3.1. Wing dataset

The datasets for the wing simulations correspond to the suction side of the NACA 4412 airfoil at an angle of attack of $5^\circ$ (Vinuesa et al. Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018; Atzori et al. Reference Atzori, Vinuesa, Stroh, Gatti, Frohnapfel and Schlatter2021, Reference Atzori, Mallor, Pozuelo, Fukagata, Vinuesa and Schlatter2023). Referred to as the Wing, the Reynolds number based on the chord length, $c$, is $Re_c = U_\infty c /\nu = 400\,000$, where $U_\infty$ denotes the free-stream velocity. After the BL is tripped at 10 % chord, the turbulent BL experiences deceleration in the streamwise direction (APG) over the domain of interest in this paper, resulting in Clauser parameter values in the range $0<\beta <40$ along the chord length up to 95 % chord. Apart from the reference Wing case, two other BLs under the same flow configuration are studied, but with control schemes employing surface suction (Wing-suction) and blowing (Wing-blowing). The control surface extends from $\xi /c = 0.25$ to $\xi /c = 0.855$, with a constant control intensity set at $0.1\,\%$ of $U_\infty$. The AMI analyses of the flow control studies are shown in Appendix B.

The wing simulations conducted in this study utilize WRLES on the open-source Nek5000 solver, which uses the spectral-element method developed by Patera (Reference Patera1984). The LES filtering technique follows the approximate deconvolution relaxation term (ADM-RT) subgrid model developed by Schlatter, Stolz & Kleiser (Reference Schlatter, Stolz and Kleiser2004). Boundary conditions at the inlet, upper and lower sides are provided by a RANS simulation, while a local-stress outflow condition is utilized for the rear side of the domain (Dong, Karniadakis & Chryssostomidis Reference Dong, Karniadakis and Chryssostomidis2014). The mesh is generated based on the wall-shear stress from RANS simulations to ensure a resolution of approximately $(\Delta \xi ^+,\Delta \eta ^+,\Delta z^+) <(18, 0.64, 11.9)$ in the turbulent region of the domain, where superscript $(+)$ denotes wall units (normalized by $\delta _\nu =\nu /u_\tau$, where $u_\tau =\sqrt {{\tau }_w/\rho }$ is the friction velocity) and $z$ represents the spanwise direction. While we refer to this as a WRLES due to the use of a subgrid-scale stress model, the grid resolution is only slightly coarser than typical DNS resolutions. Vinuesa et al. (Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018) reported that averages were taken over approximately $10$ eddy turnover times in the original simulation. A detailed analysis of statistical convergence is given in an appendix of Atzori et al. (Reference Atzori, Vinuesa, Fahland, Stroh, Gatti, Frohnapfel and Schlatter2020). The data used in this paper include additional averaging time, about twice a long as reported in the above references. For further information regarding the numerical set-up, the reader is referred to Vinuesa et al. (Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018) and Atzori et al. (Reference Atzori, Vinuesa, Fahland, Stroh, Gatti, Frohnapfel and Schlatter2020, Reference Atzori, Vinuesa, Stroh, Gatti, Frohnapfel and Schlatter2021, Reference Atzori, Mallor, Pozuelo, Fukagata, Vinuesa and Schlatter2023).

3.2. Bump dataset

The Bump dataset analysed in this paper comes from a DNS of a turbulent BL over a two-dimensional Gaussian bump performed by Balin & Jansen (Reference Balin and Jansen2021). The surface of the bump is defined by the equation $\eta (\xi ) = h \exp (-(\xi /\xi _0)^2)$ in the Cartesian coordinate system. This geometry, illustrated in figure 4(a), is designed to replicate the three-dimensional bump flow (Slotnick Reference Slotnick2019) experimentally studied by Williams et al. (Reference Williams, Samuell, Sarwas, Robbins and Ferrante2020). In the bump's surface relation, $h=0.085L$ and $\xi _0=0.195L$ are length parameters describing the bump's dimensions, with $L=0.9144$ m representing the length of the square cross-section of the wind tunnel used in the experimental set-up. The flow is characterized by a reference Reynolds number $Re_L = U_\infty L/\nu = 1\,000\,000$, where the dimensional free-stream velocity $U_\infty = 16.4$ m s$^{-1}$, matching standard sea level conditions, resulting in an incompressible flow with Mach number $M = 0.045$. This flow experiences an alternating pressure gradient due to surface curvature, resulting in a weak APG region upstream, followed by a strong FPG upstream of the bump's peak at $\xi /L = 0$. Downstream of the bump's peak, a severe APG induces incipient flow separation ($\beta \to \infty$).

The DNS was performed using the stabilized finite-element method by applying trilinear hexahedral elements and second-order-accurate, implicit time integration following Whiting (Reference Whiting1999) and Jansen, Whiting & Hulbert (Reference Jansen, Whiting and Hulbert2000). A no-slip, no-penetration BC is imposed at the bump's surface, while the top BC (at $\eta /L = 0.5)$ was modelled as an inviscid wall offset by the RANS-predicted displacement thickness described above with zero transpiration (zero velocity component normal to the surface) and zero traction. The inflow is generated by a synthetic turbulence generator (Shur et al. Reference Shur, Spalart, Strelets and Travin2014). Finally, for the outflow weak enforcement of zero pressure was applied along with zero traction. The computational grid used for DNS has a typical spacing of $(\Delta \xi ^+,\Delta \eta ^+, \Delta z^+ ) < (15, 10, 8 )$ with the minimum $\Delta \eta ^+ = 0.1$ near the surface, in wall units. The reader is referred to Balin & Jansen (Reference Balin and Jansen2021) for more information about the numerical set-up. The primary concern of the simulation was of the BL flow prior to incipient separation, so the present analysis of the resulting dataset is restricted to this region of the flow. As was the case for the discussion in Balin & Jansen (Reference Balin and Jansen2021), no analysis is made in this paper of incipient separation or other downstream aspects of the flow.

3.3. Flat-plate dataset

For comparison purposes, we also investigate a series of turbulent BLs developing over a flat plate. These simulations were conducted using WRLES, employing the ADM-RT subgrid model, similar to the approach used for the airfoil simulations. The simulations were executed using the SIMSON code, a pseudo-spectral-based solver developed by Chevalier et al. (Reference Chevalier, Schlatter, Lundbladh and Henningson2007). Specifically, we examine data from the ZPG case simulated by Eitel-Amor, Örlü & Schlatter (Reference Eitel-Amor, Örlü and Schlatter2014), as well as APG cases simulated by Bobke, Örlü & Schlatter (Reference Bobke, Örlü and Schlatter2016) and Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017), both with similar numerical set-up that results in spatial resolution $(\Delta x^+,\Delta y^+, \Delta z^+ ) = (20, 0.2-30, 10 )$ in wall units. For the flat plates with APG, the pressure gradient was imposed through the variation of the free-stream velocity at the top of the numerical domain, following the near-equilibrium definition by Townsend (Reference Townsend1976). The wall-tangential and wall-normal coordinates are aligned with the Cartesian coordinate system in these simulations, so no mapping is required.

3.4. Numerical techniques for AMI analysis

Because the AMI analysis is applied in a curvilinear coordinate system following the surface in both the Wing and Bump flows, an appropriate tensor rotation is employed to map the flow statistics from the Cartesian frame of reference to local orthogonal directions. Additionally, as detailed in § 2, we utilize the methodology developed by Griffin et al. (Reference Griffin, Fu and Moin2021) to locally reconstruct an inviscid velocity at the wall, $U_{io}(x)$, for each streamwise position and predict the location of the BL edge based on $\delta _{99}$. In the Bump case, which has the strongest curvature of the cases considered, the surface-normal rays (in the $y$ direction) were examined at each streamwise location, confirming they intersect neither within nor near the BL.

Figures 2(a) and 2(b) illustrate the reconstructed approximation to the irrotational velocity, $U$, as dashed lines alongside the mean velocity profiles, $\bar {u}$ normalized by $U_\infty$, at three distinct streamwise locations for Wing and Bump, respectively. As expected, there is a remarkable agreement between $U$ and $\bar {u}$ profiles within the outer flow, marked beyond the black symbols that indicate the prediction of $\delta _{99}$. This confirms that there is significantly non-uniform velocity in the region of the flow which satisfies the Bernoulli equation. For example, the three streamwise locations in the Bump flow are associated with the region under strong FPG $\xi /L = -0.12$, strong APG $\xi /L = 0.15$ and very close to the separation point $\xi /L = 0.22$, and even near the separation point, the local reconstruction method of $U$ and $\delta _{99}$ is quite reasonable. It is important to note that the trends observed in the computed $U_{io}$ and predicted BL edge velocity ($U_e$) are similar. The reconstructed irrotational velocity at the wall, $U_{io}$, is used throughout the rest of the paper because it was more effective at minimizing the (normalized) residual error in the AMI equation,

(3.2)\begin{equation} \epsilon(x) = \frac{|C_f - 2 {\rm RHS}|}{C_f}, \end{equation}

for both the Wing and Bump cases. Here, RHS represents the sum of all terms on the right-hand side of (2.7). Appendix A contains more specific discussion of residual errors. Note that the normalized residual error defined in (3.2) includes multiple sources of error, including both statistical convergence errors and error related to neglecting surface curvature. Errors due to neglecting terms via the standard BL assumptions, ${\mathcal {I}}^\ell$ in (2.7), are not included in the residual error.

Figure 2. Example calculations of the locally reconstructed inviscid velocity, $U$, along the wall-normal direction for (a) Wing and (b) Bump datasets. The solid lines represent the mean streamwise velocity from the simulation and the dashed lines show the inviscid velocity profiles reconstructed using the method of Griffin et al. (Reference Griffin, Fu and Moin2021). Both velocity profiles are normalized by the free-stream velocity, $U_\infty$. The solid black lines and symbols denote the location of (normalized) $\delta _{99}$.

To compute the streamwise derivatives of flow statistics required for closing the AMI budget, we utilize the second-order central finite-difference scheme, excluding the end grid points. However, it is worth noting that computing these derivatives can amplify natural turbulent noise, which tends to increase $\epsilon$ (Elnahhas & Johnson Reference Elnahhas and Johnson2022; Kianfar et al. Reference Kianfar, Di Renzo, Williams, Elnahhas and Johnson2023a). Furthermore, for the numerical wall-normal integration in the AMI analysis, we employ the midpoint scheme.

4.1. Flow over airfoil

In this section, we employ the AMI equation to analyse a turbulent BL over the suction side of a NACA 4412 airfoil at an angle of attack of $5^{\circ }$, referred to as the Wing case in table 1. The objective is to quantitatively study the interaction of turbulence and the APG in terms of ‘torques’ which compete to reshape the mean velocity profile and alter the skin friction relative to a baseline laminar solution.

Figure 3 presents the non-negligible terms of the AMI equation, (2.7), for the Wing case. The budget includes the four major terms in the AMI equation, namely laminar friction, turbulent torque, pressure gradient and mean flux. Following previous work (Elnahhas & Johnson Reference Elnahhas and Johnson2022; Kianfar et al. Reference Kianfar, Di Renzo, Williams, Elnahhas and Johnson2023a,Reference Kianfar, Elnahhas and Johnsonb), the momentum thickness is used to define the similarity Reynolds number, $Re_\ell$, where $\ell =4.54 \delta _2$, a value derived from the Blasius solution. The laminar friction term thus represents the skin friction coefficient of an equivalent Blasius BL with matching momentum thickness Reynolds number, $Re_{\delta _2}$. The other three terms represent how each effect (turbulent mixing, free-stream pressure gradient, changes in mean streamwise growth) enhances or diminishes the skin friction coefficient relative to the Blasius BL. Figure 3(a) shows the absolute contributions of each term and figure 3(b) exhibits the relative contributions (normalized by $C_f/2$). The unsteady effects and terms neglected by the BL approximation, referred to as negligible terms, are not included here. This omission is justified by their limited contribution to the skin friction coefficient away from separation, as shown in Appendix A. The streamwise-averaged residual error, $\epsilon$, is less than $6\,\%$. A similar level of residual error in the AMI equations was observed in ZPG transitional and turbulent incompressible BLs ($\approx$2 %) (Elnahhas & Johnson Reference Elnahhas and Johnson2022; Kianfar et al. Reference Kianfar, Elnahhas and Johnson2023b), as well as in high-speed ZPG turbulent BLs ($\approx$5 %) (Kianfar et al. Reference Kianfar, Di Renzo, Williams, Elnahhas and Johnson2023a). In all these cases, the residual error exhibits an oscillatory behaviour and the primary source of error was the computation of streamwise derivatives, $\partial (\cdot ) / \partial x$, which amplifies inherent statistical noise due to averaging over a finite amount of data. That is, the residual error could be further reduced with additional averaging in time. In addition, the small residual error validates our assumption on neglecting the geometrical curvature effects.

Figure 3. The AMI budget with respect to the chord length, $\xi /c$, for the Wing case (table 1): (a) the absolute budget and (b) the relative budget normalized by $C_f/2$. In (b), for scaling purposes, the black line denotes 100 % contribution to $C_f/2$. In (a,b) the faded black dotted line represents the residual of the AMI equation.

Compared with ZPG turbulent BLs, APG turbulent BLs exhibit a faster reduction of skin friction coefficient as the flow develops downstream, shown with a black line in figure 3(a). The BL is fully turbulent and its skin friction coefficient significantly exceeds that of the equivalent laminar BL. In fact the relative contribution of the laminar friction to the AMI equation is small throughout the domain of interest. In contrast, the turbulence-induced enhancement of surface friction by the turbulent torque in the AMI equation is substantial. Consistent with previous ZPG results (Elnahhas & Johnson Reference Elnahhas and Johnson2022), its relative contribution to the skin friction is near $100\,\%$ within the range of $0.2<\xi /c<0.4$, where $\beta < 2$ (weak-to-moderate pressure gradient). Interestingly, the streamwise variation of the turbulent torque is relatively small, varying by approximately 13 % from $\xi /c=0.2$ to $\xi /c=0.9$, even under strong downstream APG.

As expected, the pressure-gradient torque is negative, acting to decrease the skin friction, and strengthens rapidly as the flow approaches the trailing edge. The normalized contribution of the pressure gradient shown in figure 3(b) exhibits a range from approximately $0\,\%$ upstream to more than $-$600 % ($\beta _\ell = 6$), highlighting the profound impact of pressure gradient on the transport of momentum deficit and, consequently, wall-shear stress. Note that the relative contribution of the pressure gradient to the attenuation of the skin friction coefficient is equal to the AMI-based modified Clauser parameter, $\beta _\ell$ in (2.15). As the non-equilibrium caused by pressure gradient ramps up downstream, the mean flux term in the AMI equation opposes the pressure-gradient effects. The friction enhancement by the mean flux becomes significant for $\beta > 2$, where the impact of pressure gradient is two times greater than that of wall shear stress. While the mean flux negatively contributes to $C_f/2$ with little downstream variation in turbulent ZPG BLs (Elnahhas & Johnson Reference Elnahhas and Johnson2022; Kianfar et al. Reference Kianfar, Elnahhas and Johnson2023b), here in APG BLs, it increases and partially offsets the growing negative contribution of pressure gradient.

In addition to the baseline Wing case studied in detail here, two flow control cases are studied in Appendix B applying blowing and suction at the airfoil surface. The further application of the AMI equation to analyse the effectiveness of flow control schemes is deferred to future studies.

4.2. Flow over Gaussian bump

This section examines an incompressible fully turbulent BL over a two-dimensional Gaussian bump, referred to as Bump flow in table 1 (Balin & Jansen Reference Balin and Jansen2021). This presents a more complex scenario due to the alternate APG and FPG induced by the bump's surface curvature. Specifically, the BL experiences a significant FPG from $\xi / L = -0.29$ to the peak of the bump at $\xi / L = 0$, followed by an extreme APG downstream, leading to BL separation where ${\tau }_w \leq 0$. The Bump flow thus provides an interesting comparison with and contrast to the Wing flow analysed in § 4.1.

Within the streamwise range of interest in this paper, the averaged residual error ($\epsilon$) is maintained at $\epsilon < 10\,\%$ (equation (3.2)). Downstream of the bump's peak, characterized by an APG ($\xi /L\leq 0.18$), the averaged $\epsilon$ further diminishes to approximately $6\,\%$, followed by a noticeable escalation approaching the incipient separation region (which is therefore not analysed in this paper). There are multiple potential reasons for the observed trends in residual error. In our experience, the level of residual error is most sensitive to averaging time (statistical convergence). The residual error upstream of the incipient separation is acceptable for the present analysis, but could possibly be reduced with longer time averages, especially in this case due to the natural unsteadiness of the separation bubble. A brief analysis of the full AMI budget and BL assumptions is reported in Appendix A. Wei, Li & Wang (Reference Wei, Li and Wang2024) investigated the role of terms neglected by the BL assumption for the momentum integral equation, and future work can explore similar considerations for the AMI equation for separated flows.

Figure 4 presents the main terms in the AMI equation (2.7) for the Bump flow. To aid the interpretation of the AMI analysis, figure 4(a) illustrates the alternating APG and FPG due to the geometry of the surface. The BL first encounters weak APG as it approaches the bump, followed by relatively strong FPG on the upstream side of the bump. For $0 \leq \xi /c \leq 0.4$, the BL experiences strong APG, leading to flow separation at $\sim \xi /c=0.2$ (where the black line in figure 4(b), $C_f/2$, becomes weakly negative). All four primary terms in the AMI equation are shown in figure 4(b): laminar friction, turbulent torque, pressure gradient and mean flux. As in the case of the Wing analysis, the AMI equation is calculated relative to a laminar BL with matching $Re_{\delta _2}$, using $\ell = 4.54 \delta _2$ (derived from the Blasius solution). The contributions of unsteady effects and terms neglected by BL approximations – referred to as negligible terms – are minimal upstream of the separation point. Therefore, we do not exhibit these flow phenomena in the main paper, but they are recorded in Appendix A. Figure 4(c) focuses on the turbulent torque and splits the associated integral into contributions from different regions of the turbulent BL.

Figure 4. The AMI analysis for the BL over a Gaussian bump simulated by Balin & Jansen (Reference Balin and Jansen2021): (a) the geometry of the bump in the normalized ($\xi$–$\eta$) plane, (b) the AMI budget with respect to $\xi /L$ and (c) the contribution to turbulent torque confined within the inner layer ($\kern0.7pt y/\delta _{99} \leq 0.1$), log-law region ($\kern0.7pt y/\delta _{\nu } \geq 30$ and $y/\delta _{99} \leq 0.3$) and outer layer ($\kern0.7pt y/\delta _{\nu } \geq 50$) with respect to $\xi /L$. In (a), the dotted and solid black lines represent $\ell$ and $\delta _{99}$, respectively. In (b), the faded black dotted line denotes the residual error in the AMI equation. Moreover, asterisk symbols represent the three streamwise locations at which turbulent statistics are compared in figure 5. In (c), the full turbulent torque is provided with faded colour for the sake of comparison.

According to figure 4(b), within the FPG region on the upstream half of the bump starting at $\xi / L = -0.29$, the skin friction coefficient initially increases, reaching a peak at $\xi /L=-0.14$. According to the AMI budget, this enhancement is primarily attributed to the simultaneous increase of the turbulent torque and the pressure gradient, only partially offset by the mean flux. As the flow approaches the peak of the bump, the FPG term decreases rapidly. Balin & Jansen (Reference Balin and Jansen2021) presented compelling evidence that the BL in this dataset undergoes relaminarization with onset near $\xi / L = -0.15$. The turbulent torque increases through the early FPG region, reaches a maximum around $\xi /L = -0.2$ and decreases through the (incomplete) relaminarization region up to the top of the bump at $\xi /L = 0$.

Figure 4(c) defines three partial contributions to the turbulent torque as integrals of the Reynolds shear stress over (i) the inner layer defined as $y^* = y/\delta _{99} < 0.1$, (ii) the outer layer defined as $y^+ > 50$ based on local friction units and (iii) the log-law (overlap) layer defined as $y^+ > 30$ and $y^* < 0.3$ representing the region where the mean velocity profile is approximately logarithmic with wall-normal distance for an equilibrium wall-bounded flow. These definitions are based on Pope (Reference Pope2000). Note that the mean velocity profile departs significantly from the standard logarithmic law over most of the bump's surface (Balin & Jansen Reference Balin and Jansen2021). As defined, these three layers overlap, so the sum of the turbulent torques from each of the three is not equal to the total turbulent torque.

As is true for ZPG BLs, the outer layer is responsible for the overwhelming majority of the turbulent skin friction enhancement. The primary reason for this is that the outer layer encompasses the majority of the domain and the Reynolds shear stress integral in the AMI equation is unweighted. From the AMI perspective (comparing turbulent BLs with equivalent laminar BLs), the fluctuations in the outer layer are more influential in the sense that they transport momentum over larger distances in the wall-normal direction. Thus, it is particularly noteworthy that the inner-layer contribution to the turbulent torque remains relatively constant in the FPG region and becomes significant in the APG region upstream of incipient separation. This corresponds to the initiation and growth of an internal layer in the FPG region upstream of the apex as the turbulence in the outer layer decays. Then, the retransition process initiates relatively close to the wall. These features may be seen in the Reynolds stress profiles and visualizations in Balin & Jansen (Reference Balin and Jansen2021). Figure 4(b) shows that the significant decline in turbulent torque in the strong FPG region is accompanied by a corresponding increase in the laminar friction from $10\,\%$ to about $17\,\%$ of the local skin friction coefficient.

Just upstream of the bump's peak, while the flow is still experiencing FPG and relaminarization, pressure gradient and mean flux become smaller in magnitude and reach a plateau. However, this equilibrium is disrupted immediately downstream of the peak as the flow begins to encounter an APG. Following the upstream relaminarization, a sort of retransition to turbulence occurs. As the strength of the APG increases, the turbulent torque also increases leading to a short-lived rise in the skin friction coefficient. This phenomenon is reminiscent of transition to turbulence. However, the AMI equation reveals important differences between this APG phenomenon and the transition of a laminar BL to a turbulent one. For example, in prior studies of transitional ZPG BLs (Elnahhas & Johnson Reference Elnahhas and Johnson2022; Kianfar et al. Reference Kianfar, Elnahhas and Johnson2023b), the growth in the turbulent torque significantly outpaced the growth in skin friction, with partial offset coming from the mean flux term. Under the influence of APG, however, the mean flux does not resist the growth in the turbulent torque. On the other hand, it significantly contributes to increasing $C_f$ to counteract the substantial APG.

Furthermore, examining figure 4(c), it is evident that the initial boost in turbulent torque is not confined solely to the outer layer; the inner layer's Reynolds shear stress also experiences a significant increase. Notably, during the retransition, the increase in turbulent torque of the outer layer lags the increase closer to the wall, reflecting the growth of an internal layer in the APG region (Balin & Jansen Reference Balin and Jansen2021). As a result, the outer layer does not contain the dominant share of the turbulent torque, confirming that the skin friction enhancement through retransition is driven first by near-wall turbulence and later by turbulence further from the surface once the internal layer penetrates into the outer region of the BL. Thus, the AMI equation provides a quantitative assessment of the flow physics observed by Balin & Jansen (Reference Balin and Jansen2021) and others in terms of the skin friction coefficient.

Interesting similarities and differences between the turbulent APG BLs in the Bump (figure 4) and Wing (figure 3) cases may be observed. Both flows show the rapid growth of the opposing pressure-gradient torque and the mean flux. Prior to separation, one important difference between the two flows is that the turbulent torque increases significantly over the APG region in the Bump case whereas it is nearly constant in the APG region on the suction surface in the Wing case. The increase of the turbulent torque continues and even accelerates downstream of the peak $C_f$, driven mostly by a faster increase of Reynolds shear stress in the outer layer.

Despite the friction enhancements by both turbulent torque and mean flux, they prove insufficient, as APG forces the flow to separate at approximately $\xi /L=0.2$ in the Bump case. The trends in the pressure gradient and mean flux terms from the AMI equation reverse, but the residual error in the AMI equation becomes too large (presumably due to statistical convergence errors) to justify further interpretation of the results without further time averaging (see Appendix A). It is essential to note that the AMI equation, obtained solely from conservation equations, does not depend on the validity of the BL approximation. Therefore, accurate results from AMI can be expected with sufficient (time) averaging, even for separated flows. In proximity of the separated flow region, where $C_f \leq 0$, statistics are noisy due to an unsteady separation bubble (e.g. characterized by several frequencies (Simpson Reference Simpson1989; Na & Moin Reference Na and Moin1998)), and the BL approximations are naturally invalidated. Within that region, the amplitude of the residual error in the AMI equation increases in magnitude, shown as a faded dotted black line in figure 4(b). Even so, the effect of residual error remains significantly smaller than the turbulent torque and the pressure gradient which play the dominant role for such flows. Note that, in figure 4(c), the turbulent torque within the logarithmic region and the outer layer are omitted in the separated flow region, because the definition of $y^+$ is much less meaningful when ${\tau }_w \approx 0$.

4.3. Favourable pressure gradients: relaminarization

The relaminarization of the Bump dataset was described in Balin & Jansen (Reference Balin and Jansen2021) and in a similar simulation was analysed by Uzun & Malik (Reference Uzun and Malik2022). Here, we revisit and expound on their insights in connection with the AMI equation, mainly the behaviour of the turbulent torque (integral of the Reynolds shear stress) through the FPG region. The absolute and outer-scaled profiles of Reynolds shear stress and turbulent kinetic energy, $k = \frac {1}{2} (\overline {u^\prime u^\prime }+\overline {v^\prime v^\prime }+\overline {w^\prime w^\prime } )$, are presented in figure 5. These profiles are plotted at three distinct streamwise locations: $\xi /L=-0.2$, $\xi /L=-0.14$ and $\xi /L=-0.01$; these locations are depicted in figure 4 with asterisk symbols. As shown in figure 5(a), $-\overline {u^\prime v^\prime }$ has the largest values at $\xi /L=-0.2$, where the turbulent torque peaks. However, as the FPG acts on the BL, the peak Reynolds shear stress is shifted towards the surface. This is consistent with the observations of FPG trends in figure 4(c) and with the growth of an internal layer initiated at the start of the strong FPG region (Balin & Jansen Reference Balin and Jansen2021).

Figure 5. Reynolds shear stress and turbulent kinetic energy profiles in the bump flow with respect to $y^+$: absolute (a,c) and outer-scaled (normalized by $U_{io}^2$) (b,d). In (c,d), the dashed and dotted lines are associated with $\overline {u^\prime u^\prime }/2$ and $(\overline {v^\prime v^\prime }+\overline {w^\prime w^\prime })/2$, respectively. The streamwise location for each of the profiles is indicated by asterisks with corresponding colour in figure 4.

The turbulent torque in the AMI equation is normalized by $U_{io}$, so figure 5(b) represents the integrand of the turbulent torque. As such it represents how the turbulent enhancement of skin friction coefficient (relative to the baseline laminar case) is distributed in the wall-normal direction. In particular, it shows how the contribution of the Reynolds shear stress to the skin friction coefficient changes through the FPG region, decreasing most severely in the outer region. The fact that the outer layer accounts for the large majority of turbulent torque skin friction enhancement in the FPG region (figure 4) helps explain the significant drop in the overall turbulent torque despite the fact that the Reynolds shear stress in the inner layer remains relatively constant in an absolute sense (figure 5a).

The Reynolds shear stress responsible for skin friction enhancement in the AMI equation requires (correlated) fluctuations in both streamwise and wall-normal velocities. The streamwise kinetic energy is produced/sustained directly proportional to the produced mean shear and the existing Reynolds shear stress. The wall-normal kinetic energy, however, relies mostly on pressure redistribution for sustenance. Interestingly, the peak turbulent kinetic energy shown in figure 5(c) increases through the FPG region, reaching a maximum near the bump's peak where the turbulent torque is minimum. This peak occurs in the buffer layer and is almost entirely due to the streamwise component (shown as a dashed line in figure 5). Kinetic energy further from the surface decreases, however. The wall-normal and spanwise kinetic energy components uniformly decrease through the FPG region (shown as a dotted line in figure 5). When normalized by $U_{io}$ (figure 5d), the kinetic energy decreases through the FPG region, though not as severely as the Reynolds shear stress (figure 5b). This shows that the mean flow acceleration outpaces the growth in streamwise kinetic energy, while the other two components decay.

The following picture emerges (see also Uzun & Malik Reference Uzun and Malik2022). As the mean shear moves towards the surface under the action of the FPG, the production of the streamwise component of kinetic energy occurs more in the buffer layer, where the wall-blocking effect strongly suppresses the pressure redistribution of kinetic energy to the wall-normal component and the turbulence is predominantly single-component. Therefore the outer layer relaminarizes while the inner layer actually gains kinetic energy due to the increasing free-stream velocity (increasing total shear). Overall, the shifting of kinetic energy towards the surface affects a deactivation of turbulence or frozen turbulence as described by Narasimha & Sreenivasan (Reference Narasimha and Sreenivasan1973). That is, thinking in terms of integrals of the Reynolds stress components across the BL, the turbulence becomes one-dimensional so that the existing streamwise kinetic energy affects little momentum transport and the turbulent enhancement of skin friction.

The preceding two paragraphs largely summarize and synthesize insights from Balin & Jansen (Reference Balin and Jansen2021) and Uzun & Malik (Reference Uzun and Malik2022). What the AMI equation adds to this picture is an exact quantification of how the suppression of Reynolds stresses impacts the skin friction coefficient. The turbulent torque is an unweighted integral of the Reynolds shear stress across the BL and the outer layer encompassing a significant majority of the domain of integration experiences a severe attenuation of Reynolds shear stress. The Reynolds shear stress in the inner layer makes up a small portion of the turbulent torque at the beginning of the FPG region (as is typical of BLs in near-ZPG conditions). Therefore, even though the Reynolds shear stress maintains a near-constant magnitude in the near-wall region, the overall turbulent torque decreases dramatically, especially through relaminarization. The RANS models typically over-predict the skin friction coefficient in relaminarizing BLs and the AMI analysis of RANS results could help evaluate the relative importance of model errors related to RANS predictions of Reynolds stress profiles through the FPG region.

4.4. Adverse pressure gradient: AMI budget

Next, the focus turns to the effect of APG on turbulent BLs. In particular, it is fruitful to compare and contrast the AMI budget and statistics of the APG region of the Wing and Bump cases with a set of flat plates that experience weak-to-moderate APG, as detailed in the dataset provided in table 1. In the APG region, the reference Blasius BL is set based on matching the Reynolds number based on the displacement thickness, $Re_{\delta _1}$, by setting $\ell =1.75 \delta _1$ based on (2.11). The choice of the Blasius BL allows for clean isolation of pressure-gradient effects in a single term of the AMI equation, in contrast to choosing a Falkner–Skan BL with some APG effects baked into the laminar skin friction. The displacement thickness is selected for the AMI length scale for APG analysis, $\ell \sim \delta _1$, because of its significance in the pressure-gradient term in the momentum integral equation and the definition of non-equilibrium Clauser parameter, (2.12) and (2.14).

Based on the available data from the Wing and Bump datasets, the current investigation shows results for streamwise positions corresponding to $1400 \leq Re_{\delta _1} \leq 6500$, where $Re_{\delta _1} = U_{io} \delta _1/\nu$, and to regions where $0\leq \beta < 40$ and $0\leq \beta _\ell < 20$. For the Bump dataset, it is useful to relate this range of $Re_{\delta _1}$ to a few key landmarks. First, the apex of the bump occurs at $\xi /L = 0.00$ where $Re_{\delta _1} \approx 1360$. The small local maximum in skin friction caused by retransition occurs at $\xi /L \approx 0.05$ where $Re_{\delta _1} \approx 1850$. The analysis by Balin & Jansen (Reference Balin and Jansen2021) includes up to $\xi /L = 0.1$ where $Re_{\delta _1} \approx 3300$. Here we consider up to $\xi /L \approx 0.145$ ($Re_{\delta _1} = 6500$). We end our analysis upstream of incipient separation ($\xi /L \approx 0.19$, $Re_{\delta _1} \approx 11\,000$). The detailed physics of retransition can be sensitive to grid resolution (Shur et al. Reference Shur, Spalart, Strelets and Travin2021), given the role of smaller-scale flow structures in re-establishing a fully turbulent BL. Our analysis based on the AMI equation does not include statistics of higher order than the wall-normal integral of the Reynolds shear stress. It is instructive to explore the physical response of turbulent APG BLs to different upstream histories using to the AMI equation, even if the quantitative details of the retransition history in the Bump simulation could be sensitive to grid resolution.

The primary components of the AMI equation are illustrated in figure 6 for each dataset using the associated colours given in table 1. Notably, the laminar friction in figure 6(a) is the same across all cases due to the choice of $\ell$ based on the Blasius solution at a matched $Re_{\delta _1}$. This gives a unified baseline laminar BL with which all APG turbulent BL datasets are compared in the AMI analysis. As may be expected, the magnitude of the laminar skin friction is fairly small compared with the other terms in the AMI equation.

Figure 6. The AMI budget within the APG region with respect to $Re_{\delta _1}$. Contribution of the substantial flow phenomena impacting $C_f/2$: (a) laminar friction, (b) turbulent torque, (c) torque due to pressure gradient and (d) torque due to total mean flux. (e) The contribution to $C_f/2$ by the sum of (c) and (d).

In contrast, the turbulent torque is a dominant contribution to the skin friction throughout the APG region (figure 6b). For each of the flat plates, the streamwise variation of the turbulent torque is quite limited, with the highest variation at only $8\,\%$, associated with case m18. This is not surprising, given that the BLs in these cases experience relatively constant values of the Clauser parameter. More notably, the turbulent torques of the different flat-plate cases largely collapse on top of each other. For instance, the difference in the turbulent torque between cases $\beta 1$ and m18 never exceeds $7\,\%$, despite rather significant differences in the Clauser parameter (table 1). Meanwhile, the turbulent torque for the Wing BL also exhibits a low amount of streamwise variation, only $13\,\%$ over the investigated region, despite the Clauser parameter ranging from $\beta \approx 0$ to $\beta \approx 40$. The turbulent torque, however, shows more variation in the Bump case. As the effects of relaminarization are reversed, the early APG region, the turbulent torque experiences a substantial increase of about $63\,\%$ in the range $1400 \leq Re_{\delta _1} < 2000$. A similar rapid growth of the turbulent torque occurs during transition to turbulence (Kianfar et al. Reference Kianfar, Elnahhas and Johnson2023b). Following this retransition in the Bump case, the turbulent torque changes relatively slowly, increasing only by $20\,\%$ from $Re_{\delta _1} = 2000$ to $Re_{\delta _1} = 6500$. Together with figure 7(a), these observations suggest that the turbulent torque does not depend strongly on the strength of the APG, but is more sensitive to other details of the flow configuration such as an upstream history of relaminarization.

Figure 7. The Clauser parameter with respect to $Re_{\delta _1}$ (a) based on $\delta _1$ (the classic definition) and (b) based on $\delta ^\ell _1$ obtained from the AMI analysis. The black dots denote the matching locations at which turbulent statistics are compared according to table 2.

Figure 6(c) shows that the pressure-gradient torques in the AMI equation are generally negative because the APG acts to attenuate the skin friction coefficient. One notable exception in the present results is the positive torque of the APG ($\beta > 0$) in the range $1400 \leq Re_{\delta _1} < 2500$ for the Bump case. Mathematically, this reflects that $\delta _{1}^{\ell }$ becomes negative in this region. Physically, it suggests that the comparison with a laminar BL at the same $Re_{\delta _1}$ captures the historical influence of FPG in some way. Based on our current analysis, it is not clear whether or not this is related to relaminarization. For the flat-plate cases, there is little streamwise variation of the pressure-gradient torque, as may be expected. Further downstream in the Bump case, as also in the Wing case, the skin friction attenuation of the pressure-gradient torque grows in magnitude, competing with and eventually overtaking the magnitude of the turbulent enhancement in figure 6(b). The battle between the turbulent torque and APG torque (see figure 1) lies at the heart of the dynamics leading to BL separation, and figure 6(b,c) illustrates the ability of the AMI equation to produce a simple quantitative account of this competition.

The mean flux term in the AMI equation, figure 6(d), encapsulates the effect of mean wall-normal velocity and streamwise BL growth on the skin friction coefficient. This term often opposes (and partially cancels) the prevailing trends from other terms in the AMI equation, such as the growth of the turbulent torque during transition to turbulence. In ZPG turbulent BLs, the mean flux attenuates the skin friction by absorbing some of the turbulent torque into an increase of the BL's moment of momentum by increasing the physical thickness of the BL. In the flat-plate APG cases, the mean flux term is also negative, in opposition to the turbulent torque, which is stronger than the pressure gradient torque. The mean flux varies little in the streamwise direction for these cases, following the trend in the pressure-gradient and turbulent torques. In the Wing and Bump cases, the mean flux follows the inverse trend of the pressure-gradient torque and undergoes a sign change near the location where the pressure-gradient torque overtakes the turbulent torque in magnitude. This signals that the mean flux transitions into a regime primarily characterized by absorption of the pressure-gradient torque into streamwise changes in the moment of momentum as well as a growth in the mean wall-normal velocity. Finally, as turbulence is restored after relaminarization between $Re_{\delta _1}=1400$ and $Re_{\delta _1}=2000$ in the Bump case, the mean flux generates an increasingly negative contribution to resist the growing turbulent torque. A similar observation has been made for ZPG transitional BLs (Elnahhas & Johnson Reference Elnahhas and Johnson2022; Kianfar et al. Reference Kianfar, Elnahhas and Johnson2023b).

Given the opposite trends of pressure gradient and mean flux in figure 6(c,d), the summation of these two terms is explored in figure 6(e). In each BL case, this combination reaches an approximately streamwise-constant value (for $\ell = 1.75 \delta _1$) even for the Wing and Bump BLs undergoing strong streamwise development forced by a strong APG. This suggests a picture in which, in addition to the momentum transport of turbulence, the pressure-gradient torque must also generate significant mean wall-normal velocity and streamwise BL growth to force the BL to separate.

4.5. Adverse pressure gradients: AMI-based Clauser-like parameter

The commonly used Clauser parameter, (2.14), emerges from the momentum integral equation, (2.12), as the ratio of the pressure-gradient term and the skin friction coefficient. Similarly, the ratio of the pressure-gradient torque in the AMI equation, figure 6(c), and the skin friction coefficient leads to a new AMI-based Clauser parameter, $\beta _\ell$. This AMI-based Clauser-like parameter physically represents the fractional attenuation of the skin friction coefficient due to the direct effect of the APG imposed by the free stream, in competition with the turbulent stress. Mathematically, $\beta _\ell$ provides this streamlined interpretability by careful treatment of the linear dependence of the free-stream source/sink term on the defect velocity in (2.5) in terms of the implied impact on how momentum is (re)distributed in the wall-normal direction by the imposed free-stream pressure-gradient force (see figure 1). As such, it is of interest to explore the similarities and differences relating to the use of $\beta$ and $\beta _\ell$ to characterize APG turbulent BLs.

Figure 7 illustrates the trends of $\beta$ and $\beta _\ell$ over the range of $Re_{\delta _1}$ explored in figure 6. The overall trends of $\beta$ and $\beta _\ell$ are similar; however, $\beta$ spans from $0$ to $40$, and $\beta _\ell$ yields smaller values in the range $0 < \beta _\ell < 20$ because the $1 - y/\ell$ weighting in the integral for $\delta _1^\ell$ reduces it compared with $\delta _1$. Additionally, the growth rates of $\beta$ and $\beta _\ell$ differ, leading to different intersection (or matching) locations between the datasets. The most significant difference in behaviour between $\beta$ and $\beta _\ell$ corresponds to the Bump dataset, where the plot of $\beta _\ell$ versus $Re_{\delta _1}$ shifts to the right compared with the plot of $\beta$. The differences in BL characterization between $\beta$ and $\beta _\ell$ observed in figure 7 raises the question of how an analysis of history effects may be affected by taking into account the AMI equation. That is, if two BLs have the same Reynolds number and $\beta$ or $\beta _\ell$, to what degree do different upstream conditions alter the state of the BL at the matching point?

The matching location has been the subject of several studies examining the pressure gradient history effects on turbulent statistics. Specifically, Bobke et al. (Reference Bobke, Vinuesa, Örlü and Schlatter2017) chose the matching $Re_\tau$–$\beta$ to examine the upstream pressure gradient history effect. Some of the differences observed in that study can be accounted for using a Reynolds number based on the displacement thickness rather than the local wall shear stress. To deepen our understanding of the similarities and differences between APG BLs having various upstream histories, including the mean streamwise velocity and Reynolds stress components, matching locations for $Re_{\delta _1}$–$\beta$ and $Re_{\delta _1}$–$\beta _\ell$ between the different APG BL datasets are explored in this subsection. All such matching points are recorded in table 2.

Table 2. Matching positions between $Re_{\delta _1}$–$\beta _\ell$ and $Re_{\delta _1}$–$\beta$ for the dataset in table 1.

Furthermore, figure 8 illustrates profiles of the inner-scaled mean streamwise velocity, $\bar {u}$, and Reynolds stress components, $\overline {u^\prime _i u^\prime _j}$, for matching points V (Wing–m18), X (Bump–m18) and XI$_6$ (Wing–Bump) based on both $\beta _\ell$ and $\beta$. These corresponding locations are indicated in figure 7 with details provided in table 2. In all these plots, the $x$ axis represents the wall-normal position normalized by $\ell = 1.75 \delta _1$, while the insets present the profiles with respect to $y^+$. In figure 8 the bold lines depict the profiles at matching points for $Re_{\delta _1}$–$\beta _\ell$, and faded colours represent profiles from matching points for $Re_{\delta _1}$–$\beta$.

Figure 8. The APG history effect on inner-scaled flow statistics with respect to $y/\ell$ at matching $Re_{\delta _1}$–$\beta _\ell$ ($Re_{\delta _1}$–$\beta$ faded lines) for (a,b) V, $3064$–$1.48$ ($2872$–$3.45$); (c,d) X, $3232$–$1.57$ ($1638$–$1.54$); and (e,f) XI$_6$, $6245$–$18.51$ ($6404$–$37.54$). (a,c,e) The mean velocity and (b,d,e) the Reynolds stress components, where the solid and dotted lines represent $\overline {u^\prime u^\prime }^+$ and $-\overline {u^\prime v^\prime }^+$, respectively. The insets exhibit the same profiles versus $y^+$. Symbol $\Diamond$ denotes the wall-normal position of $\delta _{99}$.

As a representative of the comparison of the Wing and flat-plate datasets, figure 8(a,b) shows the mean velocity and Reynolds stress profiles at matching point V using both $\beta$ (faded lines) and $\beta _\ell$ (bold lines). When matching based on $Re_{\delta _1}$–$\beta _\ell$, the mean velocity profiles from the two datasets are more similar to each other than when matching $Re_{\delta _1}$–$\beta$. The increased similarity when using $Re_{\delta _1}$–$\beta _\ell$ is also evident in the Reynolds stress profile, though less complete. The streamwise variance exhibits a secondary peak in the outer layer in addition to the buffer layer peak observed near the wall. This secondary peak is strongly influenced by the pressure gradient, and its location and magnitude vary depending on the upstream pressure-gradient history (Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017). The improved similarity between the profiles from the two datasets when matching based on $\beta _\ell$ rather than $\beta$ does suggest that the AMI-based analysis does alter the perceived effect of upstream history, reducing it at least in some cases. The magnitude of the peak Reynolds stress in the outer layer is smaller for the Wing case, presumably due to the fact that it has experienced a weaker pressure gradient upstream than in the case of the flat-plate BL (see figure 7). It is worthwhile to note in passing that more robust similarity of Reynolds stress profiles was observed when comparing the Wing case with flat-plate cases having weaker pressure gradients, such as the $\beta 1$ or m16 cases (not shown).

As shown in figure 8(c,d), the differences between the Bump and flat-plate profiles are more significant, presumably due to the larger differences in upstream history stemming from the FPG and relaminarization in the Bump case. Compared with the $\beta _\ell$ matching location, the matching point using $\beta$ occurs significantly upstream, before the streamwise variance develops the secondary outer peak. The spanwise and wall-normal variances are not shown for reasons related to the readability of the figure. Therefore, the use of the AMI-based $\beta _\ell$ parameter has a larger effect on the comparison of the Bump and flat-plate datasets, significantly reducing the perceived history effects.

The matching location of the Wing and Bump datasets, XI$_6$, occurs further downstream at $Re_{\delta _1}=6242$ for $\beta _\ell$ and $Re_{\delta _1}=6404$ for $\beta$. The associated profiles for these locations are shown in figure 8(e,f). There, an inner peak of the streamwise variance is no longer observed. In the Wing–Bump comparison, there is not a significant difference between the matching based on $\beta$ or $\beta _\ell$. Note that the AMI-based analysis of the Wing–Bump comparison coincidentally leads to a significant region of overlap, permitting the investigation of a case when the two BLs share the same history for some distance along the surface.

To further investigate, figure 9 displays the skin friction coefficient at matching $Re_{\delta _1}$–$\beta _\ell$ (and $Re_{\delta _1}$–$\beta$ with faded colour) for the intersections (table 2) between the Wing and flat plates and Bump and flat plates. The comparison of the Wing BL with flat plate cases in figure 9(a) shows that both $\beta$ and $\beta _\ell$ indicate relatively weak history effects, especially in the case of $\beta _\ell$ for stronger pressure gradients. In comparison, figure 9(b) shows significant differences in $C_f$ for the matching locations between the Bump and flat plate cases when matched based on $\beta$ and $Re_{\delta _1}$, presumably due to significant differences in upstream history. On the other hand, the history effects do not seem as severe from the AMI-based perspective. Matching based on $\beta _\ell$ shows a significantly lower sensitivity of $C_f$ to differences between the Bump and flat-plate BLs.

Figure 9. Skin friction coefficient at the streamwise location of matching $Re_{\delta _1}$–$\beta _\ell$ (and $Re_{\delta _1}$–$\beta$ faded colour) according to table 2. A comparison between (a) Wing and flat plates and (b) Bump and flat plates.

Finally, the skin friction coefficient comparison is carried out for the Wing and Bump cases in figure 10. Figure 7 shows that there is a single matching point in $Re_{\delta _1}$–$\beta$ space, whereas in plots of $\beta _\ell$, the Wing and Bump lines continuously overlap each other for a segment starting at $Re_{\delta _1} = 3712$ ($\beta _\ell =3.05$) until $Re_{\delta _1} = 6245$ ($\beta _\ell =18.51$). The skin friction coefficients between the two cases converge towards each other throughout the interval with overlapping $\beta _\ell$, such that they are virtually indistinguishable by the point that $\beta$ also matches. This result shows the gradual ‘forgetting’ of upstream history (in terms of its influence on $C_f$) by the BL over the length of surface where the two BLs experience the same $\beta _\ell$. Note that figure 8(e,f) shows that the mean velocity and Reynolds stress profiles still retain some differences even after this period of forgetting.

Figure 10. Skin friction coefficients for the Wing and Bump datasets at the streamwise location of matching $Re_{\delta _1}$–$\beta _\ell$ ($Re_{\delta _1}$–$\beta$ faded colour).