Pressure-Gradient Turbulent Boundary Layers Developing Around a Wing Section

The flow around wings is of large interest, both from a scientific and from a practical/industrial point of view. The different physical mechanisms taking place, i.e., laminar-turbulent transition, wall-bounded turbulence subjected to pressure gradient and wall curvature, flow separation and turbulence in the wake, are highly coupled and therefore the resulting flow configuration is complex. As a consequence, the aeronautical industry has traditionally relied heavily on experimental findings and rules of thumb derived from experience for design purposes. A recent report by NASA [2] discusses a number of findings and recommendations regarding the present and future role of CFD (computational fluid dynamics), and points out the necessity of accurate predictions of turbulent flows with significantly separated regions. Since Reynolds-Averaged Navier–Stokes (RANS) simulations, widely used in industry, generally fail to predict such configurations, other numerical approaches such as direct numerical simulation (DNS, where all the turbulent scales are resolved) and large-eddy simulation (LES, which relies on modeling only the smallest, more universal scales in the flow) are the best options to complement experiments and gain insight into the physics taking place in wings and airfoils.

Twenty years ago Jansen [3] performed one of the first structure-resolving simulations of the flow around wings: an LES of the cambered NACA4412 profile at a Reynolds number of R e _c = 1.64 × 10⁶, based on the freestream velocity U _∞ and the chord length c. A total of three experimental datasets of the same configuration [4‐6] were used for comparison, and while in the first experiment it was found that the angle of attack of maximum lift was 13.87^∘, in the other two the reported angle was 12^∘. In his second study, Wadcock [6] claimed that the previous one suffered from a non-parallel mean flow in the wind tunnel, which caused the different critical angle of attack. The idea behind the LES by Jansen [3] was to test this numerically, although the computational resources available at the time did not allow him to obtain good agreement with the experiments. Note that his LES was based on a low-order finite-element method, and one of his main conclusions was that accurate simulations of that flow would require a high-order numerical method. Additional factors, such as low resolution in particular in the near-wall boundary-layer region, the use of explicit LES based on the dynamic model, and generally limited computational resources available at the time, also contributed to this discrepancy. A more recent example of the difficulties of matching experiments and computations of the flow around wing sections is the work by Olson et al. [7], who studied separation and reattachment locations on a SD7003 airfoil at different angles of attack at low R e _c values from 20,000 to 40,000. They performed multi-line molecular tagging velocimetry measurements, and although their implicit LES was based on a sixth-order compact finite-difference scheme, in their case they found that several facility-dependent issues (such as the freestream turbulence level) significantly affected the results, and therefore they did not achieve good agreement between the various datasets. Another numerical study on wings is the DNS of the flow around the symmetric NACA0012 wing profile carried out by Shan et al. [8] at R e _c = 100,000 and 4^∘ angle of attack. A very interesting conclusion from their work, based on a sixth-order compact finite-difference scheme, was that the backward effect of the disturbed flow on the separated region may be connected to the self-sustained turbulent flow and the self-excited vortex shedding on the suction side of the wing. Another relevant finding from their study was the fact that the vortex shedding from the separated free-shear layer was due to a Kelvin–Helmholtz instability. Direct numerical simulations of the same profile were performed by Rodríguez et al. [9] at a lower Reynolds number of R e _c = 50,000 and larger angles of attack of 9.25^∘ and 12^∘. Based on a second-order conservative scheme, they found that the massive separation observed on the suction side of the wing was due to a combination of leading edge and trailing edge stall.

Another interesting phenomenon in wings is the so-called laminar separation bubble (LSB), which takes place when the laminar boundary layer detaches from the wing surface due to the adverse pressure gradient (APG) induced by the wall curvature. In the separated region disturbances are greatly amplified, which may lead to transition to turbulence, and the resulting turbulent flow exhibits larger momentum close to the wall therefore reattaching downstream. LSBs, which lead to increased drag and may determine stall behavior [10], were studied numerically on the NACA0012 wing profile by Jones et al. [11, 12] and by Alferez et al. [13]. The work by Jones et al. was based on DNS at R e _c = 50,000, with 5^∘ angle of attack, and employed a fourth-order finite-difference numerical method. On the other hand, Alferez et al. [13] performed an LES at R e _c = 100,000 based on a second-order finite-volume method, and they considered a pitch-up motion starting from an angle of attack of 10.55^∘, up to 10.8^∘. Rosti et al. [14] performed DNSs of the flow around a NACA0012 wing profile undergoing a ramp-up motion, with angles of attack ranging from 0^∘ to 20^∘, at R e _c = 2 × 10⁴. They used a second-order finite-volume code, and performed coherent-structure and Lyapunov-exponent analyses on the flow. The impact of LSBs on the aerodynamic performance of the NACA0012 profile was reported by Gregory and O’Reilly [15], who performed measurements at R e _c = 1.44 × 10⁶ and 2.88 × 10⁶ over a range of angles of attack, and observed that in their experiments the LSB disappeared intermittently, significantly affecting the results. The backflow present in turbulent wings at R e _c = 400,000, and its varying features for increasing pressure-gradient magnitudes, was analyzed numerically by Vinuesa et al. [16].

From the perspective of wall-bounded turbulence, the boundary layers developing over the suction and pressure sides of a wing section are complex since they are affected by a pressure gradient (PG), and by wall curvature. Although the zero-pressure-gradient (ZPG) turbulent boundary layer (TBL) has received a great deal of attention in the turbulence community (good examples are the experimental studies by Österlund [17] and Bailey et al. [18], or the numerical work by Schlatter and Örlü [19] and Sillero et al. [20]), the TBL driven by a non-uniform freestream velocity U _∞ has not been studied in such level of detail. One of the first studies where PG TBLs were assessed was the work by Coles [21], where he introduced the “law of the wake”. Among other datasets, he analyzed two sets of measurements on airfoils approaching separation [22, 23] and one on an airfoil following reattachment [24]. In a more recent study, Skåre and Krogstad [25] performed measurements on a TBL subjected to a strong APG and found a second peak in the production located in the outer region of the boundary layer, which was responsible for significant diffusion of turbulent energy towards the wall. The magnitude of the pressure gradient can be quantified in terms of the Clauser pressure-gradient parameter β = δ ^∗/τ _w dP _e /dx _t , defined in terms of the displacement thickness δ ^∗, the mean wall-shear stress τ _w and the gradient of the pressure at the boundary-layer edge P _e in the direction tangential to the wing surface (direction defined by the x _t coordinate). In the experiments by Skåre and Krogstad [25] the β parameter ranged from 12 to 21, over a range of Reynolds numbers based on momentum thickness 25,000 < R e _𝜃 < 54,000. Monty et al. [26] also found interesting pressure-gradient effects in the outer region of APG TBLs in their experimental study, in which they showed that the large-scale structures from the outer flow were energized by the PG, which led to the increase in streamwise turbulence intensity across the boundary layer. Their study was limited to a lower Reynolds number range 5,000 < R e _𝜃 < 19,000, and to more moderate APGs with β values between 0.8 and 4.75. Their results were extended to production and Reynolds shear-stress by Harun et al. [27], who also analyzed the effect of an FPG and performed spectral and scale-decomposition analyses. Maciel et al. [28] developed a theory of self-similarity and equilibrium in the outer region of APG TBLs, for which they considered the Zagarola–Smits [29] outer scaling and introduced a new pressure-gradient parameter. Simple APG and FPG configurations, as well as consecutive sequences of APG and FPG, were studied experimentally over 10,000 < R e _𝜃 < 40,000 and − 0.5 < β < 0.5 by Nagib et al. [30] and Vinuesa et al. [31], respectively. Note that although most contributions are of experimental nature, PG TBLs have also been studied through DNS in the recent years: Spalart and Watmuff [32], Skote et al. [33], Lee and Sung [34], Piomelli and Yuan [35], Gungor et al. [36] and Kitsios et al. [37]. A numerical experiment by Maciel et al. [38] has revealed that in progressively stronger APGs, the coherent structures of turbulence tend to be shorter, less streaky and more inclined with respect to the wall than in ZPG. The interaction of the larger-scale motions with the outer flow, and in particular the assessment of history effects on the development of the TBL, is the focus of the recent numerical study by Bobke et al. [39].

In the present study we use a DNS database [1] of the flow around a NACA4412 wing profile, at R e _c = 400,000 and 5^∘ angle of attack, to assess the effects of APGs and FPGs on the TBLs developing around the wing section. The relevance of this work lies in the significantly higher Reynolds number compared with other studies, the additional flow complexity introduced by the cambered airfoil, and the use of high-order spectral methods for the simulations. Whereas in the previous article by Hosseini et al. [1] we focused on the numerical aspects and on the description of the computational setup, in the present work we emphasize the characteristics of the TBLs developing on both the suction and pressure sides of the wing section. This is a very interesting case from the fluid mechanics perspective, since the suction-side TBL is subjected to an exponentially-increasing APG, which significantly modifies the structure of wall-bounded turbulence. Also, given the importance of history effects on the state of the flow documented by Bobke et al. [40], it is essential to provide high-quality TBL data that can be used to evaluate such effects. On the other hand, the pressure-side TBL is subjected to a mild FPG, and its analysis provides an interesting characterization of a flow close to the widely-studied ZPG TBL, but still subjected to the effect of history.

The article is structured as follows: the details of the computational setup are provided in Section 2; the turbulence statistics of the two TBLs are discussed in Section 3; the spectral analysis performed on the TBLs developing on the suction and pressure sides of the wing is presented in Section 4; and a summary of the article together with the main conclusions can be found in Section 5.

In order to further assess the characteristics of the boundary layers developing around the wing section, their energy distribution is studied through the analysis of the inner-scaled spanwise premultiplied power-spectral density of the streamwise velocity \(k_{z} {\Phi }_{u_{t} u_{t}}^{+}\), shown at x/c = 0.4, 0.8 and 0.9 in Fig. 8 for both sides of the wing. The first feature of these spectra is the fact that all of them exhibit the so-called inner-peak of spectral density, at a wall-normal distance of around \(y_{n}^{+} \simeq 12\), and for wavelenghts of around λ z+ ≃ 120. This was also observed in the LES of a ZPG TBL by Eitel-Amor et al. [66] up to a much higher R e _𝜃 value of around 8,300, and is a manifestation of the inner peak of the streamwise velocity fluctuation profile discussed in Section 3.4. In fact, the value of this inner peak is also highly affected by the pressure gradient: on the suction side, and at x _{s s} /c = 0.4, the inner peak in spectral density is around 4, close to the value in ZPG boundary layers. As β increases this inner peak also becomes amplified, reaching a value of around 5 at x _{s s} /c = 0.8, and up to around 6 at x _{s s} /c = 0.9, behavior which again strongly resembles the one of the streamwise velocity fluctuations, and highlights the connection between the coherent structures in the boundary layer and the turbulence statistics. Moreover, the wavelength \(\lambda _{z}^{+} \simeq 120\) corresponds to the characteristic streak spacing in wall-bounded turbulence, as shown for instance by Lin et al. [67]. The power-spectral density distributions shown in Fig. 8 also show that the computational domain appears to be large enough in the spanwise direction to capture the contributions of the relevant turbulent scales in both boundary layers. However, the spanwise width of L _z = 0.1c will probably not be sufficient to simulate the boundary layers in cases with significant separation and stalled regions. Moreover, instantaneous flow visualizations [1] also suggest that the domain might not be sufficiently wide to accurately simulate the flow physics in the wake.

Regarding the spectra in the outer region of the boundary layer, on the pressure side there is a slight development with Reynolds number, leading to accumulation of energy in progressively larger scales. As will be discussed below, since this boundary layer is subjected to a mild FPG, the energy levels are slightly below the ones corresponding to a ZPG boundary layer, but the development of the outer region is comparable in terms of R e effects. Interestingly, the spectral distribution observed at x _{s s} /c = 0.4, which is subjected to a moderate APG of β = 0.6, is comparable to the one observed at x _{p s} /c = 0.9, although the Reynolds number is lower (R e _τ = 242 on the upstream location from the suction side, whereas the one on the downstream location in the pressure side is R e _τ = 317). This is a first indication of the effect of the APG energizing the large-scale motions in the outer lager in a similar way as it is done by the increase of R e. Further streamwise development on the suction side shows how an outer peak emerges at x _{s s} /c = 0.8 (subjected to a strong APG of β = 4.1), with a value of inner-scaled power-spectral density of around 4. The very strong APG found at x _{s s} /c = 0.9, where β has a value of 14.1, leads to a power-spectral density level in the outer region larger than the one in the inner region of the boundary layer, with an inner-scaled value of around 8. The connection with the streamwise velocity fluctuation profiles is again clear in the development of the outer region, since at x _{s s} /c = 0.8 the outer peak is also slightly below the inner one (but of the same magnitude as the inner peak in a ZPG boundary layer), and at x _{s s} /c = 0.9 also in the \(\overline {{u^{2}_{t}}}^{+}\) profile the outer peak is larger than the inner one. Therefore, the progressively stronger APG leads to relatively more intense large-scale motions in the flow, which on the other hand have a footprint in the near-wall region [27] responsible for the increase of energy in the buffer layer with respect to the ZPG, when scaled in viscous units. The emergence of this outer spectral peak was also observed by Eitel-Amor et al. [66] in their ZPG simulations at much higher Reynolds numbers, with an emerging outer peak at R e _𝜃 ≃ 4,400 which started to become more prominent at around R e _𝜃 ≃ 8,300. Note that in their case the spectral-density level in the outer region was significantly lower than the one in the inner region, and therefore much higher Reynolds numbers would be necessary in a ZPG boundary layer in order to reach similar levels of energy in the outer region. On the other hand, Eitel-Amor et al. [66] observed the emergence of the outer spectral peak at around λ _z ≃ 0.8δ ₉₉, whereas the results in Fig. 8 show that on the suction side of the wing the outer peak emerges at around λ _z ≃ 0.65δ ₉₉. Due to the significantly lower Reynolds numbers present on the wing, it is difficult to assess whether this difference in the structure of the outer region is due to a fundamentally different mechanism in the energizing process of the large-scale motions from APGs and high- R e ZPGs, or whether this is due to low- R e effects. In any case, and as also noted by Harun et al. [27], the effect of the pressure gradient on the large-scale motions in the flow has features in common with the effect of R e in ZPG boundary layers [64], and therefore further investigation at higher Reynolds numbers would be required to separate pressure-gradient and Reynolds-number effects.

A more quantitative assessment of the differences between the spectra computed in the wing and the ones obtained from ZPG boundary layers is shown in Fig. 9. In this figure, we subtract the ZPG \(k_{z} {\Phi }_{u_{t} u_{t}}^{+}\) contours from the ones computed in the wing, after interpolating on the same \(y^{+}_{n}\) and \(\lambda _{z}^{+}\) sets of values. It can be observed that at x _{s s} /c = 0.8 near the wall, i.e., for \(y^{+}_{n} < 10\), the APG boundary layer exhibits slightly larger energy levels than the ZPG, a fact that was also noticeable in the \(\overline {{u^{2}_{t}}}^{+}\) profile. Near the inner-peak region at \(y^{+}_{n} \simeq 12\) and λ z+ ≃ 120 (and also at longer wavelenghts with \(\lambda _{z}^{+} \simeq 200\)), the spectral-density level of the APG is slightly below the one from the ZPG, by a small difference of around 0.1. This was also observed by Harun et al. [27] in their streamwise spectra \(k_{x} {\Phi }_{u u}^{+}\) from moderate APG boundary layers, although it is also important to highlight that for shorter wavelengths, with \(\lambda _{z}^{+} < 100\), the spectral density at x _{s s} /c = 0.8 again exceeds the one from ZPG, with differences from 0.8 to 1. In fact, the \(\overline {{u^{2}_{t}}}^{+}\) profile exhibits a larger inner peak at x _{s s} /c = 0.8 than the one from ZPG, and the integrated value over all the wavelengths of the difference \(\left [ k_{z} {\Phi }_{u_{t} u_{t}}^{+} \right ]_{x_{ss}/c=0.8} - \left [ k_{z} {\Phi }_{u u}^{+} \right ]_{\text {ZPG}}\) at \(y^{+}_{n} = 12\) is ≃ 0.3. This is indeed in agreement with a slightly larger energy value in the inner peak under this β condition, and suggests that the APG also affects the structure of the near-wall region, by concentrating energy in slightly shorter wavelengths. The development of an outer peak is also noticeable at this location, with a significant difference in spectral density of around 2.5 in the outer region. These effects are also observed at x _{s s} /c = 0.9, although in this case the minimum difference between the wing spectral-density distribution and the the one of the ZPG case (also found at the location of the inner peak with y n+ ≃ 12 and \(\lambda _{z}^{+} \simeq 120\)), is positive although very close to zero. At y n+ ≃ 12 there is again concentration of energy in the wavelengths shorter than around 100, and as in the previous case the spectral density is larger for \(y^{+}_{n}<10\) than in the ZPG boundary layer. The very prominent spectral outer peak shows a large difference of around 7.5 with respect to the ZPG, which again highlights the effect of the APG energizing the outer region of the boundary layer.

Regarding the spectral-density distributions in the pressure side of the wing, the profiles at x _{p s} /c = 0.8 and 0.9 are very similar, with very small differences presumably due to Reynolds-number effects. Firstly, the maximum value is zero, which means that the TBL subjected to the slight FPG is less energetic than the ZPG when scaled in viscous units, as also observed in the \(\overline {{u^{2}_{t}}}^{+}\) profiles. The largest differences are found in the near-wall region, where the inner peak exhibits a value below the ZPG one by around 0.9, a fact that is in agreement with Piomelli and Yuan [35] who discussed the FPG effect in the stabilization of the near-wall streaks. Differences are also noticeable in the outer region, at wavelenghts of the order of the boundary-layer thickness, where the corresponding level of energy is around 0.5 units below the one of the ZPG, highlighting the presence of less energetic large-scale motions when scaled with \(u_{\tau }^{2}\).

In the present study we analyze a DNS database [1] of the flow around a NACA4412 wing section with R e _c = 400,000 and 5^∘ angle of attack. Turbulence statistics were computed at a total of 80 locations over the suction and pressure sides of the wing, and expressed in the directions tangential and normal to the wing surface. The Clauser pressure-gradient parameter β increases monotonically from ≃ 0 to 85 on the suction side, and varies non-monotonically from around 0 to − 0.25 on the pressure side. Therefore, the TBL on the suction side is subjected to a progressively stronger APG with x, whereas the pressure gradient is slightly favorable on the pressure side (with a small section subjected to a mild APG). The R e _τ curves are monotonically increasing on the pressure side, whereas on the top surface it reaches a maximum at x _{s s} /c ≃ 0.8, and decreases after this point. This is due to the fact that, although the boundary-layer thickness still increases with x, the decrease in friction velocity is much larger. Moreover, comparisons of H and C _f curves from both sides and with ZPG TBLs reveal the effect of the APG, i.e., increased boundary-layer thickness (and therefore larger H), and reduced skin-friction coefficient. The FPG has the opposite effect on the TBL, and although the magnitude of the FPG is quite small, subtle effects include decreased boundary-layer thickness, lower H, and increased C _f with respect to the ZPG.

We further assessed the effect of the APG on the TBLs by comparing inner-scaled mean profiles at x _{s s} /c = 0.4, 0.8 and 0.9 with the ZPG boundary-layer data from Schlatter and Örlü [19] at matching R e _τ . The corresponding β values are 0.6, 4.1 and 14.1, which approximately correspond to the pressure-gradient magnitudes obtained in the experiments by Vinuesa et al. [31], Monty et al. [26] and Skåre and Krogstad [25], respectively. The first effect of the APG on the mean flow is the emergence of a more prominent wake, reflected in a higher \(U^{+}_{e}\) and a larger wake parameter Π. In addition to this, the APG produces a steeper overlap region, which is characterized by lower values of the von Kármán coefficient κ and the logarithmic-law intercept B, as well as by lower inner-scaled velocities in the buffer region. These effects, which were also observed by Monty et al. [26], are due to the fact that the APG leads to relatively more intense large-scale motions in the flow (through the development of a more prominent outer region), which become shorter and more elongated, and have their footprint in their near-wall region. Also, these manifestations of the APG become more evident as β increases. Moreover, comparisons of several components of the Reynolds-stress tensor showed a progressive increase (when scaled in viscous units) in the value of the inner peak of the streamwise velocity fluctuation profile, and the development of an outer peak which in the strong APG case (β ≃ 14.1) exceeds the magnitude of the inner peak. These effects were also observed by Monty et al. [26] and Skåre and Krogstad [25]. Note that the development of a more energetic outer region with increasing β is also observed in the wall-normal and spanwise fluctuation profiles, as well as in the Reynolds-shear stress profile. Comparison of the TKE budgets also shows the differences in energy distribution across the boundary layer when an APG is present, with increased production and dissipation profiles throughout the whole boundary layer. The emergence of an incipient outer peak in the production profile is observed at β ≃ 14.1, phenomenon which was also reported by Skåre and Krogstad [25]. The increased dissipation is accompanied by larger values of the viscous diffusion and the velocity-pressure-gradient correlation near the wall in order to balance the budget. Regarding the impact of the FPG on the TBL statistics, it basically has the opposite effect as the APG, as also observed by Harun et al. [27]. And since the magnitude of β is small in the pressure side of the wing, the effect of the FPG is quite subtle at all the locations under consideration. Thus, the wake region is slightly less prominent than the one from the ZPG, and \(U^{+}_{e}\) is lower due to the increased skin friction. A higher value of κ is also observed, which leads to a less steep overlap region, and the value of the inner peak in the \(\overline {{u^{2}_{t}}}^{+}\) profile is also attenuated. This is related, together with the decrease of all the Reynolds-stress tensor components in the outer region, with the fact that the FPG leads to less energetic large-scale motions in the flow. This is also confirmed by the TKE budgets, which essentially show a decrease in production and dissipation across the boundary layer.

Analysis of the inner-scaled premultiplied spanwise spectra showed the presence of the inner spectral peak at around y n+ ≃ 12 and \(\lambda _{z}^{+} \simeq 120\), in agreement with the observations by Eitel-Amor et al. [66] in ZPG TBLs at higher R e _𝜃 of around 8,300. As the inner peak of \(\overline {{u^{2}_{t}}}^{+}\), the spectral near-wall peak increases with the magnitude of the APG, as a consequence of the energizing process of the large structures in the flow, which have their footprint at the wall. Also as a consequence of this energizing process an outer spectral peak emerges at strong APGs with β ≃ 4.1; note that this outer spectral peak corresponds to the larger outer-region values in all the components of the Reynolds-stress tensor. The spectral outer peak is observed at wavelengths of around λ _z ≃ 0.65δ ₉₉, closer to the wall than the outer peak observed at R e _𝜃 ≃ 8,300 by Eitel-Amor et al. [66] in the ZPG case, at λ _z ≃ 0.8δ ₉₉. At this point it is not possible to state whether this difference arises from low- R e effects, or from a mechanism of energy transfer to the larger scales fundamentally different between high- R e ZPG TBLs and APGs. On the other hand, the effect of the FPG on the spectral-density distributions is the opposite, i.e., to reduce energy levels both in the inner and outer regions of the boundary layer, in agreement with what was observed in the streamwise velocity fluctuation profiles.

The novelty of the present work lies in the use of high-order spectral-element methods to characterize the TBLs developing on the suction and pressure sides of a wing section, at a moderate Reynolds number of R e _c = 400,000. We have documented in detail the characteristics of the boundary layers, including Reynolds-stress-tensor components, TKE budgets and spectra. Moreover, we have provided a high-quality database for the study of PG effects on TBLs, and the assessment of the impact of history on the state of the TBL, as discussed by Bobke et al. [39]. Future studies at higher Reynolds numbers will be aimed at further assessing the connections between the effect of APGs on the large-scale motions in the flow and the effect of R e in ZPG boundary layers, as also suggested by Harun et al. [27], in order to separate pressure-gradient and Reynolds-number effects.