3.1.1. Outer layer

The global parameters of the flow and the ratio of forces in the inner and outer layers are presented in figure 2 as a function of $x/\delta _{av}$. Figure 2(a) presents the outer-layer pressure gradient parameter $\beta _{ZS}$ and the ratio of pressure force at the edge of the boundary layer ($F_{p,o}$) to the maximum of turbulent force in the outer layer ($F_{to,max}$), where the outer layer is defined as the region above $0.1\delta$. Maciel et al. (Reference Maciel, Wei, Gungor and Simens2018) demonstrated that $\beta _{ZS}$ follows the ratio of pressure to turbulence force in the outer region using six APG TBLs. This is confirmed in the present flow as $\beta _{ZS}$ and the ratio of forces follow each other very closely in the region where the pressure gradient is positive (adverse). However, $\beta _{ZS}$ and the force balance do not correspond in the FPG region, although the trend is still the same for both. This could be due, at least partly, to the delay in turbulence response to the flow changes. As presented in § 3.2, Reynolds shear stresses remain very high in the outer region of the whole momentum-gaining zone, even higher than downstream. Their expected decay only starts at $x/\delta _{av}\approx 28$. This could explain why the absolute value of the ratio of pressure force to turbulence force is low.

Figure 2. Streamwise development of the main parameters of the APG TBL. The vertical lines indicate the streamwise stations selected for the detailed analysis.

As was already presented in the introduction, in the first part of the domain, until approximately $x/\delta _{av}=12$, $\beta _{ZS}$ increases at a fairly constant rate as it was aimed for in the design of this flow. Afterwards, it decreases at a similar absolute rate until the end of the domain. Such a streamwise evolution of $\beta _{ZS}$ implies a non-equilibrium boundary layer with a constant increase in importance of the pressure force with respect to the other forces followed by a constant decrease as it can be seen with $F_{p,o}/F_{to,max}$. Until approximately $x/\delta _{av}\approx 21.5$ where the shape factor reaches its maximum (figure 2d), the flow is similar to some strong non-equilibrium APG TBL cases in the literature (Hickel & Adams Reference Hickel and Adams2008; Gungor et al. Reference Gungor, Maciel, Simens and Soria2016; Hosseini et al. Reference Hosseini, Vinuesa, Schlatter, Hanifi and Henningson2016; Gungor et al. Reference Gungor, Maciel and Gungor2022). In this region, the boundary layer constantly loses momentum ($H$ increases) because the pressure force's importance increases upstream for a long extent, as reflected by $F_{p,o}/F_{to,max}$ and $\beta _{zs}$. Afterwards, the flow is still under the effect of an APG until $x/\delta _{av}=28$. However, the flow gains momentum in this region because the pressure force's importance decreases ($\beta _{ZS}$ decreases), which is a particular behaviour not studied previously, as discussed in the introduction. The rest of the domain is an FPG TBL where the pressure gradient is maintained with a negative ${\rm d}\beta _{ZS}/{{\rm d} x}$ similar to that upstream.

Although we present the remaining parameters in figure 2 later, for now, we focus on discussing the mean momentum budgets to provide a more comprehensive description of the streamwise evolution of the force balance. Figure 3 presents the outer-scaled mean momentum budgets for various streamwise positions. These positions are identified with vertical lines in figure 2. Details of these positions, including the acceleration parameter ($K=(\nu U^2)/({\rm d}U_e/{{\rm d} x})$), are given in table 2 and the corresponding mean velocity profiles are shown in figure 4. In figure 3, the $x$-axis ranges have been selected to ensure that the reference turbulent force value ($F_{to,max}$) is approximately located at the same position in all subfigures.

Figure 3. The mean momentum budget profiles of the eight streamwise positions as functions of $y/\delta$. The budget terms are normalised with $U_{e}$ and $\delta$. The $x$-axis ranges are chosen so that the reference turbulent force value $F_{to,max}$ is approximately at the same location in all subfigures.

Table 2. The main parameters of selected streamwise positions in the present case along with the ZPG TBL of Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2013) and the near-equilibrium APG TBL cases of Kitsios et al. (Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017).

Figure 4. The mean velocity profiles for nine streamwise positions of the present case along with the ZPG TBL and near-equilibrium cases EQ1 and EQ2 as a function of $y/\delta$. The profiles are normalised with $U_e$.

The momentum budgets reveal the non-equilibrium nature of the flow since the balance of forces changes from one station to another. The budgets of the first two stations (S1 and S2) show explicitly that the pressure force increases in importance with respect to the turbulent force in the outer region. Further downstream, the pressure force's relative magnitude with respect to the other forces begins to decrease as can be seen with the momentum budgets starting from S3. This decrease continues until the pressure force becomes zero at S7. Afterwards, the pressure gradient becomes negative (FPG zone) until the end of the domain.

In the momentum budgets of S1 to S6, the inertia term indicates that the fluid is losing momentum in the streamwise direction (${\rm d}U/{{\rm d} x}<0$), except near the wall for S6. This is confirmed by the mean velocity profiles in figure 4, but it is important to note that these are normalised profiles $U/U_e$ as a function of $y/\delta$ that are representative of $U$ variation in approximate streamline directions, not $x$. This explains why the fluid has gained momentum from S5 to S6 in the normalised representation of figure 4 although figure 3 indicates that ${\rm d}U/{{\rm d} x}$ is still negative at S6 in most of the outer region.

Indeed, while it was constantly losing momentum upstream, the fluid starts to gain momentum around S5 because of the decrease of the pressure force's relative magnitude (${\rm d}\beta _{ZS}/{{\rm d} x}<0$) that began slightly after S2. The fact that momentum gain only starts at S5, and not right after S2, shows that the mean flow responds with a delay to changes in the force balance because of inertia. As explained in the introduction, it is not the sign of the pressure gradient that indicates momentum gain or loss (disequilibrium), but the streamwise evolution of the relative magnitude of the pressure force, reflected by ${\rm d}\beta _{ZS}/{{\rm d} x}$ in the outer layer and ${\rm d}\beta _{i}/{{\rm d} x}$ in the inner layer. The present flow was specifically designed to produce a zone, from S5 to S7, where the fluid gains momentum in the boundary layer (in the normalised representation $U/U_e$ as a function of $y/\delta$ of figure 4) even if the pressure gradient is still positive (APG).

The momentum changes observed in figure 4 can also be followed with the distribution of the shape factor, $H$, as presented in figure 2(d), with increasing $H$ indicating momentum loss and vice versa. The velocity defect is small at S1 ($H=1.41$), increases to become large and maximal at S5 ($H=2.87$) and then monotonously decreases downstream. The shapes of the profiles in the momentum-losing (streamwise positions S2 to S4) and momentum-gaining zones (S6 to S8) are different from each other at matching or very close shape factors as illustrated in figure 4. This happens due to the different upstream flow history in the two zones. The main difference is that the profiles in the momentum-gaining region are fuller in the inner layer due to the faster response of the flow in that layer.

To examine how the momentum gain starts in the boundary layer, we consider two streamwise positions, S5 and the position denoted as SA, where the shape factor is 2.78, located just slightly downstream of S5. As it can be seen from figure 4, the inner layer gains momentum at SA but the defect in the outer layer remains almost the same. The inner layer therefore starts gaining momentum before the outer layer even if the momentum-gaining effect of the pressure force initiates earlier in the outer layer (${\rm d}\beta _{zs}/{{\rm d} x}$ becomes negative before ${\rm d}\beta _i/{{\rm d} x}$). This implies that the inner layer reacts to the pressure gradient much more rapidly than the outer layer. This is consistent with the fact that, in the inner layer, inertia forces are small and turbulence adjusts more quickly.

As mentioned in the introduction, $\beta _{RC}$ is not an outer pressure gradient parameter, but a global pressure gradient parameter for the whole boundary layer. It is presented in figure 2(b) and a comparison of $F_{p,o}/F_{to,max}$ and $\beta _{RC}$ shows that $\beta _{RC}$ indeed is not an outer region pressure gradient parameter for large-defect TBLs. It follows in fact more closely, but not exactly, the ratio of pressure to turbulent forces in the inner region given by the inner-layer pressure gradient parameter, $\beta _i$ (figure 2c).

To enhance our understanding of how the present flow responds to disequilibrium conditions, figure 4 illustrates mean velocity profiles from the two near-equilibrium APG TBLs of Kitsios et al. (Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017). These near-equilibrium flows correspond to cases of small velocity defect ($H=1.57$), denoted herein as EQ1, and large velocity defect ($H=2.57$), denoted as EQ2. The main parameters of these flows are given in table 2. Figure 4 illustrates that S2 and EQ1, sharing similar $H$ values, exhibit comparable velocity, but the near-equilibrium profile is fuller in the inner region. However, the near-equilibrium TBL attains this state with a globally smaller local pressure force effect (smaller $\beta _{RC}$) and a smaller effect in the outer region (smaller $\beta _{ZS}$). This suggests that the current non-equilibrium flow does not attain the anticipated equilibrium state corresponding to a given local pressure force effect ($\beta _{ZS}=0.352$ and $\beta _i=0.0168$). In other words, if the flow were to respond immediately to the pressure force, S2 would exhibit a larger velocity defect than EQ1, given that $\beta _{ZS}=0.352$ at S2 compared with $\beta _{ZS}=0.104$ at EQ1.

Mellor & Gibson (Reference Mellor and Gibson1966) showed that for equilibrium boundary layers (at infinite Reynolds number) there is a unique relationship between the Rotta–Clauser defect shape factor $G$ and $\beta _{RC}$, where $G=(U_e/u_\tau )(1-1/H)$. The relationship was obtained numerically from their similarity equation computations. A plot of $G$ as a function of $\beta _{RC}$ effectively illustrates, in a single graph, the departure from equilibrium of a TBL. Such plots have been used recently, for instance, by Knopp et al. (Reference Knopp, Reuther, Novara, Schanz, Schülein, Schröder and Kähler2021), Fritsch et al. (Reference Fritsch, Vishwanathan, Todd and Devenport2022b) and Volino & Schultz (Reference Volino and Schultz2023). We also use here $G$ as a function of $\beta _{RC}$ instead of $H$ as a function of $\beta _{ZS}$ for three reasons: $G$ is less sensitive to Reynolds number effects than $H$, the pressure gradient parameter that is compatible with $G$ from the similarity analysis is $\beta _{RC}$ (not $\beta _{ZS}$), plots of $G$ as a function of $\beta _{RC}$ are commonly used.

Figure 5 presents such a plot with the current flow case, together with three types of equilibrium or near-equilibrium data to appreciate its departure from equilibrium. The latter are the equilibrium TBL data of Mellor & Gibson (Reference Mellor and Gibson1966) from their similarity analysis, an empirical correlation from Cousteix (Reference Cousteix1989) based on near-equilibrium TBL data from the 1968 Stanford conference (Kline, Coles & Hirst Reference Kline, Coles and Hirst1969) that allows a comparison at high values of $\beta _{RC}$, and six cases of near-equilibrium APG TBLs from the literature (Bradshaw Reference Bradshaw1967; East & Sawyer Reference East and Sawyer1980; Skåre & Krogstad Reference Skåre and Krogstad1994; Kitsios et al. Reference Kitsios, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2016; Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017; Kitsios et al. Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017; Sanmiguel Vila et al. Reference Sanmiguel Vila, Vinuesa, Discetti, Ianiro, Schlatter and Örlü2020b). The Mellor and Gibson equilibrium data are probably reliable; however, a turbulence closure model had to be employed for its computation. The near-equilibrium flow case at $\beta _{RC}=34.2$ of Kitsios et al. (Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017) seems to deviate from the trend given by both the case of East & Sawyer (Reference East and Sawyer1980) at $\beta _{RC}=61.6$ and Cousteix's empirical correlation, but it is impossible to know which dataset might be more biased. The departure from equilibrium of the present flow case can still be appreciated despite these discrepancies.

Figure 5. The defect shape factor as a function of $\beta _{RC}$ for the present DNS (blue and orange are for the regions where ${\rm d}\beta _{ZS}/{{\rm d} x}>0$ and ${\rm d}\beta _{ZS}/{{\rm d} x}<0$, respectively), near-equilibrium cases from the literature (Bradshaw Reference Bradshaw1967; East & Sawyer Reference East and Sawyer1980; Skåre & Krogstad Reference Skåre and Krogstad1994; Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017; Kitsios et al. Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017; Sanmiguel Vila et al. Reference Sanmiguel Vila, Vinuesa, Discetti, Ianiro, Schlatter and Örlü2020b), computations of Mellor & Gibson (Reference Mellor and Gibson1966) and correlation of Cousteix (Reference Cousteix1989).

Indeed, the response lag of the mean flow (mean velocity defect) can be seen throughout the streamwise evolution of the present case. At the start, the defect shape factor $G$ is very slightly lower than the equilibrium case for a given value of $\beta _{RC}$, indicating that the momentum-losing effect of the pressure force is not immediate. At $\beta _{RC} \approx 6$, the pressure force starts decreasing in importance in the outer region which should lead to momentum gain. However, the momentum defect continues to increase substantially due to the influence of the upstream momentum-losing effect in the outer region and the increased impact of the pressure force in the inner region. The momentum defect even becomes superior to the equilibrium one at identical $\beta _{RC}$. Figure 5 illustrates that the Rotta–Clauser parameter $\beta _{RC}$ effectively reflects the momentum-losing effect of the pressure gradient across the entire boundary layer. The defect eventually starts decreasing, but it always remains higher than the equilibrium case. In the FPG zone at the end ($\beta _{RC}<0$), the defect remains significantly higher than that in equilibrium FPG TBLs.

3.1.2. The inner layer

As described in the introduction, $\beta _i$ represents the ratio of pressure force to turbulent force in the inner layer. Figure 2(c) displays both $\beta _i$ and the ratio of pressure force to the maximum turbulent force in the inner layer, where $F_{p,w}$ is the pressure force at the wall. Although the distributions of force balance and $\beta _i$ do not follow each other as close as $\beta _{ZS}$ and the force balance in the outer layer, the trend is the same. Here $\beta _i$ shows the changes in force balance well and its behaviour is consistent with the behaviour of $C_f$ as given in figure 2(d). Figure 6 displays the force balance at the eight streamwise positions for the inner layer as a function of $y^+$. The $\beta _i$ distribution indicates that the pressure force's relative importance (as a momentum-losing force) in the inner layer increases until approximately $21\delta _{av}$ (between S4 and S5). The force balance from S1 to S4 also demonstrates this as the pressure force's relative importance in the inner layer with respect to the other forces increases and later decreases from S5 to S7, and becomes a positive force (FPG) at S8.

Figure 6. The mean momentum budget profiles of the eight streamwise positions as functions of $y^+$. The budget terms are normalised with friction-viscous scales. The $x$-axis ranges are chosen so that $F_{ti,max}$ is approximately at the same location in all subfigures.

This change in force balance affects the mean velocity significantly in the inner layer. Figure 7 displays the mean velocity as a function of $y^+$ for the eight streamwise positions along with the ZPG TBL case in the viscous sublayer. Note that the ZPG TBL curve, in black, is hidden behind the purple curve of S1 as both curves coincide in the viscous sublayer. As the pressure force's relative importance increases ($\beta _i$ increases), $U^+$ increases and deviates from the linear law. This increase is expected as it reflects the change from a linear law to a quadratic law at the wall when the pressure force increases in importance with respect to the wall shear stress force (Patel Reference Patel1973; Skote & Henningson Reference Skote and Henningson2002). This behaviour of $U^+$ in the viscous sublayer of APG TBLs was already reported before by Gungor et al. (Reference Gungor, Maciel, Simens and Soria2016). The deviation from the ZPG TBL curve reaches its maximum near the position where $\beta _i$ is the highest. After $\beta _i$ reaches its maximum, $U^+$ constantly decreases, which means the pressure force's relative importance begins decreasing. Figure 7 shows that the mean velocity reacts almost instantly to the relaxation of the pressure force in the viscous sublayer. This immediate response of the mean flow to the decrease of the pressure force's relative importance suggests that the effect of flow history is small in the viscous sublayer. This is consistent with the fact that inertia forces, which are linked to the flow history's effect, are small in the viscous sublayer.

Figure 7. The friction-viscous scaled mean velocity profile of the eight streamwise positions, the near-equilibrium cases and the ZPG TBL case in the viscous sublayer as a function of $y^+$. The black line (ZPG TBL) is hidden behind the purple line (S1) as they coincide within the viscous sublayer.

There is still a deviation from the ZPG TBL profile in S6 to S8 which are located in the region where ${\rm d}\beta _i/{{\rm d} x}$ is negative, albeit the deviation is mild. The decrease of pressure force's importance from S6 to S8 leads to a decrease in $U^+$ in the viscous sublayer. In the last position, the pressure force is positive (FPG) which makes S8 different from all other positions. It is important to highlight that the flow would have not become a ZPG TBL if we had kept maintaining ${\rm d}\beta _{ZS}/{{\rm d} x}<0$ in a longer domain. The boundary layer would have eventually relaminarised.

Now, we discuss the overlap layer. It is important to state that there is no consensus regarding the nature of the overlap layer and the presence of the logarithmic law in APG TBLs in the literature, particularly when considering large-defect APG TBLs and those in disequilibrium. Furthermore, the overlap layer may be small or non-existent due to the current flow being at a moderate Reynolds number. To investigate the potential presence of a logarithmic law, figure 8(a) displays the diagnostic plot associated with such a law. The canonical logarithmic law does not hold, even in the small-defect case S1, as well as in the equilibrium cases EQ1 and EQ2. Nonetheless, the trends with respect to the logarithmic behaviour can be analysed. Figures 8 and 9(c,d) show that the flow deviates from the logarithmic law with the traditional values of $\kappa$ and $B$, $0.41$ and $5$, respectively, even at S1 and S2 which are small-defect cases. Figure 9(a,b) illustrate that the consistently positive $\beta _i$ from S1 to S7 results in a drop of $U^+$ below the log law. Such an effect has been documented in numerous studies of non-equilibrium APG TBLs (Gungor et al. Reference Gungor, Maciel, Simens and Soria2016). What is perhaps more surprising is its presence in the near-equilibrium cases depicted in figure 9(a). This suggests that the phenomenon is, in part, a direct effect of the pressure gradient, rather than solely its disequilibrating nature (as indicated by an increasing $\beta _i$). In the case of three equilibrium APG TBLs with mild APGs, Lee (Reference Lee2017) also observed a decreasing trend of $U^+$ below the log law with increasing $\beta _{RC}$, though these cases were at very low Reynolds numbers. Past experimental studies of near-equilibrium TBLs tended to confirm a log law behaviour (Clauser Reference Clauser1954; East & Sawyer Reference East and Sawyer1980; Skåre & Krogstad Reference Skåre and Krogstad1994; Elsberry et al. Reference Elsberry, Loeffler, Zhou and Wygnanski2000), even for cases with very large velocity defects, but since the wall shear stress was not measured directly, it is impossible to know if these flows follow the classical log law. Note that the large-defect near-equilibrium case EQ2 in figure 8(a) does not show a log law behaviour, which could be due to the moderate Reynolds number of the flow.

Figure 8. Diagnostic plots for the logarithmic law (a) and the half-power law (b) as a function of $y^+$. Straight red line in the top plot indicates the traditional Kármán constant value $1/\kappa =1/0.41$.

Figure 9. (a,b) Mean velocity profile of various streamwise positions, the ZPG TBL case and equilibrium cases in the overlap layer as a function of $y^+$. The streamwise positions from S1 to S5 are given on the left and the streamwise positions from S5 to S8 on the right for the sake of clarity. (c–e) Mean velocity profiles of three streamwise positions, S1 (c), S2 (d) and S8 (e) along with the logarithmic law using the traditional constants and values obtained with the correlations of Nickels (Reference Nickels2004), which were used in Knopp et al. (Reference Knopp, Reuther, Novara, Schanz, Schülein, Schröder and Kähler2021).

Knopp et al. (Reference Knopp, Reuther, Novara, Schanz, Schülein, Schröder and Kähler2021) found similar deviations from the classical log law for an APG TBL with values of $H$ and $\beta _i$ close to those of S2 but with $Re_\theta$ an order of magnitude higher, above 20 000. But as they pointed out, the logarithmic law may exist with different values of $\kappa$ and $B$. Figure 9(c,d) show that the mean velocity at S1 and S2 approaches the logarithmic law obtained with the correlations of Nickels (Reference Nickels2004) based on $\beta _i$, which were used in Knopp et al. (Reference Knopp, Reuther, Novara, Schanz, Schülein, Schröder and Kähler2021) for APG TBLs. The wall-normal extent of the approach towards a logarithmic law is thin in part because of the low Reynolds number. However, it is thinner in S2 than in S1 even though the Reynolds number is higher. This indicates that as the pressure force strength increases, the logarithmic layer becomes thinner, eroding from above, as was found by Alving & Fernholz (Reference Alving and Fernholz1995), Knopp et al. (Reference Knopp, Reuther, Novara, Schanz, Schülein, Schröder and Kähler2021) and Knopp (Reference Knopp2022).

To illustrate the history effects on the mean flow in the overlap layer, we can consider another streamwise position (named SB) where $\beta _i$ has the same value $0.0167$ as S2 but with $H=2.40$ (between S6 and S7). The mean velocity profile of SB is shown in figure 9(b), and its log-law diagnostic curve in figure 8(a). The main differences between this position and S2 are the flow history and the sign of ${\rm d}\beta _i/{{\rm d} x}$ (positive in S2 and negative in SB). At SB, the flow does not follow the classical log law at all and it behaves completely differently from S2 due to upstream history effects. As another illustration that the non-equilibrium nature of the TBL transforms the overlap layer, $\beta _i$ is nearly zero at S7 but the mean velocity profile in the overlap region is very different from the ZPG TBL, where $\beta _i$ is nearly zero too. The behaviour of this non-equilibrium TBL clearly shows that it is not only the local balance of forces that matters, as reflected by $\beta _i$, but also the upstream history and the local type of disequilibrating effect of the pressure force (${\rm d}\beta _i/{{\rm d} x}$).

Because of these effects, the correlations of Nickels (Reference Nickels2004) and Knopp et al. (Reference Knopp, Reuther, Novara, Schanz, Schülein, Schröder and Kähler2021) do not work for the other stations from S3 to S8. Figure 9(e) shows the mean velocity profile and the logarithmic law for S8. The mean velocity profile seems to approach the modified logarithmic law between $y^+=30$ and $50$ but it cannot be considered as such because the profile is not logarithmic there as can be seen from figure 8(a). The profile at S8 might actually approach a log law between $y^+=100$ and $200$. It should also be noted that the flow at S8 is an FPG TBL so if the decrease of $\beta _{ZS}$ (and $\beta _i$ and $\beta _{RC}$) would have been maintained in a longer domain, which means stronger FPG effects, the velocity profile would have eventually risen above the log law (Patel & Head Reference Patel and Head1968; Warnack & Fernholz Reference Warnack and Fernholz1998; Dixit & Ramesh Reference Dixit and Ramesh2008). Such a rise above the log law was found in the transition from an APG TBL to an FPG TBL studied by Tsuji & Morikawa (Reference Tsuji and Morikawa1976)

We also examine the presence of the half-power law which is expected to exist for large-defect TBLs (Stratford Reference Stratford1959; Perry, Bell & Joubert Reference Perry, Bell and Joubert1966; Coleman et al. Reference Coleman, Pirozzoli, Quadrio and Spalart2017; Coleman, Rumsey & Spalart Reference Coleman, Rumsey and Spalart2018; Knopp et al. Reference Knopp, Reuther, Novara, Schanz, Schülein, Schröder and Kähler2021). Figure 8(b) shows the diagnostic plot for the half-power law. It seems there is an approach towards the half-power law in large-defect cases S4, S5 and S6. The region where this approach occurs is located above the region where the mean flow approaches the logarithmic law as discussed previously. Therefore, the wall-normal location of the half-power region is consistent with the literature (Perry Reference Perry1966; Knopp et al. Reference Knopp, Reuther, Novara, Schanz, Schülein, Schröder and Kähler2021). As with the logarithmic layer, the half-power region is thin.

3.2.1. Outer layer

The local and upstream pressure force effects on turbulence are now examined. Figure 10 presents outer-scaled Reynolds stresses normalised with $U_e$. As was reported numerous times for APG TBLs, outer-layer turbulence becomes dominant and inner-layer turbulence loses its importance as the velocity defect increases in the momentum-losing zone (Gungor et al. Reference Gungor, Maciel, Simens and Soria2016; Maciel et al. Reference Maciel, Wei, Gungor and Simens2018). Moreover, the Reynolds stress levels in the outer layer increase with increasing velocity defect when they are normalised with $U_e$. However, the continuing rise of the Reynolds stresses while the flow is under turbulence-reducing conditions (${\rm d} \beta _{zs}/{{\rm d} x}<0$), starting at S3, is indicative of the delayed response of turbulence to changes in the pressure force. This delayed response of turbulence is even more pronounced than that observed in the mean flow. As the velocity defect decreases in the outer layer, the mean shear also decreases (see figure 4) and turbulence is expected to decay. But figure 10 shows that outer turbulence keeps increasing all the way down to S7 due to the delay in response of turbulence to the changes in the pressure gradient and mean velocity. The delay in response is more pronounced for normal stresses $\langle v^2\rangle$ and $\langle w^2\rangle$, which continue to rise further downstream than $\langle u^2\rangle$. This is reasonable since their energy depends on the redistribution from $\langle u^2\rangle$ through pressure strain. The Reynolds shear stress follows the same trend as $\langle v^2\rangle$ and $\langle w^2\rangle$, with levels at S8 still higher than those of S5. It is important to remember that in an equilibrium FPG TBL or an FPG TBL monotonously evolving from a ZPG TBL, the Reynolds stresses in the outer region are lower than those of the ZPG TBL when scaled with $U_e$ (Harun et al. Reference Harun, Monty, Mathis and Marusic2013; Volino Reference Volino2020). The Reynolds stress levels at S8 are therefore extremely high for an FPG TBL and this is due to the persistent effect of the upstream flow history.

Figure 10. Reynolds stress profiles normalised with the outer scales as functions of $y/\delta$ for the eight streamwise positions, the ZPG TBL case and the near-equilibrium cases of EQ1 and EQ2.

We now compare the near-equilibrium flow cases EQ1 and EQ2 with the present flow to isolate the effect of flow history. We first focus on the small-defect cases: S2 and EQ1. Similar to our approach in analysing the mean flow, we choose S2 because both cases exhibit a comparable velocity defect. The mean velocity profile, and consequently the mean shear profile, are similar in both cases. However, the local pressure force effect at S2 is stronger, as indicated by the values of $\beta _{ZS}$ and $\beta _{RC}$ in table 2. In addition, S2 is at a higher Reynolds number. Despite the similarities in the mean velocity profiles, the Reynolds stress profiles differ. There exists a disparity in the distribution of intensity among the Reynolds stress components. Specifically, the transverse Reynolds stresses ($\langle v^2\rangle$ and $\langle w^2\rangle$) exhibit higher intensity in the case of EQ1. Furthermore, the outer maxima of all Reynolds stress components are situated farther away from the wall in EQ1. In the current flow at S2, the pressure force has reached its local level through a ramp-up process. Consequently, the flow lacks the time needed to generate as much turbulence in the middle of the outer region compared with the near-equilibrium TBL. Moreover, there is insufficient time for the flow to redistribute as much energy to the transverse Reynolds stresses.

Now, we compare the large-defect flow cases, S4 and EQ2, which exhibit similar mean velocity profiles. In this case, the disparity in Reynolds stress levels is even more pronounced. The Reynolds stress levels at S4 are considerably lower than those in the near-equilibrium case. However, unlike the small defect cases, the difference in the wall-normal positions of the outer maxima is less significant, although they still remain lower in the current non-equilibrium flow. The redistribution of energy to the transverse Reynolds stresses continues to be less important in the non-equilibrium flow.

The delayed response of turbulence can be observed more comprehensively through the streamwise evolution of the outer maxima of the Reynolds stresses and turbulent kinetic energy, $k$, as depicted in figure 11(b). For ease of comparison with the pressure force effect and the mean flow response, figure 11(a) reproduces the streamwise distributions of $\beta _{ZS}$, $\beta _{RC}$, and $H$, also available in figure 2. The starting points of the curves in figure 11(b) correspond to the positions where the outer maxima of the Reynolds stresses begin to manifest. We have already seen that the mean flow responds with a delay, as evident from the $H$ distribution in figure 11(a), which is shifted downstream by approximately 10$\delta$ compared with the distribution of $\beta _{ZS}$. As mentioned earlier, the delay is even more significant for the Reynolds stresses in the outer region and it varies among the different Reynolds stress components. The maxima of $\langle u^2\rangle$ and $k$ peak at $x/\delta _{av} \approx 27$ and $28$, respectively, which is roughly 14$\delta$ downstream of the maximum of $\beta _{ZS}$. The delay is more pronounced for $\langle v^2\rangle$ and even more so for $\langle w^2\rangle$, again underscoring the delay in energy redistribution.

Figure 11. The spatial development of $H$, $\beta _{ZS}/max(\beta _{ZS})$ and $\beta _{RC}/max(\beta _{RC})$ (a), the maxima of turbulent kinetic energy ($k$) and the Reynolds stresses (b) and source terms, which are production for $k$ and $\langle u^2\rangle$, and pressure-strain for $\langle v^2\rangle$ and $\langle w^2\rangle$ components (c) in the outer layer as a function of $x/\delta _{av}$. Data only shown when an outer maximum is present. The levels are normalised with $U_e$ and $\delta$.

Outer turbulence therefore continues to increase for a considerable distance even after the pressure force begins to diminish. Furthermore, figure 11(b) illustrates that the subsequent decay of turbulence is slower than the initial upstream turbulence growth, even if the increase and decrease in pressure force are of the same amplitude (comparable to $|{\rm d}\beta _{ZS}/{{\rm d} x}|$). This slow turbulence decay cannot be solely attributed to the prolonged survival of energetic turbulence structures convected from upstream, as this cannot explain the observed turbulence buildup where the pressure force is decreasing. A tentative explanation can be offered by analysing the source and sink terms of $k$ and Reynolds stresses. Figure 11(c) illustrates the streamwise evolution of the maxima of the source terms of $k$ and the Reynolds stresses. In the case of $k$ and $\langle u^2\rangle$, the source term is the corresponding production term whereas the main contribution comes from the pressure-strain (redistribution) term for $\langle v^2\rangle$ and $\langle w^2\rangle$. In addition, the $\langle v^2\rangle$ production is considered for $\langle v^2\rangle$, albeit its contribution is small. It is observed in this figure that the production of $k$ and $\langle u^2\rangle$ has peaked around S5. One might expect a subsequent decrease of $k$; however, as noted earlier, $k$ continues to increase up to $x/\delta _{av} \approx 28$ (figure 11b). This growth is attributed to an increase of the ratio of production to dissipation of $k$ in the middle of the boundary layer, from S5 to S7, as illustrated in figure 12. Indeed, figure 12 demonstrates that the ratio of production to dissipation increases after S5 in the approximate wall-normal range $y/\delta =0.3$ to $0.8$. Given that this wall-normal range corresponds to the zone where production is maximal, the rise in the ratio of production to dissipation results in the continued increase of $k$ between S5 and S7 even if production has decreased in absolute terms.

Figure 12. The ratio of production to dissipation of turbulent kinetic energy as a function of $y/\delta$.

As a concluding exploration of outer region turbulence, we examine the budget of turbulent kinetic energy to gain a deeper understanding of turbulence behaviour in the current non-equilibrium flow. The transport equation for the turbulent kinetic energy is given as follows:

(3.2)$$\begin{gather} 0={-} \left( U \frac{\partial k}{\partial x}+ V \frac{\partial k}{\partial y} \right) - \left( \frac{\partial U}{\partial x} \langle u^2 \rangle + \frac{\partial U}{\partial y} \langle uv \rangle + \frac{\partial V}{\partial x} \langle uv \rangle + \frac{\partial V}{\partial y} \langle v^2 \rangle \right)\nonumber\\ - \epsilon - \frac{1}{2} \left( \frac{\partial \langle u k \rangle}{\partial x} + \frac{\partial \langle v k \rangle}{\partial y} \right) + \nu \left( \frac{\partial^2 k}{\partial x^2} + \frac{\partial^2 k}{\partial y^2} \right) - \frac{1}{\rho} \left(\frac{\partial \langle pu \rangle}{\partial x} + \frac{\partial \langle pv \rangle}{\partial y} \right). \end{gather}$$

The terms are (in order) mean convection, production, dissipation, turbulent transport, viscous diffusion and pressure transport. The budgets are illustrated in figure 13. For the sake of simplicity, for the current flow, we focus only on four representative stations that depict distinct flow situations: the small-defect case S2, the largest defect cases S4 and S5, and the small-defect case in the FPG region S8 which shares the same shape factor as S2. In each plot, a near-equilibrium case (ZPG, EQ1 or EQ2) is included as a reference for comparison.

Figure 13. The turbulent kinetic energy budget for S2, S8, EQ1, ZPG, S4, S5 and EQ2 as a function of $y/\delta$. The terms are normalised with outer scales ($U_{e}$ and $\delta$). The straight, dashed, dashed-dotted and dotted lines are for the present case, EQ1, EQ2 and ZPG, respectively.

In the small-defect case S2, the turbulent kinetic energy budget, shown in figure 13, resembles that of canonical wall flows as the comparison with the ZPG TBL indicates. However, there is an accumulation of production and dissipation in the outer region. This characteristic is typical of small-defect APG TBLs that either originated as ZPG TBLs (Gungor et al. Reference Gungor, Maciel and Gungor2022) or are in near-equilibrium state (Kitsios et al. Reference Kitsios, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2016; Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017; Kitsios et al. Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017). Nevertheless, when comparing the budget of S2 with that of the near-equilibrium case EQ1, which shares the same shape factor but has smaller pressure gradient parameters $\beta _i$, $\beta _{ZS}$ and $\beta _{RC}$, the near-equilibrium case exhibits a local maximum of production around $y/\delta =0.35$, a feature absent in S2. The absence of a production maximum in S2 serves as another manifestation of the delay in the turbulence response of the current flow. Turbulent transport is also much stronger in the near-equilibrium case.

The two lower plots compare the large-defect cases S4 and S5 with EQ2. In all these large-defect cases, there is a noticeable presence of production, dissipation and turbulent transport in the middle of the boundary layer, consistent with the heightened turbulent kinetic energy observed in that region. The value of $\beta _{ZS}$ at S4 and EQ2 is almost the same, but the budget terms at S4 have not reached the levels observed in EQ2, once again indicating a delay in turbulence response. S5 is the case with the largest mean velocity defect. Even so, the budget terms are not more significant than those for EQ2.

The FPG TBL case S8 is compared with the ZPG TBL in the top-middle plot. Despite S8 being an FPG TBL, the outer maxima of the budget terms persist in the middle of the boundary layer due to the delayed response of turbulence, and their levels are much stronger than those in the ZPG TBL. An equilibrium FPG TBL with the same $\beta _{ZS}$ as S8 would exhibit a budget similar to the ZPG TBL, albeit with even lower levels in the outer region due to reduced mean shear. Although production and dissipation have decreased since S5, they remain elevated. Surprisingly, in contrast, turbulent transport at S8 is higher than at S5. Figure 10 illustrates that the Reynolds stress profiles at S8 are wider than those at S5 due to these sustained high levels of turbulent transport.

3.2.2. Inner layer

Figure 14 illustrates Reynolds stress profiles as functions of $y^+$. When these profiles are normalised with friction-viscous scales, it is observed that the Reynolds stresses increase everywhere as $\beta _i$ and the velocity defect in the momentum-losing zone (until S5). This phenomenon has been well-documented in previous studies on APG TBLs (Gungor et al. Reference Gungor, Maciel, Simens and Soria2016; Maciel et al. Reference Maciel, Wei, Gungor and Simens2018). The increase in Reynolds stress levels in such boundary layers occurs because the friction velocity is inadequate as a scaling parameter for Reynolds stresses when dealing with large-velocity-defect TBLs (Maciel et al. Reference Maciel, Wei, Gungor and Simens2018). As the value of $\beta _i$ decreases, starting from S5 and continuing onwards, the amplitude of Reynolds stresses also decreases.

Figure 14. Reynolds stress profiles normalised with friction-viscous scales as functions of $y^+$ for the eight streamwise positions, the ZPG TBL case and the near-equilibrium cases EQ1 and EQ2.

In the near-wall region where inertia effects are negligible, below $y^+=10$, if turbulence were to respond instantly to the pressure force, the Reynolds stress profiles should be similar at identical $\beta _i$. However, stations S4 and S5, with close values of $\beta _i$, demonstrate that this is not the case. The Reynolds stress levels are higher at S5 than S4 in the near-wall region. Another indication of this discrepancy in response to the pressure force is the fact that the Reynolds stress profiles at S8 are higher than those in the ZPG TBL. Since $\beta _i$ is negative at S8 (FPG TBL), the Reynolds stress levels should be lower than those of the ZPG TBL in an equilibrium situation. These two observations suggest a delay in the response of turbulence. However, this delay may not be solely local; it could also be an indirect effect. As discussed later, part of the reason lies in the influence of large-scale turbulence on the near-wall region. The delayed response of the large-scale structures affects the near-wall region.

As for a comparison with the near-equilibrium cases, figure 15 illustrates the Reynolds stress profiles of the present flow, EQ1 and EQ2 at matching values of $\beta _i$. For the current flow, two profiles are shown at each $\beta _i$ value: one closer to the beginning of the domain, where the APG turbulence-promoting effect is increasing (${\rm d}\beta _i/{{\rm d} x}>0$) and depicted in orange, and another further downstream where the APG effect is decreasing (${\rm d}\beta _i/{{\rm d} x}<0$) and shown in purple. For the small-defect cases, a comparison between EQ1 and the first position in the current flow (dashed orange) reveals, in the enlarged-view plots of figure 15, that the Reynolds stresses in the inner region match up to approximately $y^+=30$, 10 and 20 for $\langle u^2\rangle ^+$, $\langle v^2\rangle ^+$ and $\langle uv\rangle ^+$, respectively. For $\langle w^2\rangle ^+$, the profiles start diverging very near the wall and no explanation has been found for such distinctive behaviour. Overall, in the present flow, the Reynolds stresses in the inner region have not yet attained the levels observed in the near-equilibrium case EQ1. Nevertheless, these history effects remain relatively minor when compared with all other stations located further downstream.

Figure 15. Reynolds stress profiles normalised with friction-viscous scales as a function of $y^+$ for the streamwise positions of the present case along with EQ1 and EQ2 at matching $\beta _i$. The arrow indicates whether $\beta _i$ is increasing or decreasing.

Indeed, at the second small-defect station (dashed orange), much further downstream, the deviations from the near-equilibrium case are significant everywhere, even very near the wall. As for the large-defect cases, the Reynolds stresses in the inner region at the first position in the current flow (dotted orange) have not reached the levels of the near equilibrium case EQ2. Meanwhile, at the second station (dotted purple), they have exceeded those levels. This underscores again the substantial delay in turbulence response to changes in the pressure force.

Turning our attention to $\langle u^2\rangle ^+$, figure 14(a) demonstrates that the near-wall peak, observed at $y^+=15$ in the ZPG TBL, becomes less sharp and eventually disappears as the local APG pressure force effect increases. This phenomenon, evident in both non-equilibrium and near-equilibrium flows, is attributed to the significant increase in $\langle u^2\rangle ^+$ above the near-wall region. The vanishing of the near-wall peak, coupled with the rise of $\langle u^2\rangle ^+$, has been documented previously (Nagano, Tsuji & Houra Reference Nagano, Tsuji and Houra1998; Dróżdż, Elsner & Drobniak Reference Dróżdż, Elsner and Drobniak2015; Gungor et al. Reference Gungor, Maciel, Simens and Soria2016). This suggests that the small-scale near-wall turbulence activity might be obscured by the footprints of the energetic large-scale structures located above the near-wall region. To investigate this, we analyse the spectral content of $\langle u^2\rangle$ and $\langle uv\rangle$, along with wall-normal profiles of these Reynolds stresses decomposed into small and large scales.

Figures 16 and 17 depict the premultiplied $\langle u^2\rangle$ and $\langle uv\rangle$ one-dimensional spanwise spectra, respectively, as functions of $\lambda _z^+$ and $y^+$ for S2, S5 and S8. The black dashed lines in these figures delineate the wavenumber sharp filter cutoff that will later be employed to decompose small and large scales. The bottom row presents the same spectra as the top row but only up to $y^+=100$ and with chosen contour values to facilitate clearer visualisation of the energy content in the inner region.

Figure 16. The spectral distribution of $\langle u^2\rangle$ as a function of $\lambda _z^+$ and $y^+$ for S2, S5 and S8. The spectra are plotted for the whole wall-normal range (a–c) and for the region up to $y^+=100$ (d–f). The flooded-contour levels are 0.3, 0.5, 0.7 and 0.95 of the maxima of each spectra. The black-line contour is 0.1 of the maxima. The dashed black lines indicate the cutoff filter $\lambda _z^+=300$ and the red dash-dotted lines indicate $\lambda _z^+=\delta ^+$.

Figure 17. The spectral distribution of $\langle uv\rangle$ as a function of $\lambda _z^+$ and $y^+$ for S2, S5 and S8. The spectra are plotted for the whole wall-normal range (a–c) and for the region up to $y^+=100$ (d–f). The flooded-contour levels are 0.3, 0.5, 0.7 and 0.95 of the maxima of each spectra. The black-line contour is 0.1 of the maxima. The dashed black lines indicate the cutoff filter $\lambda _z^+=300$ and the red dash-dotted lines indicate $\lambda _z^+=\delta ^+$.

The spectral distribution of $\langle u^2\rangle$ at S2 exhibits an inner peak, indicative of the near-wall streaks. These streaks are narrow structures with a $\lambda _z^+$ of approximately 120, a characteristic extensively discussed in the literature (Smith & Metzler Reference Smith and Metzler1983). As the velocity defect increases from S2 to S5, the outer-layer turbulence becomes dominant, and the inner peak vanishes from the spectra. This is consistent with the literature for large-defect TBLs (Kitsios et al. Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017; Lee Reference Lee2017; Gungor et al. Reference Gungor, Maciel and Gungor2022). Despite the absence of the inner peak, it remains unclear whether the streaks have vanished or if they are obscured by the presence of more intense large-scale structures (Gungor Reference Gungor2023; Gungor, Maciel & Gungor Reference Gungor, Maciel and Gungor2024).

Following the reversal to momentum-gaining conditions induced by the pressure gradient effect (${\rm d}\beta _{ZS}/{{\rm d} x}<0$ and ${\rm d}\beta _i/{{\rm d} x}<0$), the inner peak re-emerges at S8 with a similar $\lambda _z^+$, but it is weak compared with the outer peak. This finding is significant, demonstrating that inner-layer turbulence is becoming like that of canonical flows, as expected for an FPG TBL with a moderate $\beta _i$ value, despite the presence of history effects. It is important to emphasise once again that the flow is not reverting to a ZPG TBL state, as ${\rm d}\beta _{ZS}/{{\rm d} x}<0$ is maintained. If the pressure force variation were imposed in a longer domain, the flow would eventually laminarise.

In the inner region, $\langle u^2\rangle$ is also associated with wide structures at all stations. These large-scale structures have a $\lambda _z$ on the order of $\delta$, as in the outer region. At S5, these structures carry the majority of $\langle u^2\rangle$ in the inner layer. Furthermore, at S8, the relative contribution of the large scales in the inner region is higher than at S2 due to the presence of very large and energetic structures convected from upstream. This history effect also results in the structures being wider, with respect to $\delta$, at S8 than S2. Gungor et al. (Reference Gungor, Maciel and Gungor2024) showed that these large-scale structures primarily belong to the outer layer.

The spectral distributions of $\langle uv\rangle$ in figure 17 exhibit similar results to those of $\langle u^2\rangle$. The inner peak is present in the small defect cases S2 and S8, although it is notably weaker at S8 when compared with the outer peak. The spectral distribution of $\langle uv\rangle$ at S2 is consistent with those found in the literature (Lee Reference Lee2017; Gungor et al. Reference Gungor, Maciel and Gungor2022). In the large-defect case S5, the inner peak has vanished, and $\langle uv\rangle$ is influenced significantly by large-scale structures in the inner region, although small-scale structures still contribute. Overall, the effect of the large-scale structures in the inner layer is less important for $\langle uv\rangle$ than for $\langle u^2\rangle$, a trend also noted by Gungor et al. (Reference Gungor, Maciel and Gungor2024).

To investigate small-scale properties in the inner layer, we employ a filtering process to isolate the contributions of small scales by separating them from the large scales. To achieve this, we decompose the Reynolds stresses using a wavenumber sharp filter cutoff in the spectral domain, as depicted in figures 16 and 17 with vertical black dashed lines. We set the cutoff wavelength at $\lambda _z^+=300$, chosen for its effective differentiation between small and large scales in the inner region. This value is also used in the recent study by Deshpande et al. (Reference Deshpande, van den Bogaard, Vinuesa, Lindić and Marusic2023). We have tested that the reconstructed Reynolds stress distributions are relatively insensitive, but only qualitatively (see the discussion in the following), to the specific filter cutoff value.

Figure 18 presents the Reynolds stress profiles of small scales ($\langle u^2 \rangle ^+_{SS}$ and $\langle uv\rangle ^+_{SS}$) and large scales ($\langle u^2 \rangle ^+_{LS}$ and $\langle uv\rangle ^+_{LS}$) as functions of $y^+$. The full Reynolds stress profiles are presented at the top of the figure to facilitate comparison. In contrast to the full $\langle u^2 \rangle ^+$ profiles, the small-scale $\langle u^2 \rangle ^+_{SS}$ profiles at S3 and S7 reveal an inner peak. Furthermore, the inner peak has become sharper at S1, S2 and S8. These results indicate that large-scale structures indeed influence the Reynolds stress distributions in the inner layer, leading to the obscuration of small-scale contributions and the near-wall cycle of turbulence production.

Figure 18. Reynolds stress profiles as a function of $y^+$ for the full signal (a,b), large-scale structures (c,d) and small-scale structures (e,f).

Deshpande et al. (Reference Deshpande, van den Bogaard, Vinuesa, Lindić and Marusic2023) found that the inner peak of $\langle u^2 \rangle ^+_{SS}$ collapses well across ZPG TBLs and TBLs subjected to very mild APG ($\beta _{RC}<1.7$), across various Reynolds numbers. The reported peak value of $\langle u^2 \rangle ^+_{SS}$ is $5.5$ at $y^+=15$. Recall that we employ the same filtering procedure. In the current flow, the inner peak of $\langle u^2 \rangle ^+_{SS}$ closely resembles that of Deshpande et al. (Reference Deshpande, van den Bogaard, Vinuesa, Lindić and Marusic2023) only at the first station, S1, with a small velocity defect and minimal history effects. As the velocity defect increases, the $\langle u^2 \rangle ^+_{SS}$ inner profile also increases, eventually leading to the disappearance of the peak. Positions with substantial velocity defects, specifically S4 to S6, do not exhibit inner peaks in the profiles, except for a hump observed at S6. The disappearance of the near-wall peak suggests that the near-wall cycle could be significantly attenuated or even absent in TBLs with large velocity defects.

Nevertheless, interpreting the decomposed Reynolds stresses requires caution. First, the filtering procedure based on one-dimensional spectra has inherent limitations. A more refined separation between small and large scales could be achieved by employing a filtering approach based on two-dimensional streamwise–spanwise spectra, as demonstrated by Lee & Moser (Reference Lee and Moser2019). Similar to Deshpande et al. (Reference Deshpande, van den Bogaard, Vinuesa, Lindić and Marusic2023), Lee & Moser (Reference Lee and Moser2019) reported a Reynolds number independence of the $\langle u^2 \rangle ^+_{SS}$ profile up to $y/\delta =0.2$ in channel flow cases. Yet, in their case, the peak value of $\langle u^2 \rangle ^+_{SS}$ exceeded $7$. Lastly, it is plausible that the Reynolds number in our study might not be sufficiently high to achieve a clear-cut scale separation of scales.

The inner peak of $\langle u^2 \rangle _{SS}^+$ at S7 is rather surprising because it is below that at S1, the latter closely following the ZPG TBL and small defect behaviour presented in Deshpande et al. (Reference Deshpande, van den Bogaard, Vinuesa, Lindić and Marusic2023). At S7, the pressure gradient is null. One could therefore expect the inner peak to be similar to the ZPG TBL one (and, therefore, also to the S1 peak) or even higher due to the upstream trend. The lower-level peak might suggest that the near-wall cycle has not yet fully recovered from the upstream pressure gradient effects, if it was indeed dampened or destroyed by them. It should be noted that, in contrast to the situation of $\langle u^2 \rangle _{SS}^+$, the inner peak of the small-scale Reynolds shear stress $-\langle uv \rangle _{SS}^+$ at S7 is similar to that observed at S1.

Regarding the inner peak of $\langle u^2 \rangle _{SS}^+$ at S8, it is expected that it is lower than that at S1. FPG TBLs that are in near-equilibrium or are monotonically developing from a ZPG TBL have lower $\langle u^2 \rangle ^+$ levels in the inner region (complete profile) compared with ZPG TBLs (Bourassa & Thomas Reference Bourassa and Thomas2009; Harun et al. Reference Harun, Monty, Mathis and Marusic2013; Volino Reference Volino2020), even if the difference is not as pronounced as in the outer region.

As for the small-scale Reynolds shear stress $-\langle uv \rangle _{SS}^+$, it behaves similarly to $\langle u^2 \rangle _{SS}^+$. The notable difference, as mentioned previously, is that the inner peak at S7 is similar to that at S1, as would be expected if there were no history effects. The inner peak of $-\langle uv \rangle _{SS}^+$ is associated with the near-wall production of $\langle u^2 \rangle _{SS}^+$.

The profiles of large-scale Reynolds stresses demonstrate that large scales indeed contribute significantly to the inner region, and this contribution increases with the mean velocity defect (not with $\beta _i$). As expected from the spectral distributions, $-\langle uv \rangle _{LS}^+$ is small below $y^+=10$ whereas $\langle u^2 \rangle _{LS}^+$ is significant in that region, comparable to $\langle u^2 \rangle _{SS}^+$. The large-scale structures reaching the very-near-wall region cannot have a large wall-normal velocity. They are long streaky $u$-structures rather than sweeps and ejections.

To gain a deeper understanding of inner layer turbulence behaviour, we analyse the turbulent kinetic energy budget at the streamwise positions S2, S8, S4 and S5 along with ZPG, EQ1 and EQ2. Figure 19 illustrates the budget terms, normalised with friction-viscous scales, as functions of $y^+$, with a similar format as the outer-scaled budgets in figure 13. The balance of the budget terms in the inner layer at S2 closely resembles that of canonical wall flows; however, the levels of the terms are approximately $20$–$30\,\%$ higher at S2. Another distinction is the higher level of production around $y^+=100$ relative to its peak value at $y^+=10$ in the APG case, aligning with the behaviour of the Reynolds stresses depicted in figure 14. These similarities and differences have been observed in small-defect APG TBLs, either originating as ZPG TBLs (Gungor et al. Reference Gungor, Maciel and Gungor2022) or in near-equilibrium states (Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017; Kitsios et al. Reference Kitsios, Sekimoto, Atkinson, Sillero, Borrell, Gungor, Jiménez and Soria2017).

Figure 19. The turbulent kinetic energy budget for S2, S8, EQ1, ZPG, S4, S5 and EQ2 as a function of $y^+$. The terms are normalised with friction-viscous scales. The straight, dashed, dashed-dotted and dotted lines are for the present case, EQ1, EQ2 and ZPG, respectively.

Moving from S2 to S4 and S5, the inner wall-normal distributions of the various energy transfer mechanisms become notably different. For instance, production still exhibits an inner maximum, but it is no longer dominant compared with the other terms due to high levels of turbulence being transported from the outer layer. The budget terms have higher values than those for EQ2, as can be expected given that $\beta _i$ is higher at S4 and S5. At S8, the inner budget has returned to one typical of canonical wall flows, with levels comparable to those of the ZPG TBL and smaller than at S2. However, turbulent transport remains strong above $y^+=40$ due to significant outer turbulence resulting from history effects. These history effects likely explain why the Reynolds normal stresses in the inner layer are much higher at S8 than at S2 (figure 14) despite the lower energy transfer terms.