2.1. Imposing controlled pressure-gradient histories

Figure 3 presents the experimentally measured pressure coefficient profiles, $C_{\!P}(x)$ , for various test section configurations, as listed in table 1. For each configuration, the free stream dynamic pressure was measured along the streamwise extent of the test section using a Pitot–static tube mounted at $z = 0.525$ m on the streamwise traverse system (figure 2 a). The most basic configuration corresponds to nominal ZPG conditions throughout the test section, indicated by black symbols in figure 3. Across all ZPG cases tested, variations in $U_{\infty }$ remained within 1 % along the entire test section fetch ( $C_{\!P}(x)=0\pm 0.012$ ). In figure 3 and table 1, square and circle symbols denote inlet unit Reynolds numbers of $ U_{\infty }(x=0)/\nu \approx 9 \times 10^5\ \mathrm{m}^{-1}$ and $ 6 \times 10^5\ \mathrm{m}^{-1}$ , respectively. To generate different APG configurations, either one or three low-porosity screens were added to the outlet, in addition to the high-porosity mesh. These are denoted as APG1 (green symbols in figure 3 b) and APG3 (red symbols in figures 3 a and 3 b). As shown in figure 3(b), both the APG1 and APG3 configurations exhibit nominal ZPG conditions upstream of the APG development region ( $ x \lt 9\ \mathrm{m}$ ), consistent with the set-up illustrated in figure 2(b). These cases are referred to as having minimal upstream PG history and serve as the reference cases for comparison with those with selectively imposed PG histories. The TBL statistics for the APG1 and APG3 cases were found to collapse with those from the ZPG case at $ x = 8.5\ \mathrm{m}$ , confirming the absence of significant PG-history effects upstream of this location, consistent with the findings of Deshpande et al. (Reference Deshpande, Van Den Bogaard, Vinuesa, Lindić and Marusic2023). An additional configuration, referred to as the Perturbed case, was then designed to introduce a controlled PG history which was distinct from the baseline APG cases. Specifically, this set-up modifies the pattern of open/closed air-bleed slots in the upstream region, relative to the APG3 case (figures 2 b and 2 c). The resulting $ C_{\!P}(x)$ profile is shown by blue symbols in figure 3(a). For this case, the inlet and downstream flow conditions were matched to the APG3 reference configuration (Ref. APG3), but notable differences in $ C_{\!P}(x)$ appear for $x \lt 12\ \mathrm{m}$ due to the imposed PG perturbation (i.e. APG conditions for $x\lt 5$ m). The Perturbed case was also designed to include a long relaxation region (i.e. nominally ZPG conditions for $ 5 \lt x \lt 12\ \mathrm{m}$ ) following the initial PG perturbation, enabling assessment of how the initial upstream PG perturbation modifies the TBL development compared with the reference cases. Farther downstream, the facility permits sustained APG development, over a long streamwise distance, enabling a detailed comparison of the differences in development between the Perturbed and Ref. APG3 cases under locally matched flow conditions ( $ x \gt 12\ \mathrm{m}$ ). Appendix B contains additional details about the exact configurations of open and closed slats required to generate the Perturbed and Ref. APG3 cases.

Table 1.

Characteristics of the experimental TBLs investigated in the present study.

Table 1. Long description

The table presents data for different cases, including ZPG, Ref. APG3, Perturbed, APG1, and APG3. It includes measurements such as x (m), beta, Re_tau, Re_theta, u_tau, delta_99 (mm), delta (mm), and theta* (mm). The table is divided into two main sections: from figure 3(a) and from figure 3(b). Each section lists specific cases with their corresponding values. For example, in the first section, the ZPG case at x = 5.1 m has Re_tau of 4090 and Re_theta of 8970, while the Perturbed case at the same x has Re_tau of 4060 and Re_theta of 9550. The table also includes symbols representing each case.

2.2. Streamwise velocity and friction velocity measurements

Five measurement stations were chosen, spanning a significant streamwise fetch, where both the streamwise turbulence statistics and friction velocity could be measured reliably. Specifically, the stations $x_a$ and $x_1$ were chosen because they are expected to exhibit the highest contrast in turbulence statistics between the Perturbed and reference cases (due to their relative proximities to changes in PG conditions, figure 3 a). The stations $x_b$ , $x_c$ and $x_2$ are then used to assess the longer-scale recovery of the Perturbed case, towards the reference cases, under sustained matched flow conditions. However, it is noted that the physical transitions from one PG condition to another do not occur as distinct, step-like changes. This makes the determination of the exact streamwise location where a new PG condition is established difficult, and the above distances are considered conservative estimates (see Appendix B). In the relaxation region, $x_a$ $x_b$ and $x_c$ are located approximately 2, 27 and 50 $\delta _{a}$ downstream of the establishment of ZPG conditions in the Perturbed case, respectively, where $\delta _{a}$ is the boundary-layer thickness at $x_a$ . Similarly, in the development region, $x_1$ and $x_2$ are located approximately 2 and 17 $\delta _{1}$ downstream of the re-establishment of APG conditions in the Perturbed case, respectively, where $\delta _{1}$ is the boundary-layer thickness at $x_1$ . In contrast, $x_1$ and $x_2$ are located approximately 11 and 26 $\delta _{1}$ downstream of the establishment of APG conditions in the Ref. APG3 case, respectively. The mentioned development lengths are approximate, since the PG transition can only be located within the streamwise resolution of the $C_{\!P}(x)$ measurements. With a minimum spacing of $0.5$ m between adjacent pressure measurements, the corresponding uncertainty is approximately $\pm 0.25$ m.

Streamwise velocities were measured at several wall-normal locations using a single hot-wire made in-house from Wollaston wire mounted on a Dantec 55P15 probe. The wire was etched to a sensing diameter $ d = 2.5\,{\unicode[Arial]{x03BC}} \mathrm{m}$ with fixed aspect ratio $ l/d = 200$ , giving a nominal length $ l = 0.5\,\mathrm{mm}$ and resulting viscous-scaled lengths $ l^+ = 11{-}16$ across all cases. To resolve the energetic hierarchy of scales in the TBL, the sampling time $ T_S$ was chosen to satisfy $ T_S U_{\infty }/\delta \ge 20\,000$ . The sampling frequency was fixed at 50 kHz, and the signal was low-pass filtered at 25 kHz. The resulting viscous-scaled sampling intervals $ t_S^+ = {U_{\tau }}^2 /(f_S \nu )$ ranged from 0.08 to 0.31, sufficient to resolve the relevant small-scale motions. Further details of the experimental apparatus and calibration methods are available in Deshpande et al. (Reference Deshpande, Van Den Bogaard, Vinuesa, Lindić and Marusic2023). OFI was used at each streamwise measurement location (except $x=x_c$ ) to obtain an accurate and independent estimate of $ {U_{\tau }}$ . Further details of the OFI methodology implemented in the present APG TBL experiments are provided in Marusic et al. (Reference Marusic, Abu Rowin, Lozier, Lindić, Zarei and Deshpande2025). Taking into account uncertainties associated with oil viscosity calibration, interference pattern resolution and potential contamination of the oil droplets, the estimated uncertainty in the measured friction velocity was approximately $\pm 1$ %.

2.3. Summary of present experimental dataset

Key characteristics for the TBLs investigated here, obtained from these hot-wire, OFI and $C_{\!P}$ measurements at each measurement location, along with their respective symbols, are presented in table 1. The recent approach of Lozier et al. (Reference Lozier, Deshpande, Zarei, Lindić, Rowin and Marusic2025) was employed to determine the TBL thickness, $\delta$ , which is compared with the classical $\delta _{99}$ , estimated using the method of Vinuesa et al. (Reference Vinuesa, Bobke, Örlü and Schlatter2016), in table 1. The capability of the present methodology for creating controlled PG histories can now be demonstrated quantitatively using the characteristics tabulated in table 1. First, for the cases in figure 3(a), $Re_{\tau }(x)$ is measured to be nominally matched between the Perturbed and ZPG cases in the upstream section ( $x \lt 12$ m). This is an important condition for investigating the persistence of any PG-history effects observed within the relaxation region that may result from the upstream APG perturbation. In the downstream section ( $x \gt 12$ m) where $C_{\!P}$ and $\text{d}C_{\!P}/\text{d}x$ are matched for the Perturbed and Ref. APG3 cases (figure 3 a), $Re_{\tau }(x)$ and $\beta (x)$ are also confirmed to be nominally matched (table 1). This is the critical condition for decoupling the local and upstream PG effects (Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017; Mahajan et al. Reference Mahajan, Deshpande, Gungor, Maciel and Vinuesa2025), and quantifying the influence of upstream PG perturbations on the APG TBL development in the downstream development region. Finally, in figure 3(b), although the local PG (i.e. $\beta$ ) varies, the Reynolds number is found to be nearly identical between the ZPG, APG1 and APG3 cases with matched inlet conditions, as shown in table 1. This ensures that the effects of the local APG on the TBL statistics and scaling laws can be rigorously evaluated without the additional influence of varying Reynolds numbers and/or PG-history effects.

3.1. Streamwise mean velocity

We first examine viscous-scaled mean streamwise velocity profiles ( $U^{+}$ ) measured in the relaxation region, i.e. at $x_a$ , $x_b$ and $x_c$ (figure 3 a), which are shown in figure 4(a). The ZPG case is used here as a reference. Although the flow conditions for the Perturbed case are locally matched to the ZPG case in the relaxation region (table 1), differences in their mean velocity profiles can be seen at $x_a$ attributed to the upstream APG perturbation experienced by the Perturbed case (figure 3 a). Specifically, the Perturbed mean velocity profile ( $U^{+}$ ) in the wake region lies above the ZPG profile, and the Perturbed profile in the overlap region is slightly below the ZPG profile at $x_a$ (figure 3 a). This is made clearer by subtracting the classical log-law (1.1) from these measured mean velocity profiles ( $U^{+}-U^{+}_{log}$ ), which reveals a systematic deviation from this canonical scaling behaviour in the Perturbed profile, figure 4(b). It is also noted that the mean velocity profile in the inner region remains largely unaffected by the upstream APG perturbation, consistent with the findings of Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017) and Pozuelo et al. (Reference Pozuelo, Li, Schlatter and Vinuesa2022).

Figure 4.

(a) Profiles of mean streamwise velocity, and (b) deviations from the classical log-law for the ZPG and Perturbed cases in the relaxation region. Both cases have locally matched flow conditions. The inset in (b) shows the profile at $x=x_a$ . Note the profiles for $x=x_b$ and $x=x_c$ are vertically offset.

Figure 4. Long description

Panel A: A line graph shows profiles of mean streamwise velocity (U+) against the dimensionless wall-normal coordinate (z+). The graph includes data for ZPG (gray squares) and Perturbed cases (blue squares) with three different profiles labeled as x_a, x_b, and x_c. The lines represent different theoretical models: U+ = z+ (dotted line) and U+ = (1/κ)ln(z+) + B (dashed line). Panel B: Another line graph depicts deviations from the classical log-law for the ZPG and Perturbed cases in the relaxation region. The y-axis represents U+ - (1/κ)ln(z+) - B, and the x-axis is the dimensionless wall-normal coordinate (z+). The inset shows a zoomed-in view of the profile at z+ = 100. The Perturbed case data (blue squares) are vertically offset for clarity.

Figure 5.

(a,b) Profiles of mean streamwise velocity, (c,d) indicator function and (e,f) deviations from the classical log-law for the ZPG, Ref. APG3 and Perturbed cases in the development region at (a,c,e) $x=x_1$ and (b,d,f) $x=x_2$ . All APG cases have locally matched local flow conditions, but unique PG histories. The ZPG cases have locally matched $Re_{\tau }$ .

Farther downstream, the Perturbed mean velocity profiles collapse with the reference ZPG profiles at $x_b$ and $x_c$ (figure 4 a). This collapse is further confirmed by figure 4(b), where both cases show good agreement with the classical log-law. However, differences in characteristics such as the TBL thickness ( $\delta$ , see table 1), and its growth rate ( $\text{d}\delta /\text{d}x$ ), still persist as a result of the upstream APG perturbation. For example, across the full relaxation region (i.e. between $x_a$ and $x_c$ where $x_c - x_a \approx 48\delta _a$ ) the growth rate of the Perturbed TBL is 25 % greater than the reference ZPG TBL. This effect is attributed primarily to the large-scale motions (LSMs), which are energised by the upstream APG perturbation (see figure 6 c), and leads to enhanced momentum transport in the TBL. Both observations are significant: the mean velocity profile does recover ZPG scaling, suggesting that the effect of upstream PG perturbations (like those introduced by ramps) on mean turbulence statistics can be minimised with a sufficiently long relaxation length ((Knopp et al. Reference Knopp, Reuther, Novara, Schanz, Schülein, Schröder and Kähler2021; Romero et al. Reference Romero, Zimmerman, Philip, White and Klewicki2022) see figure 1), however, the boundary-layer thickness and growth rate differ from the reference ZPG case, even after a reasonably long relaxation length, making careful comparisons challenging. These results also motivate further investigation into the quantification of PG-history effects, specifically methods/parameters which can account for the specific differences in behaviour observed between $x_a$ and $x_b$ (or $x_c$ ).

The mean velocity profiles measured in the development region, i.e. at $x_1$ and $x_2$ (figure 3 a), are shown in figures 5(a) and 5(b). Here, the Perturbed and Ref. APG3 cases have nominally matched values of $Re_{\tau }$ and $\beta$ (table 1), and the ZPG case has a matched $Re_{\tau }$ . While the mean velocity profiles near the wall ( $z^{+}\leq 20$ ) appear largely unaffected by either the local or upstream PGs at both $x_1$ and $x_2$ (Pozuelo et al. Reference Pozuelo, Li, Schlatter and Vinuesa2022), the effect of the local APG ( $\beta$ ) can be observed at $x_2$ where the Ref. APG3 and Perturbed profiles collapse and deviate significantly from the ZPG profile in the wake and overlap regions (Monty et al. Reference Monty, Harun and Marusic2011; Deshpande et al. Reference Deshpande, Van Den Bogaard, Vinuesa, Lindić and Marusic2023). However, at $x_1$ , the influence of PG history is also apparent; although both the Perturbed and Ref. APG3 profiles deviate from the ZPG case, they do not collapse, consistent with the observations of Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017). These differences at $x_1$ are attributed to the different upstream PG conditions experienced by the two boundary layers before arriving at the same local conditions (i.e. $C_{\!P}(x)$ for $x_{b}\lt x\lt x_{1}$ ). Specifically, the Perturbed profile falls between the ZPG and Ref. APG3 profiles, reflecting a ‘lag’ in APG development (compared with the Ref. APG3 case which has matched conditions) due its upstream PG history (i.e. transitioning from established ZPG conditions). These results further motivate the development of methods/frameworks for quantifying PG-history effects that can account for the differences in behaviour observed between $x_1$ and $x_2$ ; one such framework, motivated by the above observations, is proposed in Part 2.

The influence of local and upstream PGs on the classical log-law is further investigated by plotting the indicator function ( $z^{+} {\rm d}U^{+} /{\rm d}z^{+}$ ) of each mean velocity profile, as shown in figures 5(c) and 5(d). At both measurement locations, and for all PG cases, a clear plateau is observed in the overlap region ( $10^2\lesssim z^{+} \lesssim 10^3$ ), confirming the existence of logarithmic scaling in this region and a sufficient separation between the inner and wake regions in all cases. Additionally, within experimental uncertainty, no systematic changes are observed in the von Kármán coefficient ( $\kappa$ ) due to either local or upstream PGs. The differences in the mean velocity profiles in the wake region also confirm that the wake parameter ( $\varPi$ ) systematically increases with $\beta$ (figure 5 d, Monty et al. (Reference Monty, Harun and Marusic2011)), and PG history (figure 5 c, Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017)). Finally, the classical log-law (1.1) was subtracted from the measured mean velocity profiles in figures 5(e) and 5(f), in a manner analogous to figure 4(b). In the overlap region, a systematic and significant decrease in the magnitude of the additive coefficient $B$ is observed with increasing $\beta$ , especially at $x_2$ . However, PG history is also seen to affect the value of $B$ at $x_1$ , as the Ref. APG3 and Perturbed cases do not collapse in figure 5(e), consistent with the lag in APG development noted earlier. It is also noted that the mean velocity profile of the Perturbed case at $x_1$ does not collapse with the ZPG case either, as shown in figure 5(a,c,e) where differences in the wake and overlap regions are beyond experimental uncertainty. This confirms that APG conditions are well-established, and influencing the mean turbulence statistics, by this measurement location (i.e. $x_1$ is sufficiently downstream of the region with nominally ZPG conditions).

3.2. Streamwise normal stress and premultiplied energy spectra

We now examine the influence of local and upstream PGs on second-order turbulence statistics and the hierarchy of energetic turbulent motions in the TBL. To this end, it is useful to first contextualise the role of LSM in high- $Re_{\tau }$ TBLs (e.g. motions associated with attached-eddies and/or superstructures). These LSMs constitute the primary energy-containing motions in the overlap and wake regions of high- $Re_{\tau }$ TBLs, and their contributions become increasingly significant with higher Reynolds numbers (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013, Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015). In the context of APG flows, it has been established that the magnitude and extent of these LSMs and/or superstructures, in the overlap and wake regions, are influenced by local PGs (Deshpande et al. Reference Deshpande, Van Den Bogaard, Vinuesa, Lindić and Marusic2023). Recent studies (Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017; Pozuelo et al. Reference Pozuelo, Li, Schlatter and Vinuesa2022) have also demonstrated that upstream PGs can leave a persistent footprint on the LSMs in the wake region of low-Reynolds-number TBLs. Hence, it is important to decouple the influences of these local and upstream PGs on the hierarchy of inertia-dominated turbulent motions. The present high- $Re_{\tau }$ experiments are critical to achieve this objective as they offer sufficient separation between the viscous and inertia-dominated regions of the TBL, facilitating detailed examination of these influences.

To investigate the distribution of turbulent energy across different scales, the premultiplied spectra of streamwise velocity fluctuations ( $f\phi _{uu}^+$ ) are plotted as a function of their viscous time scale ( $T^+={U_{\tau }}^2/ f\nu$ ) and wall-normal location ( $z^{+}$ ) in figures 6(c) and 6(d) corresponding to the streamwise locations $x_a$ and $x_b$ , respectively. The corresponding wall-normal profiles of the streamwise normal stress ( $\overline {u^{2}}^+$ ) are also plotted in figure 6(a). Some differences in the streamwise normal stress profiles (figure 6 a) and premultiplied spectra (figure 6 c) are observed in the overlap and wake regions ( $z^{+}\gtrsim 100$ ) at $x_a$ between the ZPG and Perturbed cases, attributed to differences in their respective PG histories upstream of this measurement location (figure 3 a). In figures 6(c) and 6(d), the horizontal dash–dotted lines denote the characteristic, Reynolds-number-dependent time scale associated with LSMs in canonical TBLs. At $ x_a$ , a secondary spectral peak appears for the Perturbed case (figure 6 c), aligned with this characteristic time scale, highlighting a persistent influence of the upstream APG perturbation (Deshpande et al. Reference Deshpande, Van Den Bogaard, Vinuesa, Lindić and Marusic2023). This spectral feature also corresponds with elevated wake region turbulence intensities in the Perturbed normal stress profile, and reinforces the conclusion that the upstream PG perturbation leaves a measurable footprint at this location. Farther downstream, however, at $ x_b$ , both the normal stress profiles (figure 6 a) and premultiplied spectra (figure 6 d) for the ZPG and Perturbed cases collapse, indicating recovery from the upstream PG perturbation. This collapse is also observed at $ x_c$ ; however, analysis at that location is omitted here for brevity, citing consistency with the trends noted in the mean velocity profiles.

Figure 6.

(a) Profiles of streamwise normal stress ( $\overline {u^{2}}^+$ ) for the ZPG and Perturbed cases in the relaxation region. Premultiplied energy spectra ( $f\phi _{uu}^+$ ) at (c) $x=x_a$ and (d) $x=x_b$ . Vertical dashed lines indicate $z^{+}=0.4Re_{\tau }$ and horizontal dot–dashed lines denote $T^+=4.8\delta ^{+}/U_{\infty }^{+}$ . (b) Premultiplied spectra at a single wall-normal location corresponding to the vertical dashed lines in (c) and (d). Note that the profiles for $x=x_b$ and $x=x_c$ are vertically offset in (a) and (b).

Figure 6. Long description

Panel A: A line graph shows profiles of streamwise normal stress for ZPG and Perturbed cases in the relaxation region. The x-axis represents a dimensionless wall-normal coordinate, and the y-axis represents the dimensionless streamwise normal stress. The graph includes two data series: one for ZPG (gray squares) and one for Perturbed (blue squares). Panel B: A line graph displays premultiplied energy spectra at a single wall-normal location. The x-axis represents a dimensionless frequency, and the y-axis represents the premultiplied energy spectra. Two data series are shown: one for ZPG (gray line) and one for Perturbed (blue line). Panel C: A contour plot shows premultiplied energy spectra at a specific streamwise location for the ZPG and Perturbed cases. The x-axis represents the dimensionless frequency, and the y-axis represents the dimensionless wall-normal coordinate. Contour lines indicate different levels of premultiplied energy spectra. Panel D: Another contour plot shows premultiplied energy spectra at a different streamwise location for the ZPG and Perturbed cases, with similar axes and contour lines as in Panel C.

To further investigate the influence of the upstream PG perturbation on the hierarchy of turbulent scales, spectra corresponding to the wake region ( $z^{+}=0.4Re_{\tau }$ ; vertical dashed lines in figure 6 c,d) are plotted in figure 6(b) for $x_a$ and $x_b$ . Here, $0.4 Re_{\tau }$ is approximately equivalent to $0.5 \delta _{99}^+$ , for reference, and this particular wall-normal location was chosen because it has previously been noted for being strongly influenced by local PGs (Deshpande et al. Reference Deshpande, Van Den Bogaard, Vinuesa, Lindić and Marusic2023). While the spectra at this wall-normal location collapse at $x_b$ , the influence of the upstream APG perturbation on the Perturbed case can be seen across a broad range of time scales at $x_a$ . However, the ‘inner-peak’ of $\overline {u^{2}}^+$ and the associated characteristic wall-normal location/scale in the spectrograms ( $z^{+}=15$ , $T^+=100$ ) remain unaffected by the upstream APG perturbation at $x_a$ (figure 6 a,c), which is consistent with Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017) and Pozuelo et al. (Reference Pozuelo, Li, Schlatter and Vinuesa2022). These observations reaffirm that LSMs in the overlap and wake region are more sensitive to PG-history effects (i.e. respond slower to changing PG conditions), as compared with the small-scale motions in the inner region (Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015; Sanmiguel Vila et al. Reference Sanmiguel Vila, Vinuesa, Discetti, Ianiro, Schlatter and Örlü2020; Gungor et al. Reference Gungor, Gungor and Maciel2024; Mahajan et al. Reference Mahajan, Deshpande, Gungor, Maciel and Vinuesa2025).

We now investigate the influence of local and upstream PGs farther downstream in the development region, i.e. at $x_1$ and $x_2$ . Consistent with observations from the relaxation region, $\overline {u^{2}}^+$ profiles in the inner region collapse across all cases (figure 7 a,b). However, the effects of the local and upstream PGs are evident in the wake region where LSMs are dominant (vertical dashed lines again indicate $z^{+}=0.4Re_{\tau }$ ). At $x_2$ , the Perturbed and Ref. APG3 cases collapse, but both deviate significantly from the ZPG case across a broad wall-normal range, including the overlap and wake regions (figure 7 b), highlighting the expected effect of local APGs (Deshpande et al. Reference Deshpande, Van Den Bogaard, Vinuesa, Lindić and Marusic2023). In contrast, at $x_1$ , while the Ref. APG3 case clearly departs from the ZPG case in the wake region (figure 7 a), the Perturbed case does not collapse with the Ref. APG3 case, reflecting the differences in their PG histories upstream of the measurement location. Specifically, the Ref. APG3 case has undergone a relatively longer APG development before $x_1$ , whereas the Perturbed case has had a shorter APG exposure (after an extended development under ZPG conditions) and is therefore lagging in development at this station, consistent with the observations from the mean velocity profiles (figure 5 a,c,e). Further, the magnitude of the normal stress for the Perturbed case only deviates from the ZPG case near $z^{+}=0.4Re_{\tau }$ , demonstrating the competing influences of its ZPG history upstream of $x_1$ , and the local APG conditions (figure 3).

Figure 7.

(a,b) Profiles of streamwise normal stress ( $\overline {u^{2}}^+$ ) and (c,d) premultiplied energy spectra ( $f\phi _{uu}^+$ ) for the ZPG, Ref. APG3 and Perturbed cases in the development region at (a,c,e) $x=x_1$ and (b,d,f) $x=x_2$ . (e,f) Differences between the Ref. APG3 and Perturbed spectra ( $f\phi _{uu,\mathrm{Ref.\,APG3}}^+-f\phi _{uu,\textrm {Perturbed}}^+$ ). Horizontal dot–dashed lines denote $T^+=4.8\delta ^{+}/U_{\infty }^{+}$ , horizontal dotted lines denote $T^+=2.4\delta ^{+}/U_{\infty }^{+}$ , horizontal red lines represent $T^+=350$ , and vertical dashed lines indicate $z^{+}=0.4Re_{\tau }$ .

To further examine the impact of local and upstream PGs on the hierarchy of energetic motions, the corresponding premultiplied energy spectrograms are shown in figure 7(c,d). Both the ZPG and Ref. APG3 cases display typical distributions of energetic turbulent motions; for instance, the outer spectral peak in the ZPG case aligns with the horizontal dash–dotted line ( $T^+= 4.8\delta ^+/U_{\infty }^+ \approx 6\delta _{99}^+/U_{\infty }^+$ (Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015)), while the outer peak in the Ref. APG3 case (reflecting the effect of the local APG) aligns with the horizontal dotted line ( $T^+ = 2.4\delta ^+/U_{\infty }^+ \approx 3\delta _{99}^+/U_{\infty }^+$ (Deshpande et al. Reference Deshpande, Van Den Bogaard, Vinuesa, Lindić and Marusic2023)). However, at $x_1$ , the Perturbed case again falls between the ZPG and Ref. APG3 contours in the wake region (around the vertical dashed line). Building on the observations from the streamwise normal stress and mean velocity, this behaviour reflects an APG-development lag; the Perturbed case is responding to the established APG conditions at $x_1$ , but it has not experienced the same cumulative APG exposure as the Ref. APG3 case before $x_1$ . Farther downstream, at $x_2$ , which is located more than $17\delta _{1}$ downstream of the onset of APG conditions for both cases, the Perturbed and Ref. APG3 spectra once again collapse. The differences in specific energetic motions/scales are also visualised in figure 7(e,f), where the Ref. APG3 spectra are subtracted from the Perturbed spectra, i.e. $(f\phi _{uu}^+)_{\textit{Ref.APG3}}-(f\phi _{uu}^+)_{\textit{Perturbed}}$ . At $x_1$ , prominent differences appear in the wake region, centred around the characteristic location and scale of APG-energised motions (i.e. $z^{+} = 0.4Re_{\tau }$ and $T^+ = 2.4\delta$ ), and relatively smaller differences are also observed in the overlap region, at even larger time scales ( $T^+ = 4.8\delta$ ), supporting the notion that these LSMs take longer to respond to changes in PG conditions, i.e. they retain a longer ‘memory’ of the upstream PG history. At $x_2$ , however, no significant differences remain in the subtracted spectra, confirming that the two cases are locally well-matched, and the influences of their respective PG histories upstream of $x_1$ are undetectable, i.e. the extended development under matched APG conditions ( $x_2-x_1=15\delta _1$ ) has a stronger influence than any upstream PG history differences.

Figure 8.

Large- and small-scale contributions to the streamwise normal stress for the high- $Re_{\tau }$ Perturbed (blue) and Ref. APG3 (red) cases at (a) $x=x_1$ and (b) $x=x_2$ , with nominally matched local flow conditions ( $Re_{\tau }$ and $\beta$ ).

To further support these conclusions, and compare the present results with previous studies, we decompose the fluctuating streamwise velocity component ( $u$ ) into its small- ( $u_S$ ) and large-scale ( $u_L$ ) contributions, enabled by a clear separation between these scales as seen in figure 7(c,d) (highlighting the importance of high $Re_{\tau }$ to understand such effects). We compare the Perturbed and Ref. APG3 cases at $x_1$ and $x_2$ , with locally matched flow conditions ( $Re_{\tau } \approx 8000$ , $\beta \approx 0.64$ ; see figure 8 a,b), but qualitative differences in their upstream PG histories, as established previously.

The influence of $Re_{\tau }$ and $\beta$ on $\overline {u_S^{+2}}$ and $\overline {u_L^{+2}}$ has been well documented (e.g. Deshpande et al. Reference Deshpande, Van Den Bogaard, Vinuesa, Lindić and Marusic2023; Lozier et al. Reference Lozier, Marusic and Deshpande2024); however, the present methodology allows us to isolate the additional contribution of PG history, independent of these parameters. To perform the decomposition, a cutoff time scale ( $T_c^+ = 350$ ) was selected based on the premultiplied energy spectra, following established practices (Deshpande et al. Reference Deshpande, Van Den Bogaard, Vinuesa, Lindić and Marusic2023; Lozier et al. Reference Lozier, Marusic and Deshpande2024), such that $u_S$ includes fluctuations with $T^+ \lt T_c^+$ and $u_L$ includes those with $T^+ \geq T_c^+$ . First, the results show good collapse between the small- and large-scale contributions for both cases near the wall, particularly around the inner peak ( $z^{+} \approx 15$ ). However, there are some differences in the large-scale contributions between the Perturbed and Ref. APG 3 cases, across the overlap and wake regions, while differences in the small-scale contributions are limited primarily to the wake region, which agrees with the observations of Sanmiguel Vila et al. (Reference Sanmiguel Vila, Vinuesa, Discetti, Ianiro, Schlatter and Örlü2020). Specifically, at the reference location in the wake region $z^+ \approx 0.4Re_\tau$ , the small- and large-scale contributions to the Perturbed case are approximately 20 % less than those of the Ref. APG3 case. While these differences are measurable, and significant, they are still relatively modest compared with some previous studies (Gungor et al. Reference Gungor, Gungor and Maciel2024; Mahajan et al. Reference Mahajan, Deshpande, Gungor, Maciel and Vinuesa2025) because the imposed PG history variations in the present study are moderate. Additionally, previous APG studies at low $Re_{\tau }$ (Harun et al. Reference Harun, Monty, Mathis and Marusic2013) have found differences in large-scale contributions, not only in the wake region, but also closer to the wall ( $z^{+} \leq 10$ ), which were not observed here, highlighting the effect of limited separation between these regions/scales at low $Re_{\tau }$ . These results underscore the need to continue developing methods/frameworks that can capture these scale- and region-dependent PG-history effects, which have now been characterised for high- $Re_{\tau }$ TBLs.

Figure 9.

Profiles of the change in streamwise normal stress between $x_2$ and $x_1$ for (a) inner scaled and (b) outer scaled wall-normal distances. Differences between the premultiplied energy spectra at $x_2$ and $ x_1$ ( $f\phi _{uu,x_2}^+ - f\phi _{uu,x_1}^+$ ) for (c) the Ref. APG3 case and (d) the Perturbed case. Vertical dashed lines indicate $z/\delta =0.4$ . Horizontal dot–dashed lines denote $TU_{\infty }/\delta =4.8$ and horizontal dotted lines denote $TU_{\infty }/\delta =2.4$ .

Collectively, figures 5 and 7 also reveal a unique transitional behaviour in the Perturbed case between $x_1$ and $x_2$ , where the turbulence statistics and energy distributions initially differ from the Ref. APG3 case, but eventually collapse after both cases experience sustained development under matched PG conditions (figure 3). This invites a systematic investigation of the streamwise evolution between these locations to isolate the individual effects of $Re_{\tau }$ , $\beta$ and PG history. For each case (Perturbed, ZPG and Ref. APG3), we compute differences in the streamwise normal stress and premultiplied spectra between $x_1$ and $x_2$ : $\overline {u_{x_2}^{+2}} - \overline {u_{x_1}^{+2}}$ and $(f\phi _{uu}^+)_{x_2} - (f\phi _{uu}^+)_{x_1}$ . These results are shown in figure 9, where black squares represent the effect of increasing $Re_{\tau }$ (evolution of the ZPG case), red squares show the combined effects of increasing $Re_{\tau }$ and $\beta$ (evolution of the Ref. APG3 case) and blue squares capture the influence of $Re_{\tau }$ , $\beta$ and PG history (evolution of the Perturbed case). As such, the differences between these curves represent the decoupled contributions of each condition/parameter. Here, we note that the contribution related to PG history specifically reflects the differences in $C_{\!P}(x)$ for the Perturbed and Ref. APG3 cases between $x_b$ and $x_2$ . The normal stresses are also shown in both viscous ( $z^{+}$ ) and outer normalisation ( $z/\delta$ ) to highlight differences near the wall and in the wake region, while accounting for the change in $Re_{\tau }$ between $x_1$ and $x_2$ . As expected, increasing $Re_{\tau }$ raises the normal stress magnitude slightly in the overlap and wake regions (figure 9 a,b and Marusic et al. (Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015)). The addition of a steadily increasing APG condition then further energises turbulent motions around $TU_{\infty }/\delta \approx 2.4$ and $z/\delta = 0.4$ (figure 9 c) corresponding to an increase in the normal stress for the Ref. APG3 case in the wake and overlap regions. In contrast, the Perturbed case (figure 9 d) shows significantly higher energisation between $x_1$ and $x_2$ , with a broader time scale distribution and a peak location shifted closer to the wall ( $z/\delta \approx 0.25$ ), reflecting a more rapid development/energisation of APG-related scales on account of its upstream PG history, i.e. the relaxation region. These results also suggest that there could be another delay/lag between the APG-energisation in the overlap and wake regions, consistent with the APG-energisation originating primarily in the wake region, and then propagating into the overlap region, for example, after some time. Overall, the results reinforce the need for sufficiently long APG development regions when studying the streamwise evolution of high- $Re_{\tau }$ APG TBLs, or when developing models to capture local APG effects, especially when there are changing upstream PG conditions.

5.1. PG-history effects in the wake region

It can be argued that the wake region is the part of the TBL most directly/significantly affected by both local PGs and PG history; therefore, a clear examination of the wake region is essential for identifying/quantifying PG-history effects. To demonstrate this, we calculate the indicator functions for the dataset of Preskett et al. (Reference Preskett, Virgilio, Jaiswal and Ganapathisubramani2025), specifically for TBLs with locally ZPG conditions and matched inlet conditions, but systematically varying PG histories (figure 12 b). The corresponding indicator functions for the present ZPG, Ref. APG3 and Perturbed cases at $x_1$ are also shown in figure 12(d). In both datasets, there are differences in the magnitude and shape of the wake region profiles that are solely attributable to differences in their PG histories, since all other flow conditions are matched. First, different PG histories lead to significant differences in the wake strength, $\varPi$ , which Preskett et al. (Reference Preskett, Virgilio, Jaiswal and Ganapathisubramani2025) also recognised and quantified as a function of the deviation of $\varPi$ from its ZPG value, $\Delta \varPi = \varPi - \varPi _{\textit{ZPG}}$ . However, the wake region profile is also ‘stretched’ under the influence of different PG histories, as indicated by the shift in the peak of the indicator function relative to the reference ZPG dataset at matched $Re_{\tau }$ from Marusic et al. (Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015). This suggests that characterising PG-history effects requires accounting for more than changes in $\varPi$ alone, and that an additional parameter is needed to capture this wall-normal stretching of the mean velocity profile. This is highlighted in figure 12(b,d), where the shape of the mean velocity profile is a function of a history-dependent wake-stretching parameter, $C_{H_W}$ , where the subscript $H_W$ denotes the PG-history effect on the wake region. Quantification, and further discussion, of this parameter will be presented in Part 2.

5.2. PG-history effects in the overlap region

As discussed earlier, at both low and high $Re_{\tau }$ , the mean velocity profile in the overlap region exhibits a measurable shift from the classical log-law when a local APG is imposed. The PG history further influences this shift (Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017; Pozuelo et al. Reference Pozuelo, Li, Schlatter and Vinuesa2022), even when the local flow conditions are nominally ZPG (Gungor et al. Reference Gungor, Gungor and Maciel2024). The direction and magnitude of this shift also vary across datasets, leading to either a relatively larger or smaller displacement than that observed for the present APG cases with minimum PG history, as shown in figure 11(a). A schematic illustration of the deviation of the mean velocity from the classical log-law is provided in figure 11(b), highlighting the behaviour of cases with minimal PG-history effects (where $B$ is a function of $\beta$ only) and cases with the additional influence of PG history. As such, in figure 11(b) this deviation is considered to be a function of $\beta$ , and an additional PG-history parameter, $C_{H_{\textit{i}}}$ , where the subscript $H_i$ denotes the PG-history effect on the inner and overlap regions. These combined observations of the unique behaviours of $B$ and the wake region velocity profile, due to local and upstream PGs, across a range of datasets in the literature, form the basis for developing a new composite mean velocity profile formulation which accounts for the local conditions ( $Re_{\tau }$ and $\beta$ ) and provides new insight into the influence of PG history, which will be presented in Part 2 (Zarei et al. Reference Zarei, Lozier, Deshpande and Marusic2026).