In this study, a new well-resolved large-eddy simulation of an incompressible near-equilibrium adverse-pressure-gradient (APG) turbulent boundary layer (TBL) over a flat plate is presented. In this simulation, we have established a near-equilibrium APG over a wide Reynolds-number range. In this so-called region of interest, the Rotta-Clauser pressure-gradient parameter beta exhibits an approximately constant value of around 1.4, and the Reynolds number based on momentum thickness reaches Re-theta = 8700. To the best of the authors' knowledge, this is to date the highest Re-theta achieved for a near-equilibrium APG TBL under an approximately constant moderate APG. We evaluated the self-similarity of the outer region using two scalings, namely the Zagarola-Smits and an alternative scaling based on edge velocity and displacement thickness. Our results reveal that outer-layer similarity is achieved, and the viscous scaling collapses the near-wall region of the mean flow in agreement with classical theory. Spectral analysis reveals that the APG displaces some small-scale energy from the near-wall to the outer region, an effect observed for all the components of the Reynolds-stress tensor, which becomes more evident at higher Reynolds numbers. In general, the effects of the APG are more noticeable at lower Reynolds numbers. For instance, the outer peak of turbulent-kinetic-energy (TKE) production is less prominent at higher Re. Although the scale separation increases with Re in zero-pressure-gradient TBLs, this effect becomes accentuated by the APG. Despite the reduction of the outer TKE production at higher Reynolds numbers, the mechanisms of energisation of large scales are still present.

In this study, a new well-resolved large-eddy-simulation (LES) of an incompressible near-equilibrium adverse-pressure-gradient (APG) turbulent boundary layer (TBL) over a flat plate is presented. In this simulation, we have established a near-equilibrium APG over a wide Reynolds-number range. In this so-called region of interest (ROI), the Clauser–Rotta pressure-gradient parameter β exhibits an approximately constant value of around 1.4, and the Reynolds number based on momentum thickness reaches Re_θ = 8700. To the authors’ knowledge, this is to date the highest Re_θ achieved for a near-equilibrium APG TBL under an approximately constant moderate APG. We evaluated the self-similarity of the outer region using two scalings, namely the Zagarola–Smits and an alternative one based on edge velocity and displacement thickness. Our results reveal that outer-layer similarity is achieved, and the viscous scaling collapses the near-wall region of the mean flow in agreement with classical theory. Spectral analysis reveals that the APG displaces some small-scale energy from the near-wall to the outer region, an effect observed for all the components of the Reynolds-stress tensor, which becomes more evident at higher Reynolds numbers. Generally, the effects of the APG are more noticeable at lower Reynolds numbers. For instance, the outer peak of turbulent-kinetic-energy (TKE) production is less prominent at higher Re. While the scale separation increases with Re in zero-pressure-gradient (ZPG) TBLs, this effect becomes accentuated by the APG. Despite the reduction of the outer TKE production at higher Reynolds numbers, the mechanisms of energization of large scales are still present.

In this study, we exploit the Renard-Deck identity [J. Fluid Mech. 790, 339 (2016)] to decompose the mean friction drag in adverse-pressure-gradient turbulent boundary layers (APG-TBLs) into three components, associated with viscous dissipation, turbulence kinetic energy production, and spatial growth of the flow, respectively. We consider adverse-pressure-gradient turbulent boundary layers developing on flat plates and airfoils, with friction Reynolds numbers in the range 200 < Re-tau < 2000, and with Rotta-Clauser pressure-gradient parameters (beta) ranging from 0 to 50. The effects of Reynolds number, adverse pressure gradient, and the pressure-gradient history on the contributing components are individually investigated, and special attention is paid to the comparisons with zero-pressure-gradient turbulent boundary layers (ZPG-TBLs). Our results indicate that the inner peaks of the dissipation and production terms are located at y(+) approximate to 6 and y(+) approximate to 16.5, respectively, and their outer peaks scale with the 99% boundary-layer thickness (delta(99)), i.e., y/delta(99) approximate to 0.7 and 0.53, respectively. These results are independent of the friction Reynolds number, the magnitude of beta, and its development history. Moreover, the spatial-growth component is negative in the investigated APG-TBLs, and its magnitude increases with beta.

For adverse-pressure-gradient turbulent boundary layers, the study of integral skin-friction contributions still poses significant challenges. Beyond questions related to the integration boundaries and the derivation procedure, which have been thoroughly investigated in the literature, an important issue is how different terms should be aggregated. The nature of these flows, which exhibit significant in-homogeneity in the streamwise direction, usually results in cancellation between several contributions with high absolute values. We propose a formulation of the identity derived by Fukagata, Iwamoto \& Kasagi (Phys. Fluids, vol. 14, 2002, pp. 73--76), which we obtained from the convective form of the governing equations. A new skin-friction contribution is defined, considering wall-tangential convection and pressure gradient together. This contribution is related to the evolution of the dynamic pressure in the mean flow. The results of the decomposition are examined for a broad range of pressure-gradient conditions and different flow-control strategies. We found that the new formulation of the identity allows to readily identify the different regimes of near-equilibrium conditions and approaching separation. It also provides a more effective description of control effects. A similar aggregation between convection and pressure-gradient terms is also possible for any other decomposition where in-homogeneity contributions are considered explicitly. 

The widest scales in turbulent channel flows are studied through the use of three periodic channel-flow simulations at friction Reynolds number Re_τ=550. The length and height of the channels are the same in all cases (L_x/h=8π and L_y/h=2 respectively), while the width is progressively doubled: L_z/h = {4π, 8π, 16π}. The domain width has an effect on the turbulence statistics of a similar order as the error of convergence. Note that a channel flow similar to the smaller one from Del Álamo, Jiménez, Zandonade & Moser (J.~Fluid Mech., vol. 500, 2004, pp. 135--144), which was averaged over a very long time, was used for the comparison of the results. The one-dimensional spanwise spectrum of the streamwise velocity is performed with the aim of assessing the domain-size effect on the widest scales. Our results indicate that 90% of the total streamwise energetic fluctuations is recovered without a significant influence of the size of the domain. The remaining 10% of the energy reflects that the widest scales in the outer layer are the ones most significantly affected by the spanwise length of the domain.