3.2.1. $\varPsi$-momentum budget in the SCS

Assessing the momentum equation budget in the SCS allows us to characterise the dominant forces acting on a fluid packet following the streamlines. In figure 6(a), we show the $\varPsi$-momentum budget in the near-wall region at $x/L = -0.35$, the location of the maximum APG. We observe that the momentum transfer due to the turbulent and viscous fluxes (defined in the second line of (3.4)) is the most dominant. The small difference between the two momentum fluxes is balanced by the pressure gradient close to the wall. The advection term is much smaller than the pressure gradient near the wall. The $\varPsi$-momentum budget far from the wall is shown in figure 6(b). We observe that momentum fluxes due to viscous effects are negligible in this region. The momentum flux due to the Reynolds shear stress term is almost balanced by the total pressure gradient term (advection + pressure gradient), and the slight differences which are not visible on this scale are attributed to the momentum flux due to $\overline {u'_\psi u'_\psi }$. A net negative value of advection momentum transfer is observed, indicating deceleration of the streamlines in the mild APG region. The budget behaviour in both the inner and outer regions of the boundary layer is similar at other locations in the APG region.

Figure 6. The $\varPsi$-momentum budget stations at $x/L = -0.35$ in APG region. The $x$-axis is normalised in (a) inner units and (b) outer units, whereas the $y$-axis is normalised in inner units. Line types: blue, $\bar {u}_{\psi } \,{\rm d} \bar {u}_{\psi }/{\rm d}\psi$; black, ${\rm d}\bar {p}/{\rm d}\psi$; green, ${\rm d}\overline {u'_{\psi } u'_{\psi }}/{\rm d}\psi$; orange, ${\rm d}\overline {u'_{\psi } u'_{\phi }}/{\rm d}\phi$; red, $- (\overline {u'_{\psi }u'_{\psi }} - \overline {u'_{\phi }u'_{\phi }})/L_a$; cyan, $-2\overline {u'_{\psi } u'_{\phi }}/{R_\psi }$; magenta, negative of viscous terms; yellow, budget balance. Note that the difference in $y$-axis limits for (a,b) highlight the difference in magnitude of these terms in the inner and outer regions of the flow.

In figure 7, we show the $\varPsi$-momentum budget in the near-wall region for three different $x/L$ locations corresponding to the start of the FPG region, close to the location of the maximum FPG and near the end of the FPG region. We observe that the momentum fluxes due to viscous forces, turbulence and pressure gradient balance very close to the wall. The advection term grows significantly after $n^+ > 10$ and this increase appears to have a log-linear dependence on $n^+$. The behaviour in the outer region of the boundary layer is shown in figure 8. The outer region of the boundary layer is dominated by momentum flux due to the pressure gradient, advection and turbulence terms. In this region, there is a positive value of the advection momentum flux, which indicates a net acceleration of the streamlines in the FPG region. Like the APG region, the pressure gradient, advection and Reynolds shear stress terms are mostly balanced with the next largest term (not quite visible on this scale) being the $\overline {u'_\psi u'_\psi }$ term. From these results, we observe that the dominant terms of the momentum equation budget in both the APG and FPG regions are: viscous flux, turbulence shear flux, pressure gradient and advection flux.

Figure 7. The $\varPsi$-momentum budget at several $x/L$ stations in the FPG region: (a) $x/L = -0.29$; (b) $x/L = -0.15$; (c) $x/L = -0.05$. Both axes are normalised by inner units. Line types: blue, $\bar {u}_{\psi } \,{\rm d} \bar {u}_{\psi }/{\rm d}\psi$; black, ${\rm d}\bar {p}/{\rm d}\psi$; green, ${\rm d}\overline {u'_{\psi } u'_{\psi }}/{\rm d}\psi$; orange, ${\rm d}\overline {u'_{\psi } u'_{\phi }}/{\rm d}\phi$; red, $- (\overline {u'_{\psi } u'_{\psi }} - \overline { u'_{\phi } u'_{\phi }})/L_a$; cyan, $-2\overline {u'_{\psi } u'_{\phi }}/{R_\psi }$; magenta, negative of viscous terms; yellow, budget balance.

Figure 8. The $\varPsi$-momentum budget at several stations in the FPG region: (a) $x/L = -0.29$; (b) $x/L =-0.15$; (c) $x/L = -0.05$. The $x$-axis is normalised in outer units, whereas the $y$-axis is normalised in inner units. Line types: blue, $\bar {u}_{\psi } \,{\rm d} \bar {u}_{\psi }/{\rm d}\psi$; black, ${\rm d}\bar {p}/{\rm d}\psi$; green, ${\rm d}\overline {u'_{\psi } u'_{\psi }}/{\rm d}\psi$; orange, ${\rm d}\overline {u'_{\psi } u'_{\phi }}/{\rm d}\phi$; red, $- (\overline {u'_{\psi } u'_{\psi }} - \overline {u'_{\phi } u'_{\phi }})/L_a$; cyan, $-2\overline {u'_{\psi } u'_{\phi }}/{R_\psi }$; magenta, negative of viscous terms; yellow, budget balance.

3.2.2. $\varPhi$-momentum budget in the SCS

The $\varPhi$-momentum budget enables the assessment of forces normal to the streamlines acting on a fluid packet moving along the streamlines. In figure 9(a), we show the $\varPhi$-momentum budget in the near-wall region for the APG region of the flow. At $x/L = -0.35$, we observe fluxes due to $\overline {u'_\phi u'_\phi }$ term and pressure gradient balance each other until $n^+ \approx 10$. The effects of curvature are observed from $n^+ \geq 10$ onwards in figure 9(a,b). We observe that the difference between the momentum flux due to the pressure gradient and $\overline {u'_\phi u'_\phi }$ term is non-zero. The net difference is equal to the advection normal to the streamlines.

Figure 9. The $\varPhi$-momentum budget at $x/L = -0.35$ in the APG region. The $x$-axis is normalised in (a) inner units and (b) outer units, whereas the $y$-axis is normalised in inner units. Line types: black, $\bar {u}_{\psi }^2/{R_\psi }$; blue, ${\rm d}\bar {p}/{\rm d}\phi$; green, ${\rm d} \overline {u'_{\psi } u'_{\phi }}/{\rm d} \psi$; orange, ${\rm d}\overline {u'_{\phi } u'_{\phi }}/{\rm d} \phi$; red, $-2\overline {u'_{\psi } u'_{\phi }}/L_a$; cyan, $-(\overline {u'_{\psi } u'_{\psi }} - \overline {u'_{\phi } u'_{\phi })}/{R_\psi }$; magenta, negative of viscous terms; yellow, budget balance.

In figures 10 and 11, we show the $\varPhi$-momentum budget at three different $x/L$ locations in the FPG region of the flow. We observe that the FPG region shows similar behaviour as the APG region until $x/L = -0.14$. For $-0.14 \leq x/L \leq 0$, the difference between momentum fluxes due to $\overline {u'_\phi u'_\phi }$ and pressure gradient results in a net acceleration of streamlines. This behaviour reduces the momentum in the outer region of the boundary layer and brings it into the inner region of the boundary layer leading to the decay of turbulence in the outer region of the boundary layer. From the $\varPhi$-momentum budget, we observe that the pressure gradient normal to the streamlines and the $\overline {u'_\phi u'_\phi }$ term are the dominant components for the momentum budget that governs the acceleration of streamlines towards or away from the wall.

Figure 10. The $\varPhi$-momentum budget at several $x/L$ stations in FPG region: (a) $x/L = -0.29$; (b) $x/L = -0.15$; (c) $x/L = -0.05$. Both axes are normalised by inner units. Line types: black, $\bar {u}_{\psi }^2/{R_\psi }$; blue, ${\rm d}\bar {p}/{\rm d}\phi$; green, ${\rm d} \overline {u'_{\psi } u'_{\phi }}/{\rm d} \psi$; orange, ${\rm d}\overline {u'_{\phi } u'_{\phi }}/{\rm d} \phi$; red, $-2\overline {u'_{\psi } u'_{\phi }}/L_a$; cyan, $-(\overline {u'_{\psi } u'_{\psi }} - \overline { u'_{\phi } u'_{\phi }})/{R_\psi }$; magenta, negative of viscous terms; yellow, budget balance.

Figure 11. The $\varPhi$-momentum budget at several $x/L$ stations in FPG region: (a) $x/L = -0.29$; (b) $x/L = -0.15$; (c) $x/L = -0.05$. The $y$-axis is normalised by inner units whereas the $x$-axis is normalised by outer units. Line types: black, $\bar {u}_{\psi }^2/{R_\psi }$; blue, ${\rm d}\bar {p}/{\rm d}\phi$; lime, ${\rm d} \overline {u'_{\psi } u'_{\phi }}/{\rm d} \psi$; orange, ${\rm d}\overline {u'_{\phi } u'_{\phi }}/{\rm d} \phi$; red, $-2\overline {u'_{\psi }u'_{\phi }}/L_a$; cyan, $-(\overline {u'_{\psi } u'_{\psi }} - \overline {u'_{\phi } u'_{\phi }})/{R_\psi }$; magenta, negative of viscous terms; yellow, budget balance.

3.3.1. Inner region

We draw insights from the $\varPsi$-momentum budget to obtain a non-local approximation for the Reynolds shear stress. From the $\varPsi$-momentum budgets shown in § 3.2.1, we observe that only a few terms contribute to the momentum fluxes: viscous stress, Reynolds shear stress and pressure gradient, thereby leading to a net acceleration or inertia along the streamlines. Therefore, the $\varPsi$-momentum budget (3.4) reduces to

(3.7)\begin{equation} \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} \approx-\frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} - \frac{\partial \overline{u'_{\psi} u'_{\phi}}}{\partial \phi} + \nu \frac{\partial^2 \bar{u}_{\psi}}{\partial \phi^2}. \end{equation}

Rearranging the equation and integrating the equation along $\phi$ holding $\psi$ fixed results in

(3.8)\begin{equation} -\overline{u'_{\psi} u'_{\phi}} + \nu \frac{\partial \bar{u}_{\psi}}{\partial \phi} \approx \int_{0}^{\phi} \left( \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} + \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} \right) \,{\rm d} \phi' + \frac{\tau_w}{\rho}. \end{equation}

Sufficiently far away from the wall, the viscous term is negligible, and the equation simplifies to

(3.9)\begin{equation} -\overline{u'_{\psi} u'_{\phi}} (\phi) \approx \int_{0}^{\phi} \left( \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} + \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} \right) \,{\rm d} \phi' + \frac{\tau_w}{\rho}. \end{equation}

Integrating along $\phi$ is quite challenging and far more complicated than $n$ lines.

To understand the potential error incurred by integrating along $n$ lines, in figure 12, we plot the angle between the streamlines and the wall, $\varDelta _{\theta }$. In the mild APG region, the streamlines within the boundary layer deviate away from the wall. The first profile at $x/L=-0.5$ shows a boundary layer edge streamline angle of $0.007$ rad. As the APG increases $\varDelta _\theta$ grows but then, as the pressure gradient relaxes to zero just past the APG–FPG switch, the streamlines become nearly parallel to the surface in the boundary layer. In the FPG region, the streamlines deflect toward the wall. We observe that in both the APG and FPG regions, the streamlines deviate from the surface by less than $0.005$ rad ($0.3^{\circ }$) below $y^+=100$ and less than $0.05$ rad ($3^{\circ }$) throughout. Thus, in the integral analysis that follows, the SCS integrands will be sampled and integrated along $n$ lines to obtain approximations. The small difference in the $n$ and $\phi$ direction means the SCS and the WCS are nearly aligned which will also allow us to apply the analysis in both the SCS and the WCS which is important since the WCS is where simulations and modelling are more practical.

Figure 12. The angle of deflection (in radians) of streamlines from the bump surface in (a) APG region and (b) FPG region. Here $\star$ indicates the edge of the boundary layer. Line types for APG region at $x/L$: violet, $-0.5$; cerulean, $-0.45$; green, $-0.39$; orange, $-0.35$; mahogany, $-0.3$. Line types for FPG region at $x/L$: violet, $-0.29$; royal blue, $-0.25$; cyan, $-0.2$; sea green, $-0.15$; green, $-0.1$; yellow orange, $-0.079$; red orange, $-0.05$; mahogany, $-0.006$.

In the inner region of the boundary layer the close alignment of the SCS with WCS allows (3.9) to be approximated by

(3.10)\begin{equation} -\overline{u'_{\psi} u'_{\phi}} (n) \approx \int_{0}^{n} \left( \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} + \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} \right) \,{\rm d} n' + \frac{\tau_w}{\rho}, \end{equation}

where in this approximation the integral is taken along a curve where $s$ is fixed (a straight line ray normal from the wall at a given point on the surface corresponding to $s$).

The first term on the right-hand side of the above equation accounts for the effect of streamline acceleration, and the second term on the right-hand side accounts for pressure gradient effects. We can define an approximation,

(3.11)\begin{equation} u^2_{*,i} (n) \stackrel{\text{def}}= \int_{0}^{n} \left( \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} + \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} \right) \,{\rm d} n' + \frac{\tau_w}{\rho}, \end{equation}

which could normalise the Reynolds shear stress in the SCS as

(3.12)\begin{equation} \frac{-\overline{u'_{\psi} u'_{\phi}}}{u_{*,i}^2} \approx 1. \end{equation}

The above equation indicates that the normalised shear stress in the SCS is independent of Reynolds number, pressure gradient and acceleration effects. By assuming that the misalignment of SCS and WCS is sufficiently small, we can extend this analysis to the Reynolds shear stress in the WCS.

(3.13)\begin{equation} \frac{-\overline{u'_{s} u'_{n}}}{u_{*,i}^2} \approx 1. \end{equation}

If the acceleration and pressure gradient terms are negligible, $u^2_{*,i} \approx u^2_{\tau }$, indicating that the Reynolds shear stress in the WCS normalises independently of the Reynolds number and follows the classical Reynolds shear stress normalisation used for zero pressure gradient turbulent boundary layers. When the acceleration term is negligible but pressure gradient effects are still significant in the inner region of the boundary layer, the approximation reduces to

(3.14)\begin{equation} u^2_{*,i} (n) \approx \int_{0}^{n} \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} \,{\rm d} n' + \frac{\tau_w}{\rho} . \end{equation}

Assuming that the pressure gradient is constant in the wall-normal direction, we attain the following:

(3.15)\begin{equation} u^2_{*,i} (n) \approx \frac{n}{\rho} \frac{\partial \bar{p}}{\partial \psi} + u_{\tau}^2 , \end{equation}

which is similar to the hybrid velocity normalisation presented in Sekimoto et al. (Reference Sekimoto, Kitsios, Atkinson and Soria2019) and Romero et al. (Reference Romero, Zimmerman, Philip and Klewicki2022). This observation indicates that, subject to appropriate assumptions of negligible advection or pressure gradient effects, the derived approximation reduces to other commonly used normalisations for the Reynolds shear stress (e.g. $u_\tau$ when both absent).

We compare the Reynolds shear stress profiles in the APG region with different normalisations in figure 13. We observe that the friction velocity normalisation does not collapse at all $x/L$ stations. At $x/L = -0.5$ and $-0.45$, the pressure gradient is slight, and the Reynolds shear stress profiles collapse and agree well with theoretical estimates of $-\overline {u'_{s} u'_{n}}/u_{\tau }^2 \approx 1$ sufficiently far away from the wall in the inner region of the boundary layer. As the APG increases for later downstream locations, and as $u^2_{\tau }$ does not fully account for it, the results deviate from theoretical estimates and the curves do not collapse. On the other hand, the use of $u^2_{*,i}$ for normalisation accounts for the effect of pressure gradients and the resulting acceleration, thereby collapsing the velocity profiles to the theoretical values $-\overline {u'_{s} u'_{n}}/u_{*,i}^2 \approx 1$ in the inner region of the boundary layer. This analysis indicates that the normalised Reynolds shear stress in the inner region of the boundary layer depends on streamwise acceleration, streamwise pressure gradient and wall shear stress. Unfortunately, $u^2_{*,i}$ decreases faster than the rate for $-\overline {u'_{s} u'_{n}}$ in the outer region of the boundary layer, forming a singularity that makes the normalisation ineffective far away from the wall. Furthermore, we also tested the approximation in (3.15) and observed that it gives a good collapse of stress profiles in the inner region in the early parts of the mild APG region. However, it fails to collapse the stress profile at $x/L \approx -0.3$. This observation indicates that accounting for all significant terms is essential to the success of the normalisation.

Figure 13. Reynolds shear stress in WCS in the APG region using (a) $u^2_{\tau }$ and (b) $u^2_{*,i}$. Vertical lines indicate that the quantity asymptotically goes to $\pm \infty$. Line types for APG region at $x/L$: violet, $-0.5$; cerulean, $-0.45$; green, $-0.39$; orange, $-0.35$; mahogany, $-0.3$.

When extracting physical insight from simulations, it is important to be certain that the results are not significantly influenced by boundary conditions. In particular, the collapse of the stress profiles when using $u^2_{*,i}$ throughout the APG region shown in figure 13, including at $x/L=-0.5$, greatly bolsters confidence in our assertion that the STG boundary condition is not introducing artifacts into this simulation since it seems highly unlikely for those artifacts to also lead to this collapse. Rather, we maintain that the flow has matured quickly into correct turbulence under the weak APG at the inflow which then allows the simulation to accurately reflect the physics of the smoothly increasing and then smoothly decreasing APG. Further, as the FPG region is even further downstream, we are confident it is also free of STG inflow artifacts.

We compare the Reynolds shear stress profiles in the WCS in the FPG region with different normalisations in figure 14. We observe that for all $x/L$ stations in the FPG, the friction velocity normalisation does not collapse the shear stress profiles in the inner region of the boundary layer. On the other hand, the new Reynolds shear stress approximation leads to a much better collapse of the Reynolds shear stress profiles in the inner region. Note that the flow is experiencing pressure gradients leading to acceleration, the presence of an internal layer, and concave and convex curvature effects in this region. Even with such complex flow physics, the Reynolds shear stress profiles collapse well. Even though the collapse is good, we expected it to be better in the buffer region $n^+ < 30$. While deriving the approximation, we assumed that the viscous effects are negligible sufficiently far from the wall. This assumption may not hold in the FPG region where the strong pressure gradient effects lead to the laminarescent stage, where the boundary layer adjusts towards starting the relaminarisation process. Therefore, we would expect viscous effects to be important even further away from the wall, and these effects should also be incorporated into the approximation. From (3.8) and assuming the misalignment between the SCS and WCS is sufficiently small, we attain

(3.16)\begin{equation} -\overline{u'_{\psi} u'_{\phi}} (n) \approx \int_{0}^{n} \left( \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} + \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} \right) \,{\rm d} n' + \frac{\tau_w}{\rho} - \nu \frac{\partial \bar{u}_{\psi}}{\partial \phi}, \end{equation}

leading to a new approximation, $u^2_{**,i}$,

(3.17)\begin{equation} \frac{ -\overline{u'_{s} u'_{n}}}{u^2_{**,i}} \approx 1, \quad \text{where} \ u^2_{**,i} (n) \stackrel{\text{def}}= \int_{0}^{n} \left( \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} + \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} \right) \,{\rm d} n' + \frac{\tau_w}{\rho} - \nu \frac{\partial \bar{u}_{\psi}}{\partial \phi} . \end{equation}

Figure 14. Reynolds shear stress in WCS in the FPG region normalised by (a) $u^2_{\tau }$ and (b) $u^2_{*,i}$. Vertical lines indicate that the quantity asymptotically goes to $\pm \infty$. Line types for FPG region at $x/L$: violet, $-0.29$; royal blue, $-0.25$; cyan, $-0.2$; sea green, $-0.15$; green, $-0.1$; yellow orange, $-0.079$; red orange, $-0.05$; mahogany, $-0.006$.

We show the Reynolds shear stress normalised using $u^2_{**,i}$ in figure 15. The $u^2_{**,i}$ normalisation does not significantly affect the already good collapse of stress profiles in the APG region compared with $u^2_{*,i}$. However, it substantially improves the collapse of Reynolds shear stress profiles in the FPG region in both the buffer and log regions ($10 \leq n^+ \leq 100$) of the boundary layer compared with $u^2_{*,i}$ normalisation. These results indicate that for flows where viscous effects become substantial, the normalisation of the Reynolds shear stress must be adjusted to account for the thick near-wall viscous region. Furthermore, with a better collapse of Reynolds shear stress profiles with this proposed approximation, it appears that the upstream history effects and development of internal layers are also accounted for.

Figure 15. Reynolds shear stress in WCS in the (a) APG region and (b) FPG region normalised by $u^2_{**,i}$. Vertical lines indicate that the quantity asymptotically goes to $\pm \infty$. Line types for APG region at $x/L$: violet, $-0.5$; cerulean, $-0.45$; green, $-0.39$; orange, $-0.35$; mahogany, $-0.3$. Line types for FPG region at $x/L$: violet, $-0.29$; royal blue, $-0.25$; cyan, $-0.2$; sea green, $-0.15$; green, $-0.1$; yellow orange, $-0.079$; red orange, $-0.05$; mahogany, $-0.006$.

Even though these new approximations show good collapse of the Reynolds shear stress, they should not be expected to and do not show good collapse for Reynolds normal stresses. Therefore, we derive the approximation for $\overline {u'_n u'_n}$ stress by a similar procedure. In this case, we consider the $\varPhi$-momentum budget, (3.5) and remove all the equation components that had negligible influence as observed in § 3.2.1,

(3.18)\begin{equation} \frac{\bar{u}_{\psi}^2}{{R_\psi}} \approx-\frac{1}{\rho} \frac{\partial \bar{p}}{\partial \phi} - \frac{\partial \overline{u'_{\phi} u'_{\phi}}}{\partial \phi}. \end{equation}

Rearranging this equation and integrating it along $\phi$, we obtain

(3.19)\begin{equation} \overline{u'_{\phi} u'_{\phi}} (\phi) \approx- \int_{0}^{\phi} \left( \frac{\bar{u}_{\psi}^2}{{R_\psi}} + \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \phi} \right) \,{\rm d} \phi' \approx- \int_{0}^{\phi} \frac{\bar{u}_{\psi}^2}{{R_\psi}} \,{\rm d} \phi' + \frac{\bar{p}_w - p}{\rho}. \end{equation}

To simplify the calculation, we again assume near-alignment of the SCS and WCS and using ${R_w}$ in place of ${R_\psi }$, which are both valid as shown in figure 12

(3.20)\begin{equation} \overline{u'_{\phi} u'_{\phi}} (n) \approx- \int_{0}^{n} \frac{\bar{u}_{\psi}^2}{{R_w}} \,{\rm d} n' + \frac{\bar{p}_w - p}{\rho}. \end{equation}

Therefore, we can define a new approximation for $\overline {u'_\phi u'_\phi }$

(3.21)\begin{equation} u^2_{\gamma,i} (n) \stackrel{\text{def}}=- \int_{0}^{n} \frac{\bar{u}_{\psi}^2}{{R_w}} \,{\rm d} n' + \frac{\bar{p}_w - p}{\rho}, \end{equation}

which satisfies the following equation in the near-wall region:

(3.22)\begin{equation} \frac{ \overline{u'_{\phi} u'_{\phi}}}{u^2_{\gamma,i}} \approx \frac{ \overline{u'_{n}u'_{n}}}{u^2_{\gamma,i}} \approx 1, \end{equation}

which gives a normalisation for $\overline {u'_n u'_n}$. The expression for $u^2_{\gamma,i}$ indicates that $\overline {u'_n u'_n}$ dynamically depends on the centrifugal force and local pressure. In the absence of the streamline curvature effects, the centrifugal force term is expected to be negligible, resulting in a dependence on the difference between the local pressure and the pressure at the wall.

The normalised $\overline {u'_n u'_n}$ stress for the APG and FPG regions are shown in figures 16 and 17, respectively. We observe that the $u^2_{\tau }$ normalisation does not lead to a collapse of stress profiles in both the APG and FPG regions. Using $u^2_{\gamma,i}$ for normalising the stress profile, we get a much better collapse and agreement with the theoretical estimates in the inner boundary layer region for both the APG and FPG regions of the flow. These results indicate that including the local pressure gradient and centrifugal force terms is important to account for the dynamics of these quantities in the normalisation. Our analysis leads to the derivation of approximations for $\overline {u'_{s} u'_{n}}$ and $\overline { u'_{n} u'_{n}}$. However, these derived approximations are specific to $\overline { u'_{s} u'_{n}}$ and $\overline {u'_{n} u'_{n}}$, and do not extend well to other Reynolds stress components: $\overline {u'_{s} u'_{s}}$ and $\overline {u'_z u'_z}$.

Figure 16. Plots of $\overline {u'_n u'_n}$ in APG region normalised by (a) $u^2_{\tau }$ and (b) $u^2_{\gamma,i}$. Vertical lines indicate that the quantity asymptotically goes to $\pm \infty$. Line types for APG region at $x/L$: violet, $-0.5$; cerulean, $-0.45$; green, $-0.39$; orange, $-0.35$; mahogany, $-0.3$.

Figure 17. Plots of $\overline {u'_n u'_n}$ in FPG region normalised by (a) $u^2_{\tau }$ and (b) $u^2_{\gamma,i}$. Vertical lines indicate that the quantity asymptotically goes to $\pm \infty$. Line types for FPG region at $x/L$: violet, $-0.29$; royal blue, $-0.25$; cyan, $-0.2$; sea green, $-0.15$; green, $-0.1$; yellow orange, $-0.079$; red orange, $-0.05$; mahogany, $-0.006$.

3.3.2. Outer region

Before deriving the approximations for the Reynolds stresses in the outer region using integral analysis, we first define some commonly used normalisations. The normalisation of the Reynolds stresses in the outer region is often represented using the defect law for the Reynolds stress tensor (Clauser Reference Clauser1956; Castillo & George Reference Castillo and George2001):

(3.23)\begin{equation} \frac{\overline{u'_i u'_j}}{R_{ij,o}} = f_{ij} \left(\frac{n}{\delta}\right), \end{equation}

where $R_{ij,o}$ is the normalisation for $ij$th component of the Reynolds stress tensor in the outer region of the boundary layer. If the normalisation for Reynolds stresses is based on the same velocity normalisation in both the inner and outer region of the boundary layer (Clauser Reference Clauser1954), then the friction velocity is used for normalising the stresses in the outer region and $R_{ij,o} = u_{\tau }^2$. Similarly, two velocity normalisation approaches can be taken for the Reynolds stresses, where $R_{ij,i} = u_{\tau }^2 \ne R_{ij,o}$. Using similarity analysis, Castillo & George (Reference Castillo and George2001) derived a normalisation for the Reynolds stresses in the outer region of the turbulent boundary layer in the WCS as

(3.24a–c)\begin{equation} R_{sn,o} = \bar{u}_s^2 \frac{\partial \delta}{\partial s}, \quad R_{ss,o} = \bar{u}_s^2, \quad R_{nn,o} = \bar{u}_s^2. \end{equation}

Brzek et al. (Reference Brzek, Turan, Anderson and Castillo2005) showed that this normalisation does not collapse the stress profiles in the presence of pressure gradients. Another common outer region velocity normalisation is Zagarola–Smits normalisation,

(3.25)\begin{equation} U_{ZS} = \bar{u}_{s,e}\frac{\delta^*}{\delta}, \end{equation}

where $\bar {u}_{s,e}$ is the velocity at the edge of boundary layer and $\delta ^*$ is the displacement thickness. Brzek et al. (Reference Brzek, Turan, Anderson and Castillo2005) showed that Zagarola–Smits normalisation ($R_{ij,o} = U_{ZS}^2$) works better for normalising the Reynolds stresses in the presence of pressure gradients. For the bump flow, two components of the normalised Reynolds stress profiles in the APG region of the flow are shown in figure 18. The two Reynolds stress profiles collapse well when normalised by $U_{ZS}$ in the mild APG of the boundary layer. As shown in figure 19, this behaviour is not observed for the strong FPG region of the flow where the stress profiles do not collapse for either of the two stress components. In the strong FPG region, the outer region peak of stress components increases until $x/L = -0.1$, close to the maximum $\beta$ location. Downstream of this location, there is a decrease in the outer region peak. A case could be made for the inclusion of upstream effects. However, several studies (Wosnik & George Reference Wosnik and George2000; George Reference George2006) have shown that using the $ZS$ normalisation already incorporates these effects. These results indicate that a local normalisation may not work well for the strong FPG regions of the flow, therefore integral-analysis-based approximations for Reynolds stresses need to be considered.

Figure 18. Reynolds stress tensor components: (a) $\overline {u'_n u'_n}$ and (b) $-\overline {u'_s u'_n}$ using $U^2_{ZS}$ in the mild APG region of the boundary layer. Line types for APG region at $x/L$: violet, $-0.5$; cerulean, $-0.45$; green, $-0.39$; orange, $-0.35$; mahogany, $-0.3$.

Figure 19. Reynolds stress tensor components: (a) $\overline {u'_n u'_n}$ and (b) $-\overline {u'_s u'_n}$ normalised using $U^2_{ZS}$ in the strong FPG region of the boundary layer. Line types for FPG region at $x/L$: violet, $-0.29$; royal blue, $-0.25$; cyan, $-0.2$; sea green, $-0.15$; green, $-0.1$; yellow orange, $-0.079$; red orange, $-0.05$; mahogany, $-0.006$.

The momentum budget in the SCS indicates that the viscous terms are negligible in the outer region of the boundary layer, and the $\psi$-momentum equation reduces to

(3.26)\begin{equation} \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} \approx-\frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} - \frac{\partial \overline{u'_{\psi} u'_{\psi}}}{\partial \psi} - \frac{\partial \overline{u'_{\psi} u'_{\phi}}}{\partial \phi}. \end{equation}

Rearranging the equation and integrating it from $\phi$ to freestream, we get

(3.27)\begin{equation} - \overline{u'_{\psi} u'_{\phi}} (\phi) \approx \int_{\phi}^{\infty} \left( \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} + \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} + \frac{\partial \overline{u'_{\psi} u'_{\psi}}}{\partial \psi} \right) \,{\rm d} \phi'. \end{equation}

From the discussion in § 3.3.1, while the SCS and WCS become less well aligned in the outer layer (as large as $3^{\circ }$) that misalignment may not be significant enough to spoil the approximation, and thus the integral in (3.27) may still be approximated using the wall-normal coordinate as the integrand. This assumption reduces the expression to

(3.28)\begin{equation} - \overline{u'_{\psi} u'_{\phi}} (n) \approx \int_{n}^{\infty} \left( \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} + \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} + \frac{\partial \overline{u'_{\psi} u'_{\psi}}}{\partial \psi} \right) \,{\rm d} n'. \end{equation}

Classically, ${\partial \overline {u'_{\psi }u'_\psi }}/{\partial \psi }$ is often assumed to be insignificant relative to the other terms of the equation. The $\psi$-momentum equation budgets shown in figure 8 indicated that this term is small compared to the momentum flux due to the pressure gradient, advection and the Reynolds shear stress. However, as this term is non-zero, it may still affect the approximation of the Reynolds shear stress terms. Therefore, we compare two approximations,

(3.29)\begin{equation} \frac{-\overline{u'_{\psi} u'_{\phi}}}{u_{*,o}^2} \approx \frac{-\overline{u'_{s} u'_{n}}}{u_{*,o}^2} \approx 1, \quad u^2_{*,o} (n) \stackrel{\text{def}}= \int_{n}^{\infty} \left( \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} + \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} + \frac{\partial \overline{u'_{\psi} u'_{\psi}}}{\partial \psi} \right) \,{\rm d} n', \end{equation}

and

(3.30)\begin{equation} \frac{-\overline{u'_{\psi} u'_{\phi}}}{u_{**,o}^2} \approx \frac{-\overline{u'_{s} u'_{n}}}{u_{**,o}^2} \approx 1, \quad u^2_{**,o} (n) \stackrel{\text{def}}= \int_{n}^{\infty} \left( \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} + \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} \right) \,{\rm d} n', \end{equation}

where $u^2_{*,o}$ includes the influence of ${\partial \overline {u'_{\psi } u'_{\psi }}}/{\partial \psi }$ term, whereas $u^2_{**,o}$ neglects this influence. This approximation varies locally with wall-normal locations due to the dependence of the lower integral limit on wall-normal distance. The upper limit of the integral is set as the inviscid core where the magnitude of the integrand is negligible.

The Reynolds shear stresses in the WCS normalised by $u^2_{*,0}$ and $u^2_{**,o}$ in the APG region are shown in figure 20. The profiles are plotted until $n/\delta = 0.9$ to avoid singularities observed at $n/\delta \approx 1.0$ as the approximation decreases to zero faster than the decrease in the Reynolds shear stresses. Both approximations result in a good collapse of stresses at different $x/L$ locations. Normalised Reynolds shear stresses shown in figure 21 show a better collapse than the Zagarola–Smits normalisation results shown in figure 19. We observe that $u^2_{*,o}$ collapses the Reynolds shear stress much better than $u^2_{**,o}$ indicating that the influence of ${\partial \overline {u'_{\psi } u'_{\psi }}}/{\partial \psi }$ is non-negligible, unlike many other simpler boundary layer flows. From a turbulence modelling perspective, the term ${\partial \overline {u'_{\psi } u'_{\psi }}}/{\partial \psi }$ is generally unknown and often neglected while simplifying boundary layer equations. This term is included for RANS modelling but is frequently approximated using the linear eddy-viscosity hypothesis. The non-negligible influence of ${\partial \overline {u'_{\psi } u'_{\psi }}}/{\partial \psi }$ indicates that the accurate modelling of this term will be required to obtain a better approximation of the flow solution.

Figure 20. Reynolds shear stress in WCS in the APG region of the flow normalised by (a) $u^2_{*,o}$ and (b) $u^2_{**,o}$. Vertical lines indicate that the quantity asymptotically goes to $\pm \infty$. Line types for APG region at $x/L$: violet, $-0.5$; cerulean, $-0.45$; green, $-0.39$; orange, $-0.35$; mahogany, $-0.3$.

Figure 21. Reynolds shear stress in WCS in the FPG region of the flow normalised by (a) $u^2_{*,o}$ and (b) $u^2_{**,o}$. Vertical lines indicate that the quantity asymptotically goes to $\pm \infty$. Line types for FPG region at $x/L$: violet, $-0.29$; royal blue, $-0.25$; cyan, $-0.2$; sea green, $-0.15$; green, $-0.1$; yellow orange, $-0.079$; red orange, $-0.05$; mahogany, $-0.006$.

A similar integral analysis conducted with the $\phi$-momentum equation results in the approximation

(3.31)\begin{equation} \frac{\overline{u'_{\phi} u'_{\phi}}}{{u_{\gamma,o}^2}} \approx \frac{\overline{u'_{n} u'_{n}}}{{u_{\gamma,o}^2}} \approx 1, \quad u^2_{\gamma,o} (n) \stackrel{\text{def}}= \int_{n}^{\infty} \left( \frac{\bar{u}_{\psi}^2}{{R_w}} + \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \phi} \right) \,{\rm d} n', \end{equation}

where the upper limit of the integral is in the inviscid core. The results obtained using this approximation are shown in figure 22. We observe a singularity at $n/\delta \approx 0$ and $n/\delta \approx 1.0$ as the approximation reduces to zero faster than the decrease in $\overline {u'_n u'_n}$. In addition, we observe that the approximation collapses the profiles well in the APG region. In the FPG region, the approximation collapses the profiles better than the Zagarola–Smits normalisation (shown in figure 19). However, this collapse is not as close as in the APG region, indicating that neglected budget terms in this calculation can combine to have first-order effects that are not included in the above approximation.

Figure 22. Streamwise normal stress in WCS in the (a) APG region and (b) FPG region of the flow normalised by $u^2_{\gamma,o}$. Vertical lines indicate that the quantity asymptotically goes to $\pm \infty$. Line types for APG region at $x/L$: violet, $-0.5$; cerulean, $-0.45$; green, $-0.39$; orange, $-0.35$; mahogany, $-0.3$. Line types for FPG region at $x/L$: violet, $-0.29$; royal blue, $-0.25$; cyan, $-0.2$; sea green, $-0.15$; green, $-0.1$; yellow orange, $-0.079$; red orange, $-0.05$; mahogany, $-0.006$.

3.4.1. Reynolds shear stress approximations

From the momentum budgets shown in § 3.2.1, we observe that the streamwise pressure gradient varies slowly in the wall-normal direction. Therefore, the streamwise pressure gradient contribution to the Reynolds shear stress approximation (3.11) can be approximated as

(3.32)\begin{equation} \int_0^n \frac{1}{\rho} \frac{\partial \bar{p}}{\partial \psi} \,{\rm d}n' \approx \left.\frac{n}{\rho} \frac{\partial \bar{p}}{\partial \psi} \right\vert_w = \frac{n}{\rho} \frac{\partial \bar{p}_w}{\partial s}. \end{equation}

On the other hand, the advection term in the Reynolds shear stress approximation varies significantly in the near-wall region. One approach to model the advection term is to assume a particular velocity profile for $\bar {u}^+_{\psi }$ such as the Reichardt velocity profile (Reichardt Reference Reichardt1951)

(3.33)\begin{equation} \bar{u}_{\psi}^+\approx \bar{u}_{\psi, Rei}^+= \frac{1}{\kappa} \ln (1 + \kappa n^+) + 7.8 \left[ 1 - \exp\left( \frac{- n^+}{11}\right) - \frac{n^+}{11} \exp \left(\frac{-n^+}{3} \right) \right]. \end{equation}

A similar approach was employed to construct empirical wall laws for turbulent flows subject to APGs in Galbraith, Sjolander & Head (Reference Galbraith, Sjolander and Head1977) and Knopp (Reference Knopp2022). We have found that assuming the Reichardt velocity profile in the advection term leads to a collapse of Reynolds shear stresses in the mild APG region, but it fails to lead to a collapse in the strong FPG region. This is not surprising as the mean velocity profiles strongly depart from the Reichardt velocity profile in the strong FPG region.

An alternative approach is to construct an empirical model for the advection term using the DNS data. In the momentum budgets shown in § 3.2.1, the advection term in both the APG and FPG regions of the flow appears to exhibit a log-linear relationship in the inner region of the boundary layer. Using this observation, a log-linear profile fit for the advection term in the inner region of the boundary layer is proposed:

(3.34)\begin{equation} \bar{u}_{\psi} \frac{\partial \bar{u}_{\psi}}{\partial \psi} \approx \begin{Bmatrix} (u^3_{\tau }C/\nu) \; \ln (n^+- 8), & n^+> 9\\ 0, & \text{otherwise} \end{Bmatrix}, \end{equation}

where the constant $C$ varies with $x/L$ stations and is optimised based on the DNS advection term. The comparison of the advection term obtained using DNS and the log-linear advection term approximation is shown in figure 23. As the log-linear fit adequately represents the variation of the advection term in the inner region of the boundary layer, we can use it in the approximation.

Figure 23. Advection term in (a) APG and (b) FPG regions of the flow normalised by inner region units and compared to log-linear approximation as shown in (3.34). Line types for APG region for DNS profiles at $x/L$: violet, $-0.5$; cerulean, $-0.45$; green, $-0.39$; orange, $-0.35$; mahogany, $-0.3$. Line types for log-linear advection approximation at $x/L$: violet, $-0.5$; cerulean, $-0.45$; green, $-0.39$; orange, $-0.35$; mahogany, $-0.3$. Line types for FPG region at $x/L$: violet, $-0.29$; royal blue, $-0.25$; cyan, $-0.2$; sea green, $-0.15$; green, $-0.1$; yellow orange, $-0.079$; red orange, $-0.05$; mahogany, $-0.006$. Line types for log-linear advection approximation at $x/L$: violet, $-0.29$; royal blue, $-0.25$; cyan, $-0.2$; sea green, $-0.15$; green, $-0.1$; yellow orange, $-0.079$; red orange, $-0.05$; mahogany, $-0.006$.

To arrive at a closed Reynolds shear stress approximation, we also need a model for $C$. In this direction, we next examine how $C$ varies with non-dimensional pressure gradient $(p^+ = ({\nu }/{u_{\tau }^3})({\partial p_w}/{\partial s}))$ and propose a model for $C$, referred to as $C_{mod}$, of the form

(3.35)\begin{equation} C_{mod} = A p^++ B, \end{equation}

where $B = 0$ to satisfy $C_{mod} =0$ for zero-pressure gradient turbulent boundary layers and

(3.36)\begin{equation} A = \underset{\bar{A}}{\text{argmin}} \| C - \bar{A} p^+ \|^2. \end{equation}

The resulting model was found to be $C_{mod} = -0.195 p^+$. The variation of $C$ obtained from the DNS and the linear fit approximation is shown in figure 24(a). These results indicate the linear fit agrees well in the early part of the mild APG region. However, a significant departure from this fit is observed in the latter part of the APG region and the strong FPG region as the DNS computed values of $C$ draw a loop, indicating non-local effects. Although this departure from the linear fit is not correlated with the pressure gradient directly, it correlates well with the non-dimensionalised second derivative of pressure $(\partial p^+/\partial s^+ = ({\nu ^2}/{u_{\tau }^4})({\partial ^2 p_w}/{\partial s^2}))$ as shown in figure 24(b). The key observation from this correlation is that non-local pressure gradient effects can be accounted for by considering the second derivative of the pressure gradient. Therefore, we augment the model for $C$ to be linearly dependent on both $p^+$ and $\partial p^+/\partial s^+$. The resulting model form is

(3.37)\begin{equation} C_{mod-2} = A p^++ B \frac{\partial p^+}{\partial s^+}, \end{equation}

where

(3.38)\begin{equation} A,B = \underset{\bar{A},\bar{B}}{\text{argmin}} \left\| C - \bar{A} p^+- \bar{B} \frac{\partial p^+}{\partial s^+} \right\|^2. \end{equation}

The resulting model was found to be

(3.39)\begin{equation} C_{mod-2} =-0.195 p^+- 530.914 \frac{\partial p^+}{\partial s^+}. \end{equation}

We observe that the coefficient of $p^+$ does not change even when we add the dependence on $\partial p^+/\partial s^+$. The comparison of $C_{mod-2}$ and $C$ obtained from DNS is presented in figure 25, which show that $C_{mod-2}$ adequately represents the variation of $C$ for this flow by accounting for the non-local effects. Note that the addition of higher-order derivatives of $p^+$ or the selection of higher degree polynomials (instead of linear) can further improve the accuracy of the model for $C$. However, both approaches may lead to overfitting of the model to the bump data and the latter requires expensive computations of accurate higher-order derivatives. Using the final attained model $C_{mod-2}$, the Reynolds shear stress is normalised by

(3.40)\begin{equation} u^2_{aD} (n) \stackrel{\text{def}}= u_{\tau}^2 + \frac{n}{\rho} \frac{\partial \bar{p}}{\partial s} + u_{\tau}^2 C_{mod-2} I_1 \left(\frac{n u_{\tau}}{\nu}\right), \end{equation}

where the integral, $I_1 (n u_{\tau }/\nu ) = I_1 (n^+)$, is given as

(3.41)\begin{equation} I_1 (n^+) = \begin{Bmatrix} \int_{9}^{n^+} \ln (n^{+'} - 8) \, {\rm d} n^{+'}, & n^+> 9 \\ 0, & \text{otherwise} \end{Bmatrix} , \end{equation}

which can be computed analytically.

Figure 24. Variation of the (a) approximation constant $C$ and modelled value $C_{mod}$ with $p^+$ and (b) error in the linear approximation and $\partial p^+/\partial s^+$ at several $x/L$ locations upstream of flow separation. Line types for (a): black, for $C$ obtained from DNS; and blue, for $C_{mod}$. Line types for (b): black, for $\partial p^+/\partial s^+$ obtained from DNS; and red, for normalised error in the linear fit $(= -\tfrac {1}{500} (C - C_{mod}))$.

Figure 25. Variation of the approximation constant $C$ and modelled value $C_{mod-2}$ with $p^+$. Line types: black, for $C$ obtained from DNS; and blue, for $C_{mod-2}$.

Reynolds shear stresses normalised with $u_{aD}^2$ for several $x/L$ locations are shown in figure 26. We observe that this proposed normalisation collapses the Reynolds shear stress profiles well in the inner region of the boundary layer, that is $n+ < 100$, for both the APG and FPG regions of the flow. We anticipate the approximation can be used in wall-modelled LES and RANS as this is the range of wall-normal locations at which boundary conditions are applied in such approaches.

Figure 26. Reynolds shear stress in (a) APG and (b) FPG regions normalised by $u^2_{aD}$. Vertical lines indicate that the quantity asymptotically goes to $\pm \infty$. Line types for APG region at $x/L$: violet, $-0.5$; cerulean, $-0.45$; green, $-0.39$; orange, $-0.35$; mahogany, $-0.3$. Line types for FPG region at $x/L$: violet, $-0.29$; royal blue, $-0.25$; cyan, $-0.2$; sea green, $-0.15$; green, $-0.1$; yellow orange, $-0.079$; red orange, $-0.05$; mahogany, $-0.006$.

3.4.2. $\overline {u'_n u'_n}$ normal stress approximation

We can approximate $\overline {u'_n u'_n}$ using a similar approach to that we followed for approximating Reynolds shear stress. In this case, the influence of curvature brought about through the centrifugal force term needs to be accounted for accurately. Unlike the case of the advection term in the Reynolds shear stress approximation, assuming the Reichardt velocity profile (3.33) in the centrifugal force term leads to a suitable approximation

(3.42)\begin{equation} u^2_{\gamma R} (n) \stackrel{\text{def}}=- \frac{\nu u_{\tau}}{{R_w}} I_2\left(\frac{n u_\tau}{\nu}\right) + \frac{\bar{p}_w - p}{\rho} \end{equation}

for the $\overline {u'_n u'_n}$ normal stress for both the APG and FPG regions of the flow, where $I_2(n u_\tau /\nu ) = I_2(n^+)$ is given by

(3.43)\begin{equation} I_2(n^+) = \int_{0}^{n^+} (\bar{u}^+_{\psi,Rei})^2 \,{\rm d} n^{+'} \end{equation}

and can be computed analytically. Note the approximation given by (3.42) is closed and depends only on local and wall quantities.

The $\overline {u'_n u'_n}$ stress profiles normalised by $u^2_{\gamma R}$ in both the APG and FPG regions of flow are shown in figure 27. The results indicate that the normalisation collapses the profiles well in the inner region of the boundary layer for both types of pressure gradient. The assumption of Reichardt's velocity profile does not significantly affect the accuracy of the approximation for $n^+ < 80$. Therefore, this approximation works adequately to account for the centrifugal force terms in the approximation.

Figure 27. Plots of $\overline {u'_n u'_n}$ stress in (a) APG and (b) FPG regions normalised by $u^2_{\gamma R}$. Vertical lines indicate that the quantity asymptotically goes to $\pm \infty$. Line types for APG region at $x/L$: violet, $-0.5$; cerulean, $-0.45$; green, $-0.39$; orange, $-0.35$; mahogany, $-0.3$. Line types for FPG region at $x/L$: violet, $-0.29$; royal blue, $-0.25$; cyan, $-0.2$; sea green, $-0.15$; green, $-0.1$; yellow orange, $-0.079$; red orange, $-0.05$; mahogany, $-0.006$.

These proposed semi-empirical approximations for the Reynolds shear stress and the $\overline {u'_n u'_n}$ stress were found to be adequate for the turbulent boundary layer in the presence of pressure gradients and curvature effects considered in this article. We believe other such flows could exhibit similar behaviour and additional studies with different pressure gradients are warranted to understand the applicability of these approximations for such flows. At $Re_L = 1$ million, the model based on the $Re_L = 2$ million flow agrees well for acceptable wall-normal distance relevant for wall-modelled boundary conditions in the APG region and also in the FPG region prior to where laminarisation occurs. Subject to validation on several other flows with pressure gradients and curvature effects, these approximations offer closed-form expressions for the Reynolds shear stress and the $\overline {u'_n u'_n}$ stress in the inner region of the boundary layer at wall-normal distances where wall-modelled large eddy simulations and RANS simulations apply wall-modelled boundary conditions. Further, these expressions involve variables that would be available from the interior solution. The work presented here is meant to be a starting point for the community to evaluate (3.40) and (3.42) on other available datasets, extend their generalisability and incorporate them in novel wall-modelling approaches.