Reference Jiménez and Pinelli1999). Although the overall shapes of the maps are similar, the introduction of the APGs induces larger wavelength peaks with $\unicode[STIX]{x1D706}_{z}^{+}=120$ in the near-wall region (see the black dashed lines) for the mild and moderate APGs in figure 14(b,c). The increased wavelength of the inner peaks in the APG flows indicates that the average spanwise spacing between the streaks increases (Lee & Sung Reference Lee and Sung2009). In addition, as the APGs are applied (for $m=-0.15$ and 0.25), the emergence of a secondary peak in the outer layer, referred to as an ‘outer peak’, is clear, corresponding to the presence of more energetic large-scale motions in the APG flows. However, for ZPG flows, scale separation is not clear due to the low Reynolds number in the present study. As the pressure gradient increases to $m=-0.32$ , the strength of the outer peak increases significantly, and the peak masks the inner peak in figure 14(d) due to its strong influence on the wall. It should be noted that the rise in the outer peak with an increase in $\unicode[STIX]{x1D6FD}$ is not due to the scaling of the energy spectra with the friction velocity, instead originating from the APG effect. The streamwise variation of the energy spectra in figure 15 shows that the outer peak observed in the equilibrium region for $m=-0.32$ is created immediately when the flow moves from the ZPG to the APG. As the flow moves along the downstream direction, the strength of the outer peak increases continuously, whereas the inner peak weakens continuously due to the direct influence of the outer peak. Similarly to the outer peak, the wavelength for the inner peak of the ZPG flow rapidly shifts to a larger wavelength for an APG flow, indicating that structural modifications occur simultaneously in both the inner and outer layers.

Figure 15.

Premultiplied spanwise energy spectra of the streamwise velocity fluctuations $k_{z}\unicode[STIX]{x1D6F7}_{uu}/u_{\unicode[STIX]{x1D70F}}^{2}$ for the APG flow with $m=-0.32$ at (a) $x_{r}/\unicode[STIX]{x1D703}_{in}=75$ , (b) $x_{r}/\unicode[STIX]{x1D703}_{in}=450$ , (c) $x_{r}/\unicode[STIX]{x1D703}_{in}=1125$ and (d) $x_{r}/\unicode[STIX]{x1D703}_{in}=2025$ . The contour level is varied from 0.2 to 2.6 in intervals of 0.2. The black and white dashed lines are shown to simplify the recognition of the inner and outer peak locations in the maps.

The presence of outer peaks in the spanwise energy spectra of the streamwise velocity fluctuations was previously reported in wall-bounded turbulent flows at a high Reynolds number (Hoyas & Jimenez Reference Hoyas and Jimenez2006; Bernardini & Pirozzoli Reference Bernardini and Pirozzoli2011). In particular, Hoyas & Jimenez (Reference Hoyas and Jimenez2006) reported that for a turbulent channel flow at $Re_{\unicode[STIX]{x1D70F}}=2003$ , an outer peak in the log layer with $\unicode[STIX]{x1D706}_{z}/\unicode[STIX]{x1D6FF}=1.4$ is aligned with an inner peak along a linear ridge, indicative of the linear growth of the spanwise length scale with the distance from the wall. Compared with previous results, however, the spectra in figure 14 reveal that the wall-normal locations of the outer peaks under the APGs (white dashed lines) are not situated at the geometrical centre of the log layer (Mathis et al. Reference Mathis, Hutchins and Marusic2009). Moreover, the peak locations are dependent on the APG strength when normalized by the outer scale. It is noted that the outer peak locations for $m=-0.25$ and $-0.32$ are similar to those of the Reynolds stress peaks in the outer layer in figure 4, and the scaling of the outer peak locations with the inner variables is consistent with the observation of Mathis et al. (Reference Mathis, Hutchins and Marusic2009) for a ZPG TBL flow with a high Reynolds number. In figure 14, the variation of the spanwise wavelengths associated with the outer peaks with an increase in the pressure gradient indicates that as the APG becomes stronger, larger scales are more energized in the outer layer, and the emergence of horizontal contour lines for low contour levels in the outer layer suggests that the enhanced large-scale energy in the outer layer is transferred to other wavelengths. For the ZPG and mild APG flows, the low contour levels (for example, the outermost contour level, 0.2) are sloped in the maps, indicating a shift to viscous decay at a longer wavelength as the wall is approached (Wu et al. Reference Wu, Baltzer and Adrian2012).

In figure 14(c,d), the horizontal contour lines are also found at high contour levels at approximately $\unicode[STIX]{x1D706}_{z}/\unicode[STIX]{x1D6FF}=0.8{-}1.0$ , suggesting that the large-scale energy in the outer layer is directly transferred to large-scale structures in the near-wall region. The stronger large-scale energy in the outer layer than that in the near-wall region for $m=0.32$ implies that the presence of outer peaks under APGs is not due to the linear scale growth of the small scale near the wall into the outer layer. The direct connection of the large-scale structures in the outer region and in the near-wall region in the spectra is consistent with the wall footprint of log layer large-scale structures on the inner region (Abe et al. Reference Abe, Kawamura and Choi2004; Hutchins & Marusic Reference Hutchins and Marusic2007a ). In addition, the enhanced large-scale energy in the outer layer under APGs is consistent with a previous experimental study by Harun et al. (Reference Harun, Monty, Mathis and Marusic2013), and the large-scale energy transfer from the outer layer to the near-wall region is similar to our observation in the Reynolds stress profiles in figure 4. Although the outer peak is regarded as a symptom of the large-scale organization of the velocity field in the log layer in a ZPG TBL flow (Hutchins & Marusic Reference Hutchins and Marusic2007a ), the present findings based on the correlations and energy spectra demonstrate that the increase in the large-scale energy in the outer layer for APG TBL flows is mostly due to the presence of more energized large-scale motions in the outer layer rather than to the frequent large-scale organization of velocity fluctuations.

Figure 16.

Identical to figure 14 but for premultiplied spanwise energy spectra of the Reynolds shear stress, $k_{z}\unicode[STIX]{x1D6F7}_{-uv}/u_{\unicode[STIX]{x1D70F}}^{2}$ . The contour level is varied from $-$ 0.6 to $-$ 0.05 in intervals of 0.05.

The spanwise spectrum maps for the Reynolds shear stress are presented in figure 16. Although the inner peak for the spectrum of the streamwise velocity fluctuations is dominantly characterized by approximately concentric circular contours in the ZPG flow, that in the spectrum of the Reynolds shear stress shows an inclined slope with a linear ridge between the inner and outer peaks, indicating linear spanwise growth of structures related to the Reynolds shear stress with the distance from the wall. The inner peak of the Reynolds shear stress spectra (the black dashed lines) is observed at a higher wall-normal position ( $y^{+}=30$ ) when $\unicode[STIX]{x1D706}_{z}^{+}=100$ compared with that for the spectra in figure 14, and the characteristics of the inner peaks are not affected by the APG strength. However, as the pressure gradient increases, the wall-normal locations for the outer peaks (the white dashed lines) move far from the wall, and larger scales of structures are more energized. Compared with the spectra of the streamwise velocity fluctuations, the outer peak locations in figure 16 are observed to be further from the wall due to the attenuation of the wall-normal velocity fluctuations near the wall, and the energy at the outer peak locations is weaker for the Reynolds shear stress spectrum map due to the less coherent structure of the Reynolds shear stress than the streamwise velocity-fluctuating structure in the outer layer.

Figure 17.

Premultiplied spanwise energy spectra of the (a,c) second- and (b,d) fourth-quadrant Reynolds shear stresses. (a,b) Zero pressure gradient flow and (c,d) APG flow, $m=-0.32$ . The contour level is varied from $-$ 0.6 to $-$ 0.05 in intervals of 0.05.

To examine the relative contributions of the second- and fourth-quadrant Reynolds shear stresses to the spanwise energy spectra in figure 16, quadrant spectrum maps are calculated by multiplying the quadrant fractions to the energy spectra of the Reynolds shear stress,

(3.3) $$\begin{eqnarray}\displaystyle [k_{z}\unicode[STIX]{x1D6F7}_{-uv}/u_{\unicode[STIX]{x1D70F}}^{2}]_{Qi}=\frac{\langle -u^{\prime }v^{\prime }\rangle _{Qi}}{\langle -u^{\prime }v^{\prime }\rangle }k_{z}\unicode[STIX]{x1D6F7}_{-uv}/u_{\unicode[STIX]{x1D70F}}^{2},\quad i=2\text{ and }4. & & \displaystyle\end{eqnarray}$$

Figure 17 shows the spanwise energy spectrum maps for the second- and fourth-quadrant Reynolds shear stresses in the equilibrium region. For simplicity, the APG flow data for $m=-0.32$ are only presented in (c,d), with the ZPG flow data in (a,b). A comparison between the Q2 and Q4 spectral energy levels shows that the Q2 ejection motions play an important role in the generation of the near-wall energy peaks for both the ZPG and APG flows (a,c), although there is a slight increase in the large-scale energy of Q2 events in the outer layer under the strong APG. For the Q4-event Reynolds shear stress in figure 17(b,d), it is clear that the Q4 sweep motions contribute most to the creation of the outer peaks for the ZPG and APG flows. In particular, the large-scale Q4-event motions become increasingly important for the induction of the outer peaks, consistent with our observations in figures 5 and 6.

3.2.4 Two-point amplitude modulation

For a ZPG TBL flow, it is fairly well known that large-scale structures in the log layer modulate the amplitude of small-scale motions in the near-wall region, and as the Reynolds number increases, the amplitude modulation (AM) effect becomes progressively stronger due to the superimposition of increased large-scale activity onto the near-wall cycle (Bandyopadhyay & Hussain Reference Bandyopadhyay and Hussain1984; Mathis et al. Reference Mathis, Hutchins and Marusic2009). For an APG TBL flow, Harun et al. (Reference Harun, Monty, Mathis and Marusic2013) estimated the degree of the AM and showed that near-wall $u^{\prime }$ -structures are highly modulated by the outer-region large-scale structures compared with those in a ZPG TBL flow, suggesting that the large-scale structures become more dominant in an APG flow. However, there remains the issue of which large-scale features (that is, spatial organization or energy) contribute to the enhancement of the AM effects in an APG TBL flow. In the present study, the degree of interaction between the inner and outer layers in APG TBL flows with an increase in the pressure gradient is estimated using the two-point AM covariance, which measures the modulated and modulating signals in the near-wall and outer regions simultaneously (Bernardini & Pirozzoli Reference Bernardini and Pirozzoli2011). The two-point AM coefficient is defined using the correlation between the large-scale velocity-fluctuating signal ( $u_{L}$ ) at $y_{1}$ and the spectral-filtered large-scale envelope of the small-scale velocity-fluctuating signal ( $u_{EL}$ ) at $y_{2}$ , $\langle u_{L}(y_{1})u_{EL}(y_{2})\rangle$ . To separate the large-scale and small-scale motions, a spanwise wavelength of $\unicode[STIX]{x1D706}_{z}/\unicode[STIX]{x1D6FF}=0.5$ is chosen as the cutoff wavelength (figure 14), and the envelope of the small-scale motion is extracted using the Hilbert transform.

Figure 18.

Two-point AM covariance normalized by the local friction velocity in ZPG and APG flows: (a) $m=0$ , (b) $m=-0.15$ , (c) $m=-0.25$ and (d) $m=-0.32$ . The contour level is varied from $-$ 0.14 to $-$ 0.14 in intervals of 0.02, and the solid and dashed lines represent the positive and negative contour values respectively. The straight dashed lines are included to indicate the secondary outer peak locations.

Figure 18 shows the two-point AM covariance in the equilibrium region as the APG strength increases. For the ZPG flow (figure 18 a), a clear symmetric contour is shown with respect to the diagonal line. However, an anti-symmetric contour emerges for a mild APG flow due to a secondary peak in the log layer at approximately $y_{1}/\unicode[STIX]{x1D6FF}=0.15$ and $y_{2}^{+}=5$ . As described above, this peak results from the nonlinear AM signature due to the imprints of large-scale structures in the log layer on the near-wall region. As the APG increases further, the AM effect becomes stronger and the secondary outer peak location becomes closer to the wall ( $0.1<y/\unicode[STIX]{x1D6FF}<0.4$ ). It is noted that for $m=-0.32$ , the modulation of the near-wall small-scale motions is induced by the large-scale structures residing over a wide range of wall-normal locations from the wall ( $y/\unicode[STIX]{x1D6FF}<0.4$ ), and thus strong contour lines in the near-wall region extend to the outer layer. The enhanced interaction between the inner and outer layers with an increase in the pressure gradient indicates that large-scale energy intensification in the outer layer is important to induce the active AM of the near-wall small-scale motions rather than the outer-region large-scale organization of streamwise velocity-fluctuating structures (figures 10 and 11). In addition, the existence of outer peaks for the AM effects in the log layer suggests that although the large-scale energy is enhanced in the vicinity of $0.2<y/\unicode[STIX]{x1D6FF}<0.4$ (figure 12) in the APG TBLs, the strong AM of the small-scale motions in the near-wall region is created by the log region large-scale structures, similarly to a TBL flow with a high Reynolds number (Mathis et al. Reference Mathis, Hutchins and Marusic2009).

However, in an experimental study of APG TBL flows, Harun et al. (Reference Harun, Monty, Mathis and Marusic2013) showed that the wall-normal location of the zero-crossing for small-scale AM increases with an increase in the pressure gradient. Moreover, the wall-normal location coincides with the large-scale energetic peak location based on a one-point correlation coefficient at a wall-normal location. For the one-point correlation, a typical velocity-fluctuating signal at a wall-normal location is decomposed into large- and small-scale components, and a large-scale signal and large-scale envelope of the small-scale signal at the wall-normal location are employed to compute the correlation based on the observation that the large-scale signal at the wall-normal location is assumed to be the signature of a large-scale event in the log layer (Hutchins & Marusic Reference Hutchins and Marusic2007a ; Mathis et al. Reference Mathis, Hutchins and Marusic2009). The two-point AM coefficient employed in the present study shows that a positive coefficient is observed up to the edge of the boundary layer, although the peak is found in the log layer. This result suggests that small-scale motions near the wall (here, at $y^{+}=5$ ) are modulated by the large-scale motions residing throughout the boundary layer. Contrary to the present results, the presence of negative values in the one-point correlation coefficient far from the wall is due to the strong small-scale signal within the negative large-scale excursion (Harun et al. Reference Harun, Monty, Mathis and Marusic2013).