The widest spanwise 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 = 8 π and L y / h = 2 , respectively), while the width is progressively doubled: L z / h = { 4 π , 8 π , 16 π } . The effects of increasing the domain width cannot be determined with statistical significance in our simulations, since the difference in the statistics between the simulations is of the same order as the errors of convergence. A channel flow similar to the smaller one [Del Álamo et al., “Scaling of the energy spectra of turbulent channels,” J. Fluid Mech. 500, 135-144 (2004)], which was averaged over a very long time, was used as a reference. The one-dimensional spanwise spectrum of the streamwise velocity is computed 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. The power-spectral density for kz = 0 remains constant even if the size of the domain in the spanwise direction is increased up to four times the standard spanwise length, indicating that wide, spanwise coherent structures are not an artifact of domain truncation.

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 et al. (2002), 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.

With the availability of new high-Reynolds-number (Re) databases of turbulent bound-ary layers (TBLs) it has been possible to identify in detail certain regions of the boundary layer with more complex behavior. In this study we consider a unique database at moderately-high Re, with a nearconstant adverse pressure gradient (APG) [Pozuelo et al., J. Fluid Mech. 939, A34 (2022)], and perform spectral analysis of the Reynolds stresses, focusing on the streamwise component. We assess different regions of the APG TBL, comparing this case with the zero-pressure-gradient (ZPG) TBL, and identify the relevant scaling parameters as well as the contribution of the scales of different sizes. The small scales in the near-wall region up to the near-wall spectral peak have been found to scale using viscous units. In APG TBLs, the largest scales close to the wall have a better scaling with the boundary-layer thickness (899), and they are significantly affected by the APG. In the overlap and wake regions of the boundary layer, the small energetic scales exhibit a good scaling with the displacement thickness (8*) while the larger scales and the outer spectral peak are better scaled with the boundary-layer thickness. Also, note that the wall-normal location of the spectral outer peak scales with the displacement thickness rather than the boundary layer thickness. The various scalings exhibited by the spectra in APG TBLs are reported here, and shed light on the complex phenomena present in these flows of great scientific and technological importance.

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.

Turbulent boundary layers are present in many aspects of life, from the weather and wind currents to transportation, production of energy or mixing processes. Understanding turbulent motions can allow for an improvement and development of technical devices, techniques or diagnosis of phenomena where a fluid flow is in the turbulent regime. From the economical and environmental perspectives, knowledge of turbulent boundary layers may help to reduce the drag on aerodynamic surfaces in transportation, thus leading to a reduction in fuel consumption and emissions. It is also possible to enhance the production of energy from wind sources or the harvest of tidal energy. Otherturbulent motions that have recently impacted society are those related to diffusion such as the transport of aerial diseases, or the motions of air in the respiratory system. In all those examples, external pressure gradients or those produced by the curvature of wall surfaces affect the turbulent structures and thus, the outputs that we study such as drag, transport of substances, energetic output, etc. A relevant case is that of adverse pressure gradient, which enhances the wall-normal convection and redistributes the turbulent energy across the turbulent boundary later. In this work, we study a canonical case of an adverse-pressure-gradient turbulent boundary layer, which is the flow over a flat plate, under near-equilibrium adverse-pressure-gradient conditions. We have extended the previous datasets on flat-plate boundary layers under adverse pressure gradients which were obtained at low Reynolds numbers, with a new numerical simulation reaching high Reynolds numbers, comparable to those of experimental campaigns. This new data set allowed us to study both adverse-pressure-gradient and Reynolds-number effects, where the thicker boundary layer exhibits a clear separation of turbulent scales. The influence of the size of the domain and the wider turbulent scales are analyzed through a set of turbulent channel-flow simulations and the spectral analysis of the Reynolds stresses in high Reynolds numbers turbulent boundary layers. The impact of the wider scales was analyzed and scaling factors were found for different regions of the spectra of the Reynolds stresses. In particular, we propose a new scaling for the energy of the small scales that have been advected to the outer region of the boundary layer.

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. 

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.

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.