3.1. The new composite profile formulation

The observations described above motivate a series of modifications to the formulation of Nickels (Reference Nickels2004). Specifically, the present formulation will introduce a velocity overshoot in the inner region, reformulate the wake function to be consistent with the physically motivated definition of $\delta$ adopted here and incorporate recent physical insights regarding PG effects in the overlap region (at high $ \textit{Re}_{\tau }$ ) and PG-history effects in the wake region. Accordingly, we propose a modified composite profile expressed as

(3.1) \begin{align} U^{+} = U^{+}_{\textit{inner}} \big(z^{+},p_x^+,z^{+}_{c} \big)+U^{+}_{ov\textit{erlap}} \big(z^{+},p_x^+,z^{+}_{c},\eta ,C_{H_w} \big)+U^{+}_{w\textit{ake}} \big(\eta ,C_{H_w},\varPi \big), \end{align}

with the specific modifications to each component of the formulation described below.

3.1.1. Addition of a wake-stretching parameter

First, an additional parameter, $C_{H_w}$ , is introduced as a ‘wake-stretching’ parameter to account for PG-history effects observed in the wake region. The coefficients in the wake function are also revised to ensure consistency with the boundary-layer thickness definition of Lozier et al. (Reference Lozier, Deshpande, Zarei, Lindić, Rowin and Marusic2025). Accordingly, the outer-scaled wall-normal coordinate $\eta$ is rescaled within the wake-region expression ( $U^{+}_{w\textit{ake}}$ ), resulting in an effective coordinate

(3.2) \begin{align} \eta _{\textit{eff}} = 1.455 C_{H_w} \eta . \end{align}

3.1.2. Addition of an overshoot function

The overshoot function is included here primarily because it influences the inferred lower bound of the logarithmic region; in particular, the commonly assumed onset near $z^+ \approx 70$ typically underestimates the start of the logarithmic region (see figure 13). If this effect is not accounted for, part of the buffer region may be interpreted as logarithmic scaling which can bias fitted quantities such as $\kappa$ , $B$ and $U_{\tau }$ . To account for the velocity overshoot observed in the buffer region, a Gaussian-type function is used following the approach of Monkewitz et al. (Reference Monkewitz, Chauhan and Nagib2007) and Chauhan et al. (Reference Chauhan, Monkewitz and Nagib2009), who incorporated a similar term into the inner-region expression of Musker (Reference Musker1979). The overshoot function is written as

(3.3) \begin{align} f(z^{+}_{c}) \exp \left ( \frac {-\ln ^2(z^+/38)}{0.835} \right ) \!, \end{align}

where $f$ represents the maximum overshoot relative to the baseline profile of Musker (Reference Musker1979). For ZPG flow, $f \approx 0.2$ , consistent with the findings of Chauhan et al. (Reference Chauhan, Monkewitz and Nagib2009). However, this value depends on the PG (i.e. varies with $z^{+}_{c}$ ), as demonstrated in Appendix B. The contribution of the overshoot function is also restricted to the inner region, $9 \leqslant z^{+} \leqslant 150$ , with its maximum occurring at $z^{+} = 38$ . This ensures that the inner-region expression asymptotes to $U^{+}=z^{+}_{c}$ with increasing $z$ (figure 3), consistent with the framework of Nickels (Reference Nickels2004) and maintaining compatibility with the overlap-region formulation.

Figure 3.

Contributions of the inner (3.7), overlap (3.8) and wake (3.9) expressions to the present composite profile as a function of (a) the wake-history factor $C_{H_w}$ , (b) $z^{+}_{c}$ and (c) $\varPi$ . A reference dataset with $ \textit{Re}_{\tau }=5000$ was used to generate these profiles.

Figure 3. Long description

Panel A: A line graph shows the mean velocity profile contributions of inner, overlap, and wake expressions as a function of the wake-history factor. The x-axis is labeled z+ and ranges from 10^1 to 10^4. The y-axis is labeled U+ and ranges from 0 to 35. Three lines represent different values of the wake-history factor: 0.7, 1.0, and 1.3. Panel B: A line graph shows the mean velocity profile contributions as a function of z+. The x-axis is labeled z+ and ranges from 10^1 to 10^4. The y-axis is labeled U+ and ranges from 0 to 35. Three lines represent different values of z+: 12.0, 10.8, and 9.5. Panel C: A line graph shows the mean velocity profile contributions as a function of z+. The x-axis is labeled z+ and ranges from 10^1 to 10^4. The y-axis is labeled U+ and ranges from 0 to 35. Three lines represent different values of Pi: 0.5, 1.0, and 1.5.

3.2. Linking local and upstream PGs to $z^{+}_{c}$

As demonstrated in Part 1 and by Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017), PG-history effects can produce differences in the overlap-region mean velocity profiles of APG TBLs even when the local flow conditions ( $ \textit{Re}_{\tau }$ and $\beta$ ) are matched. These differences appear primarily as variations in the additive constant $B$ (or equivalently $z^{+}_{c}$ , see Nickels Reference Nickels2004). To capture this behaviour, we introduce the concept of an effective PG parameter

(3.4) \begin{align} \beta _{\textit{eff}} = \beta C_{H_i}, \end{align}

which combines the local PG parameter, $\beta$ , with a PG-history parameter, $C_{H_i}$ , to represent the combined influence of local and upstream PGs on the inner and overlap regions.

Importantly, $C_{H_i}$ is not introduced as an additional free parameter of the composite profile. Rather, it is obtained from the fitting process and provides a means of linking both local and upstream PG effects to $z^{+}_{c}$ , thereby offering physical insight into the overall flow conditions experienced by the TBL. To model this relationship, we follow the framework of Nickels (Reference Nickels2004), as described by (1.10) and (1.11). While $p_x^+$ plays an important role in the near-wall shear-stress expression for APG TBLs (1.10) (particularly for $z^+\lt 5$ ), the resulting model for the sublayer thickness (1.11) does not capture the changes in $z^{+}_{c}$ (or equivalently $B$ ) observed for high- $ \textit{Re}_{\tau }$ TBLs with moderate APGs in Part 1. Accordingly, we propose that $z^{+}_{c}$ is more appropriately described as a function of $\beta _{\textit{eff}}$ , consistent with the results of Part 1 and previous studies. This leads to a modified form of (1.11)

(3.5) \begin{align} c\, \beta _{\textit{eff}}\, z_c^{+3} + z_c^{+2} - R_c^2 = 0, \end{align}

whose root predicts $z^{+}_{c}$ for a given $\beta _{\textit{eff}}$ . This model yields excellent agreement with experimental data for both low- and high- $ \textit{Re}_{\tau }$ APG TBLs when $c=0.0145$ , as demonstrated by the dashed line in figure 8(a).

The history parameter $C_{H_i}$ can then be expressed in terms of $z^{+}_{c}$ and $\beta$ using (3.5)

(3.6) \begin{align} \begin{array}{ll} C_{H_i} = \dfrac {144 - z_c^{+2}}{c \beta z_c^{+3}} & \text{for } \beta \neq 0; \\[12pt] C_{H_i} = \dfrac {12}{z_c^{+}} & \text{for } \beta = 0. \end{array} \end{align}

To avoid the singularity that arises for nominally ZPG cases ( $\beta =0$ ), the second expression compares the fitted $z^{+}_{c}$ directly with the canonical ZPG value $z^{+}_{c}=12$ . For APG TBLs with minimal upstream PG-history effects, both history parameters introduced here should be approximately unity ( $C_{H_i} \approx 1$ and $C_{H_w} \approx 1$ ). Deviations from unity, therefore, indicate a significant influence of PG history on the TBL.

Notably, $C_{H_i}$ and $C_{H_w}$ are also independent parameters that can be interpreted in terms of the characteristic time scales associated with different PG effects. Specifically, $C_{H_i}$ reflects the relatively rapid response of the inner region to changes in PG conditions, whereas $C_{H_w}$ represents the slower adjustment of the wake region, consistent with the observations reported in Part 1.

Incorporating these modifications, the proposed composite profile formulation becomes

(3.7) \begin{align} U_{\textit{inner}}^+ &= z^{+}_{c} \Bigg [ 1 - \Bigg ( 1 + 2 \left ( \frac {z^+}{z^{+}_{c}} \right ) + \frac {1}{2} \left ( 3 - p_x^+ z^{+}_{c} \right ) \left ( \frac {z^+}{z^{+}_{c}} \right )^2 - \frac {3}{2} p_x^+ z^{+}_{c} \left ( \frac {z^+}{z^{+}_{c}} \right )^3 \Bigg ) e^{- \tfrac {3z^+}{z^{+}_{c}}} \Bigg ] \nonumber \\[5pt]&\quad + \left (0.23 + 0.5 \tanh \left (31 \, \frac {144 - {z^{+}_{c}}^2}{{z^{+}_{c}}^3}\right )\right ) \exp \left ( \frac {-\ln ^2(z^+/38)}{0.835} \right )\!, \\[-28pt] \nonumber \end{align}

(3.8) \begin{align} U^+_{ov\textit{erlap}} = \frac {\sqrt {1+p_x^+z^{+}_{c}}} {6\kappa } \ln \left ( \frac {1+(0.6(z^+/z^{+}_{c}))^6} {1 + \eta _{\textit{eff}}^6} \right )\!, \\[-28pt] \nonumber \end{align}

(3.9) \begin{align} U^+_{w\textit{ake}} = \frac {2\varPi }{\kappa } \left ( 1 - \exp \left ({\frac {-15(\eta _{\textit{eff}}^4 + \eta _{\textit{eff}}^8)}{1 + 15\eta _{\textit{eff}}^3}} \right ) \right )\!. \\[0pt] \nonumber \end{align}

Figure 3 illustrates the influence of the three free parameters ( $C_{H_w}$ , $z^{+}_{c}$ and $\varPi$ ) on the resulting composite mean velocity profile. Variations in $C_{H_w}$ and $\varPi$ primarily reshape the wake region, influencing both the extent and magnitude of the wake. In contrast, variations in $z^{+}_{c}$ modify the overlap region (through shifts in the mean velocity) and the inner region (through the magnitude of the buffer-region overshoot and the near-wall profile shape). We also note that the present composite profile is only intended to model the mean velocity profile for $z \leqslant \delta$ , and does not attempt to reach the external inviscid region beyond $\delta$ , where the mean velocity is not necessarily constant in PG flows (Coleman et al. Reference Coleman, Rumsey and Spalart2018).

3.3. Note for high-Reynolds-number TBLs

In the high- $ \textit{Re}_{\tau }$ limit, the proposed composite profile formulation simplifies, particularly in the overlap region. Specifically, as $ \textit{Re}_{\tau } \rightarrow \infty$ , $p_x^+ \rightarrow 0$ and $\eta _{\textit{eff}} \rightarrow 0$ , while $z^{+} \rightarrow \infty$ (since $\delta ^+ \rightarrow \infty$ ). In this limit, the inner-region expression asymptotes to $U^{+}=z^{+}_{c}$ for $z^{+} \gg z^{+}_{c}$ . Consequently, the overlap-region expression (3.8) reduces to

(3.10) \begin{align} U^+ = \frac {1}{\kappa } \ln (z^+) + \underbrace { z^{+}_{c}-\frac {1}{\kappa } \ln (z^{+}_{c}/0.6) }_{B}, \end{align}

which has the form of the classical logarithmic law. Here, $\kappa$ remains constant, while the additive constant $B$ depends only on $z^{+}_{c}$ (i.e. PG effects), consistent with the results of Part 1. For ZPG TBLs, when $z^{+}_{c}=12$ , (3.10) reduces to the classical log law with $\kappa =0.39$ and $B=4.3$ , consistent with Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013).

3.4. Fitting method

We now outline practical considerations for applying the proposed composite mean velocity profile to APG TBL datasets. The primary inputs may be expressed in normalised form as $U^+$ , $z^+$ , $ \textit{Re}_{\tau }$ and $p_x^+$ (or equivalently in dimensional form as $U$ , $z$ , $U_\tau$ , $\delta$ , ${\rm d}P/{\rm d}x$ , $\nu$ and $\rho$ ). The three parameters optimised when fitting the composite profile to a dataset are the sublayer thickness $z^{+}_{c}$ (which also determines $\beta _{\textit{eff}}$ and $C_{H_i}$ ), together with the wake parameters $\varPi$ and $C_{H_w}$ .

For all cases considered here, the fitted parameters were determined simultaneously using a nonlinear least-squares curve-fitting procedure applied to the measured mean velocity profiles over the range $z^+\lt 1.25\mathrm{Re}_{\tau }$ . The procedure minimises the sum of the squared differences between the measured and composite values of $U^+$ , with equal weighting assigned to all retained data points. The upper limit is imposed because measurements beyond the boundary-layer edge correspond to the external inviscid region, where the mean velocity may not necessarily approach a constant value (Coleman et al. Reference Coleman, Rumsey and Spalart2018). To facilitate application of this method, an example implementation is provided as a notebook. Normalised mean velocity data, together with the corresponding flow parameters described above, can be supplied as inputs. The notebook then performs the nonlinear fit to determine the optimal values of the three free parameters in the composite profile formulation (3.1) and plots the resulting fitted profile.

The fitting procedure was tested for all datasets considered in this study (table 2) to confirm that the optimisation remained robust across a wide range of flow conditions. In addition, for TBLs with minimal or well-controlled PG-history effects (i.e. where $C_{H_i}=C_{H_w}=1$ ), the composite profile may alternatively be used to estimate $U_\tau$ and $\delta$ , as discussed in § 5. An implementation of this procedure is also provided in a separate notebook, where dimensional mean velocity data can be supplied when $U_\tau$ and $\delta$ are not known a priori.

4.1. Overlap region

We first examine trends in $z^{+}_{c}$ , a key parameter governing the overlap-region behaviour of the mean velocity profile. As discussed by Nickels (Reference Nickels2004), and demonstrated experimentally in Part 1, $z^{+}_{c}$ is related to the additive constant of the classical log law and therefore governs the magnitude of the mean velocity in the overlap region. This parameter is also expected to depend on the local PG according to the model proposed in (3.5). As such, the optimised values of $z^{+}_{c}$ for cases with minimal PG history are plotted as a function of $\beta$ in figure 8(a). Excellent agreement is observed between the $z^{+}_{c}$ values obtained from the composite profile fitting method and the tuned model (3.5; when $C_{H_i}\approx 1$ ), shown by the dashed black line in figure 8(a,b). Notably, $z^{+}_{c}$ decreases systematically with increasing APG, independent of Reynolds number, consistent with the results of Part 1. Additionally, these results agree more closely with the proposed $\beta$ -dependent model (3.5) as compared with the original $p_x^+$ -based model (1.11) of Nickels (Reference Nickels2004) (which was tested, although not shown here).

Figure 8.

Variation of $z^{+}_{c}$ with $\beta$ for (a) cases with minimal PG history and (b) cases with PG-history effects. The dashed line represents the proposed model (3.5), and the grey lines represent $C_{H_i}$ levels. (c) Schematic of the mean velocity profile for reference high- $ \textit{Re}_{\tau }$ ZPG and APG datasets depicting ‘history-enhanced’ and ‘history-damped’ shifting behaviours.

Next, we consider cases with known PG-history influences, as shown in figure 8(b). For these datasets, the optimised $z^{+}_{c}$ values deviate from the model of (3.5), indicating a measurable influence of their PG histories, and $C_{H_i} \neq 1$ . Physically, this $C_{H_i}$ indicates the cumulative influence of upstream PG history on the overlap-region structure of the mean velocity profile. To further aid in the interpretation of $C_{H_i}$ , we consider the composite mean velocity profiles of the reference high- $ \textit{Re}_{\tau }$ ZPG (black) and APG (red) TBLs with minimal PG history, shown schematically in figure 8(c). The reference APG profile exhibits the expected ‘downward’ shift relative to the reference ZPG profile in the overlap region, consistent with (3.5). However, if PG-history effects are introduced (while maintaining the same local $\beta$ ), the profile may instead lie in between the reference ZPG and APG cases, corresponding to $C_{H_i}\lt 1$ (yellow shaded regions in figure 8). We refer to this behaviour as history-damped shifting. Conversely, the upstream PG history may enhance the downward shift of the mean velocity profile in the overlap region, exceeding that of the reference APG profile, corresponding to $C_{H_i}\gt 1$ (red shaded regions in figure 8). We refer to this behaviour as history-enhanced shifting. It is also noted that if the profile instead lies above the reference ZPG profile, then $C_{H_{i}}\lt 0$ (i.e. indicating behaviour akin to an FPG TBL).

While minor deviations from the model curve (e.g. that fall within the representative uncertainty of $\pm 0.1$ ) should not necessarily be interpreted as evidence of significant PG-history effects, some cases do show clear evidence of strong PG-history effects. For example, the datasets of Gungor et al. (Reference Gungor, Gungor and Maciel2024) and Preskett et al. (Reference Preskett, Virgilio, Jaiswal and Ganapathisubramani2025) contain cases with nominally ZPG conditions (i.e. $\beta =0$ locally), which have experienced significant upstream PG variations. These cases appear as the yellow and orange downward triangles in figure 8(b), lying at $\beta =0$ but with $z^{+}_{c} \neq 12$ . Their corresponding mean velocity profiles are also shown in figure 7, specifically case G2 from Gungor et al. (Reference Gungor, Gungor and Maciel2024) ( $z^{+}_{c}=8.2$ ) and case K4 from (Preskett et al. Reference Preskett, Virgilio, Jaiswal and Ganapathisubramani2025) ( $z^{+}_{c}=12.6$ ), illustrating the non-canonical behaviour of these profiles despite having locally ZPG conditions.

These cases with $C_{H_i} \neq 1$ can also be mapped to an effective PG parameter $\beta _{\textit{eff}}$ using the fitted $z^{+}_{c}$ and (3.5). Flows with matched $\beta _{\textit{eff}}$ will therefore exhibit similar overlap-region behaviours (i.e. the same additive constant $B$ ). In this way, $\beta _{\textit{eff}}$ is interpreted as the effective PG conditions experienced by the boundary layer after accounting for both the local PG and the influence of its upstream PG history. This framework, therefore, provides a simple means of interpreting/characterising the influences of different upstream PG histories on the mean velocity profile.

The different patterns of $z^{+}_{c}(\beta )$ observed within individual datasets further reflect the unique PG histories/developments of each case (e.g. differences in their $\beta (x)$ distributions arising from ramp or roof geometries in experiments). These PG-history effects become clearer when considering specific APG TBLs with matched local conditions ( $ \textit{Re}_{\tau }$ and $\beta$ ) but different PG histories. The corresponding $z^{+}_{c}$ values for such cases from the numerical datasets of Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017) and Pozuelo et al. (Reference Pozuelo, Li, Schlatter and Vinuesa2022), together with the high- $ \textit{Re}_{\tau }$ experimental dataset from Part 1, are shown in figure 9(a,b). In figure 9(a), differences in the upstream PG development (see inset $\beta (x)$ distributions) produce clear differences in $z^{+}_{c}$ despite having matched local flow conditions. In other words, the influences of the local and upstream PGs combine to produce different effective PG conditions, $\beta _{\textit{eff}}$ . Specifically, for the higher- $ \textit{Re}_{\tau }$ data points (labelled 2), the case of Pozuelo et al. (Reference Pozuelo, Li, Schlatter and Vinuesa2022) exhibits history-damped shifting ( $\beta _{\textit{eff}}\lt \beta$ on account of its PG history; orange curve in the inset), whereas the case of Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017) exhibits history-enhanced shifting ( $\beta _{\textit{eff}}\gt \beta$ on account of its PG history; black curve in the inset).

In figure 9(b), the cases labelled ZA and ZD were designed to introduce measurable PG-history effects relative to the reference minimal PG-history cases (ZA-Ref. and ZD-Ref.) from Part 1. For ZA, an imposed upstream APG perturbation produces history-enhanced shifting relative to the reference ZPG case (ZA-Ref.), corresponding to $\beta _{\textit{eff}}\gt 0$ , as expected. In contrast, delayed APG development in ZD, relative to the reference APG case (ZD-Ref.) leads to history-damped shifting, although $z^{+}_{c}$ remains less than the canonical ZPG value of 12 (i.e. $\beta \gt \beta _{\textit{eff}} \gt 0$ ). Finally, despite differences in upstream PG history, both ZE and ZE-Ref. exhibit overlap-region shifts consistent with the local $\beta$ , following (3.5).

Together, these observations demonstrate that the composite profile, fitting procedure and PG-history characterisation framework proposed here provide a consistent way to describe the combined influences of local and upstream PGs across a wide range of APG TBLs. While the parameter $C_{H_i}$ characterises PG-history effects on the overlap-region velocity magnitude, the wake region can exhibit additional history-dependent behaviours. We therefore next examine how PG history influences the wake region through the parameter $C_{H_w}$ .

Figure 9.

Variation of $z^{+}_{c}$ with $\beta$ for cases with matched local flow conditions ( $ \textit{Re}_{\tau }$ and $\beta$ ) but different PG histories from the datasets of (a) Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017) and Pozuelo et al. (Reference Pozuelo, Li, Schlatter and Vinuesa2022), and (b) Zarei et al. (Reference Zarei, Lozier, Deshpande and Marusic2026).

Figure 9. Long description

Panel A: A scatter plot shows the variation of zc+ with beta for cases with matched local flow conditions. The x-axis represents beta, ranging from 1.0 to 2.2, and the y-axis represents zc+, ranging from 10.2 to 11.6. Data points from Bobke et al. (2017) and Pozuelo et al. (2022) are plotted, with different symbols and colors representing different datasets. An inset graph shows the variation of beta with Reτ. Panel B: Another scatter plot shows the variation of zc+ with beta for cases with matched local flow conditions from the dataset of Zarei et al. (2026). The x-axis represents beta, ranging from 0 to 1.5, and the y-axis represents zc+, ranging from 10.8 to 12.2. Data points for perturbed and reference cases are plotted with different symbols and colors.

4.2. Wake region

As established, the wake region can be strongly influenced by both local and upstream PGs (Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017; Gungor et al. Reference Gungor, Gungor and Maciel2024; Preskett et al. Reference Preskett, Virgilio, Jaiswal and Ganapathisubramani2025). However, in contrast to the inner and overlap regions, which adjust relatively rapidly to changes in the local flow conditions, the wake region is typically associated with larger-scale motions, and therefore may retain a longer memory of the upstream PG history. This behaviour motivates the introduction of the independent wake-history parameter $C_{H_w}$ in the composite profile formulation, which captures history-dependent stretching of the wake-region mean velocity profile, as described in Part 1. Consequently, it is also challenging to formulate a general wake function that accounts simultaneously for a wide range of local APG conditions and PG-history effects, and incorporates an independent measure of $\delta$ . To demonstrate the robustness of the present composite formulation, both PG-history parameters (i.e. $C_{H_w}$ and $C_{H_i}$ ) obtained from a subset of the datasets considered here are plotted as a function of $ \textit{Re}_{\tau }$ in figure 10.

Figure 10.

Variation of (a) $C_{H_w}$ and (b) $C_{H_i}$ for all datasets considered here.

For cases with minimal PG histories and/or near equilibrium conditions, including both experimental and numerical ZPG and APG datasets, $C_{H_w} \approx C_{H_i} \approx 1$ , largely independent of $ \textit{Re}_{\tau }$ and $\beta$ . Here, the representative uncertainties in $C_{H_w}$ and $C_{H_i}$ obtained from the composite profile fitting method were estimated to be $\pm 0.05$ and $\pm 0.1$ , respectively (shaded regions in figure 10). This behaviour is consistent with previous studies, such as Chauhan et al. (Reference Chauhan, Monkewitz and Nagib2009) and Krug et al. (Reference Krug, Philip and Marusic2017), which employed constant coefficients in their wake functions, although these functions were primarily valid for ‘well-behaved’ TBLs.

When significant PG-history effects are present, deviations (beyond their respective uncertainties) appear in the wake parameter ( $C_{H_w}$ ) and/or the inner/overlap parameter ( $C_{H_i}$ ). This behaviour, for instance, is evident for the cases ZA and ZD discussed in § 4.1. Another example is the non-equilibrium APG dataset of Volino (Reference Volino2020) (the magenta diamonds), which shows very large deviations of both $C_{H_i}$ and $C_{H_w}$ from unity in figure 10. However, in some datasets, the influence of PG history is (significantly) more pronounced in $C_{H_w}$ rather than in $C_{H_i}$ ; for example, the orange downward triangles corresponding to Preskett et al. (Reference Preskett, Virgilio, Jaiswal and Ganapathisubramani2025). This observation highlights the importance of considering $C_{H_w}$ when characterising/interpreting PG-history effects, and justifies the inclusion of $C_{H_w}$ as an additional fitting parameter in the present composite formulation. Without this parameter, the wake function cannot reproduce the observed variations in the wake-region mean velocity profile across datasets with different PG histories, leading to systematic deviations in the wake-region scaling of up to $\pm 20\,\%$ . Further, these results also highlight the importance of incorporating an independent measure of $\delta$ , since the wake function can scale with both $C_{H_w}$ and $\delta$ through $\eta _{\textit{eff}} = 1.455\ C_{H_w}\ z/\delta$ . This observation is consistent with Lozier et al. (Reference Lozier, Deshpande, Zarei, Lindić, Rowin and Marusic2025), who showed that the ratio of $\delta /\delta _{99}$ (effectively a measure of wake stretching) remains nearly constant for TBLs with minimal PG history but deviates when PG-history effects are present.

Finally, it should be noted that variations in tripping conditions and/or surface roughness changes, can also influence the history parameters $C_{H_w}$ and $C_{H_i}$ , as discussed further in Appendix C.

Figure 11.

Comparison between fitted and directly measured values of (a) the friction velocity $U_{\tau }$ , (b) the wake strength $2\varPi /\kappa$ and (c) the TBL thickness $\delta$ for the datasets from Part 1.

Figure 11. Long description

Panel A: A scatter plot compares fitted and measured values of friction velocity. The x-axis represents measured friction velocity (U_t,exp) and the y-axis represents fitted friction velocity (U_t,fit). The plot includes a dashed line representing y = x. Panel B: A scatter plot compares fitted and measured values of wake strength. The x-axis represents measured wake strength (2π/κ)_exp and the y-axis represents fitted wake strength (2π/κ)_fit. The plot includes a dashed line representing y = x. Panel C: A scatter plot compares fitted and measured values of boundary layer thickness. The x-axis represents measured boundary layer thickness (δ_exp) and the y-axis represents fitted boundary layer thickness (δ_fit). The plot includes a dashed line representing y = x.

5.1. Revisiting previous datasets

Accurate measurement of $U_{\tau }$ and/or $\delta$ is often challenging in APG TBL experiments, especially for those at high $ \textit{Re}_\tau$ . Accordingly, one of the primary utilities of the proposed composite profile is to estimate these key boundary-layer characteristics from datasets in which they are not directly measured. To achieve this, the curve-fitting procedure described in § 3.4 is applied using the composite profile formulation together with (3.5). In this case, the inputs consist of the non-normalised mean velocity profile ( $U$ and $z$ ), the streamwise PG ( ${\rm d}P/{\rm d}x$ ), and the fluid properties ( $\nu$ and $\rho$ ). The PG-history parameters must also be specified; for well-behaved TBLs, these typically satisfy $C_{H_i}=C_{H_w}=1$ . The fitting procedure then provides optimised values of $U_{\tau }$ , $\varPi$ and $\delta$ , from which other boundary-layer properties may be determined.

To evaluate the accuracy of this approach, the method is applied to the datasets from Part 1, which are characterised by minimal PG history ( $C_{H_i} \approx C_{H_w} \approx 1$ ) and for which $U_{\tau }$ and $\delta$ were independently measured. The comparison is shown in figure 11. The values obtained from the composite-profile fitting procedure agree closely with the experimentally measured values and fall within the range of typical experimental uncertainties for all three quantities considered.

This demonstrates the utility of the composite profile as a tool for revisiting datasets in which direct measurements of $U_{\tau }$ were not reported. As an example, the method is applied to the dataset of Marusic & Perry (Reference Marusic and Perry1995), where the APG flow evolves from a well-developed ZPG region upstream (minimum PG history), and $U_{\tau }$ was originally estimated using the Clauser method since near-wall measurements were limited. Despite the absence of near-wall data, the composite profile matches the experimental velocity profiles well (figure 12 a). Comparing the resulting friction velocities with those obtained using the Clauser method (figure 12 b) reveals clear differences for cases with increasing $\beta$ , while the ZPG case shows good agreement. For the strongest APG case (A6 with $\beta \approx 5$ ), the Clauser method underestimates $U_{\tau }$ by approximately 10 %. This behaviour is expected, since the Clauser method is reliable primarily for canonical TBLs with negligible PGs. The optimised parameters obtained from this procedure are listed in table 2. In addition to providing improved estimates of the friction velocity, the composite profile also yields a well-resolved representation of the near-wall velocity profile, which was experimentally inaccessible for these cases (yellow-shaded region in figure 12 a).

Figure 12.

(a) Experimental mean velocity data and corresponding composite profiles from Marusic & Perry (Reference Marusic and Perry1995). The yellow-shaded region indicates the wall-normal extent which was not measured experimentally. Dashed lines represent (1.2) and dash-dotted lines represent (1.3). (b) Comparison of $U_{\tau }$ values obtained from the Clauser method and the composite profile.

5.2. Improving accuracy of the indicator function

The indicator function (5.1) is also of particular interest because a logarithmic mean velocity profile produces a plateau, with a value of $1/\kappa$ , providing a direct criterion for identifying regions with logarithmic scaling (i.e. the start and end points), estimating $\kappa$ , and selecting an appropriate interval over which $B$ may be evaluated. Overall, the indicator function enables a clear examination of the influences of $ \textit{Re}_{\tau }$ , APG strength and PG history on the classical log law. In practice, however, evaluating the indicator function from experimental data, particularly single-point hot-wire measurements, is challenging because it requires accurate wall-normal derivatives of the measured mean velocity profile.

Experimental noise, small measurement errors and sparse wall-normal resolution make it difficult to draw firm conclusions about the existence of a logarithmically scaled region or to identify its start and end points. Similarly, numerical datasets require very high wall-normal resolution to accurately resolve the indicator function, making them computationally expensive at high $ \textit{Re}_{\tau }$ (Nagib et al. Reference Nagib, Vinuesa and Hoyas2024).

In contrast, the composite profile provides an analytical representation of the mean velocity profile that can be differentiated analytically, yielding a smooth expression for the indicator function. For example, in the overlap region, the indicator function can be written as

(5.1) \begin{align} z^+ \frac {{\rm d}U^+}{{\rm d}z^+} = \frac {\gamma z^+}{\kappa } \boldsymbol{\cdot }\frac {0.6 \, \xi ^5 \, Re_\tau (1 + \eta _{\textit{eff}}^6) - \eta _{\textit{eff}}^5 \, z^{+}_{c} (1 + \xi ^6)}{z^{+}_{c} \, Re_\tau \, (1 + \xi ^6)(1 + \eta _{\textit{eff}}^6)};\quad \xi = \frac {0.6 z^+}{z^{+}_{c}};\quad \gamma = \sqrt {1 + p_x^+ z^{+}_{c}}. \end{align}

In the high- $ \textit{Re}_{\tau }$ limit, this expression reduces to $z^+ {{\rm d}U^+}/{{\rm d}z^+} = 1/\kappa$ , consistent with the classical log-law behaviour expected in the overlap region of high- $ \textit{Re}_{\tau }$ TBLs.

Figure 13.

Indicator functions computed from composite profiles fitted to (a) the low- $ \textit{Re}_{\tau }$ ZPG LES dataset of Eitel-Amor et al. (Reference Eitel-Amor, Örlü and Schlatter2014), (b) the high- $ \textit{Re}_{\tau }$ ZPG experimental datasets of Marusic et al. (Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015) and the high- $ \textit{Re}_{\tau }$ moderate-APG experimental datasets from Part 1 with (c) minimal PG histories or (d) imposed PG-history effects. Horizontal dashed lines represent $1/\kappa$ .

Figure 13. Long description

Panel A: One line graph shows indicator functions computed from composite profiles fitted to the low-zero pressure gradient (ZPG) LES dataset. The x-axis is labeled z+ on a logarithmic scale, and the y-axis is labeled z+ dU+ / dz+. The graph includes data points and a composite fit line. Panel B: One line graph displays indicator functions for high-ZPG experimental datasets and high-moderate adverse pressure gradient (APG) experimental datasets. The x-axis is labeled z+ on a logarithmic scale, and the y-axis is labeled z+ dU+ / dz+. Multiple lines represent different datasets. Panel C: One line graph shows indicator functions for datasets with minimal pressure gradient (PG) histories. The x-axis is labeled z+ on a logarithmic scale, and the y-axis is labeled z+ dU+ / dz+. Different lines represent varying conditions. Panel D: One line graph depicts indicator functions for datasets with imposed PG-history effects. The x-axis is labeled z+ on a logarithmic scale, and the y-axis is labeled z+ dU+ / dz+. Different lines represent varying conditions.

The indicator function also provides an additional criterion for assessing the quality of the composite-profile fit. Figure 13(a) compares the indicator function computed directly from the well-resolved ZPG LES of Eitel-Amor et al. (Reference Eitel-Amor, Örlü and Schlatter2014) with that obtained from the corresponding composite profile, demonstrating excellent agreement.

Further insight can be obtained by considering indicator functions fitted to high- $ \textit{Re}_{\tau }$ experimental datasets. For example, the plateau associated with the overlap region becomes increasingly pronounced as $ \textit{Re}_{\tau }$ increases, reflecting the growing separation between the inner and wake regions and therefore a larger overlap-region extent. This behaviour is shown in figure 13(b) for the high- $ \textit{Re}_{\tau }$ ZPG experimental dataset of Marusic et al. (Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015). For the highest- $ \textit{Re}_{\tau }$ case (M8), the pronounced plateau corresponding to $1/\kappa$ indicates a clear overlap region following classical logarithmic scaling. In contrast, at lower $ \textit{Re}_{\tau }$ (e.g. case M1 in figure 13 b and the cases in figure 13 a), a well-defined plateau is not observed.

The effect of increasing APG on the indicator function at approximately $ \textit{Re}_{\tau }=10\,000$ is illustrated in figure 13(c) using experimental datasets from Part 1 with $\beta _{\textit{eff}}=$ 0, 0.66 and 1.44. For moderate APGs at high $ \textit{Re}_{\tau }$ , the indicator function still exhibits a clear plateau, indicating that the overlap region still follows logarithmic scaling. However, as $\beta _{\textit{eff}}$ increases, the wake strengthens and the plateau terminates earlier, reducing the extent of the logarithmically scaled region, which is consistent with the findings of Knopp (Reference Knopp2022). Increasing $\beta _{\textit{eff}}$ also reduces the near-wall velocity gradient, decreasing the first peak of the indicator function, while simultaneously increasing the buffer-region overshoot, which deepens the local minimum (figure 13 c).

Figure 13(d) further illustrates the influence of PG history on the indicator function and the extent of the overlap region. The case corresponding to $C_{H_i}=0.57$ exhibits a slightly longer overlap region than the case with $C_{H_i}=1$ , despite having matched $ \textit{Re}_{\tau }$ and $\beta$ .

The composite profile and the associated indicator function can also be used to estimate the number of decades of $z^{+}$ over which logarithmic scaling occurs in the overlap region by measuring the plateau extent within a given tolerance (e.g. $1/\kappa \pm 1\,\%$ ). Predicted variations in the number of decades of the log law are shown as a function of $\beta _{\textit{eff}}$ and $ \textit{Re}_{\tau }$ in figure 14(a). In this analysis, the following approximations of Perry et al. (Reference Perry, Marusic and Jones2002) were employed: $\varPi = 0.68(\beta + 0.5)^{0.75}$ and $\delta / \delta ^+ = 7.912 / (1 + 0.1861 \beta )$ , which are considered valid for $\beta \lt 6$ . The results demonstrate competing, but expected trends: the logarithmic extent decreases with increasing $\beta$ at fixed $ \textit{Re}_{\tau }$ , but increases with $ \textit{Re}_{\tau }$ at fixed $\beta$ . The red dashed line denotes a representative minimum of 1/3 of a decade in $z^+$ , roughly corresponding to the predicted logarithmic extent of a ZPG TBL at $ \textit{Re}_{\tau }=3000$ . Such predictions provide useful guidelines for selecting experimental parameter ranges for future investigations of logarithmic scaling in APG TBLs.

Figure 14.

Variation in (a) the decades of $z^{+}$ which follow the classical log law and (b) the value of $\kappa$ as a function of $ \textit{Re}_{\tau }$ and $\beta _{\textit{eff}}$ . The red dashed line indicates 1/3 of a decade.

The composite profile also predicts the slope of the mean velocity profile in the overlap region, i.e. the von Kármán coefficient. According to (3.8), $\kappa$ varies as $\kappa /0.39 = 1/\sqrt {1 + p_x^{+} z^{+}_{c}}$ . Predicted variations of $\kappa$ with $\beta _{\textit{eff}}$ are shown as a function of $ \textit{Re}_{\tau }$ in figure 14(b). At low $ \textit{Re}_{\tau }$ , $\kappa$ varies significantly with $\beta _{\textit{eff}}$ , whereas for $ \textit{Re}_{\tau } \gtrsim 10^4$ the curves approach $\kappa =0.39$ (within typical experimental uncertainty) regardless of $\beta _{\textit{eff}}$ . This indicates that the $p_x^+$ terms in (3.7) and (3.8) become negligible at sufficiently high $ \textit{Re}_{\tau }$ , consistent with (3.10). It also suggests that the hierarchy of attached eddies in the overlap region remains largely unchanged at high $ \textit{Re}_{\tau }$ , even under moderate APGs or PG-history effects, consistent with Woodcock & Marusic (Reference Woodcock and Marusic2015).

Figure 15 shows the estimated wall-normal extents of the log law, in decades of $z^+$ , for the datasets in table 1, based on indicator functions derived from the fitted composite profiles. The red dashed line again denotes the approximate threshold of 1/3 of a decade in $z^+$ which is required to clearly identify logarithmic scaling. Cases below this threshold are highlighted in yellow, datasets with previously noted PG-history effects in the overlap region are highlighted in green, and cases satisfying both conditions are highlighted in red. This comparison highlights the uniqueness of the dataset presented in Part 1, which simultaneously maintains a significant overlap region (high $ \textit{Re}_{\tau }$ and moderate $\beta$ ) and controlled PG history effects, enabling a clearer investigation of the logarithmic scaling in APG TBLs.

Figure 15.

Decades of log law for the datasets listed in table 1. The red dashed line indicates the threshold of 1/3 of a decade. Cases below this threshold are highlighted in yellow, datasets with noted PG-history effects in the overlap region are highlighted in green and cases with both conditions are highlighted in red.

A further utility of the composite profile is that it provides an analytical expression from which inflection points ( ${\rm d}^{2}U/{\rm d}z^{2}=0$ ) can be identified. This is useful for testing the framework of Schatzman & Thomas (Reference Schatzman and Thomas2017), which links APG-induced energisation of wake-region motions to a shear instability associated with an inflectional mean velocity profile. The availability of an analytical composite profile, therefore, facilitates identification of these inflection points and further assessment of this mechanism for APG TBL dynamics.